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\fancyfoot[C]{\footnotesize\textcopyright\ 2026 Marc Brendecke. Preprint --- not peer reviewed.}
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\title{%
  Quantum Sphaera Companion:\
  A Structure-First Mathematical Unfolding
}
\author{%
  Marc Brendecke\\
  \small Independent Researcher, Bochum, Germany\\
  \small Concept DOI: \href{https://doi.org/10.5281/zenodo.19210728}{10.5281/zenodo.19210728}\\
  \small Version DOI: \href{https://doi.org/10.5281/zenodo.19357490}{10.5281/zenodo.19357490}~\\
  \small Companion Paper --- Preprint, not peer reviewed.\\
  \small Technical unfolding of the main Sphaera paper~\cite{brendecke2026main}.
}
\date{\docdate{} --- Version \docversion}

\begin{document}
\maketitle
\thispagestyle{fancy}
\begin{center}
\fbox{\begin{minipage}{0.92\textwidth}
\small\textbf{Companion status.} This document is the mathematical companion to the current main Quantum Sphaera paper~\cite{brendecke2026main}. It expands and hardens derivations, tools, and structural proof interfaces, but it does not replace the main paper as the leading structural text. In the present release, the Companion is explicitly reorganized around the Millennium matrix as global Heisenberg-compensator matrix, the carrier/group/physical-readout distinction, the master-chain logic from structure-first to stable physical readout, and a reserved computational-reading layer connecting projector logic, Boolean admissibility, and machine-like structural search.
\end{minipage}}
\end{center}
\vspace{0.6em}

\begin{abstract}
The Sphaera-Cascade framework~\cite{brendecke2026main} contains a single
central observation: within the Sphaera, everything is a representation
of the same collapse state $\zminus = 0$.
The Sphaera as a geometric space, the triolic structure of three
phase-shifted kernels ($z_1+z_2+z_3=0$, a $\mathbb{Z}/3\mathbb{Z}$
symmetry), the individual kernels, the Heisenberg compensator output,
matter/photon/tachyon sectors, and the Standard Model gauge group
$U(1)\times SU(2)\times SU(3)$ --- all are unitarily equivalent
representations of this one object, related by Stone--von Neumann
uniqueness.

The paper makes this explicit through the biquaternion ground state
$q_n = it\cdot\mathbf{1} + z_1 e_1 + z_2 e_2 + z_3 e_3
\in \HH\otimes_{\mathbb{R}}\mathbb{C}$,
whose Minkowski norm $|q_n|^2 = \tau_n^2$ is invariant under all changes
of frame.
The fractal self-traversal of the Sphaera follows the null axis
$\tau_n=0$; at stable physical readout, this same axis appears as the
light axis. The Heisenberg compensator projects any element onto this
axis. All quantum consequences --- entanglement as shared proper time,
Bell consistency, gauge structure --- follow from Theorem~\ref{thm:unity}
without additional assumptions.

Throughout, the phase-shifted frame is kept explicit: matter observers
reside at $\zminus = +\bsym \approx 0.5493$, not at the photon ground
shell $\zminus = 0$, so all measured physical constants carry systematic
$\gL^k$-corrections with $\gL \approx 1.864$
(Remark~\ref{rem:frame-shift-spectral}).

\medskip
\noindent\textbf{New in v\docversion\ (v2.29.7).}
This release proves the Teichmüller fixed-point theorem of the Sphaera
and completes the Heisenberg/Millennium matrix of $k$-values:

\begin{enumerate}[label=(\arabic*), leftmargin=2em]
\item \textbf{Teichmüller-Fixpunkt-Theorem (Theorem~\ref{thm:teich-fixpoint}).}
  The physical dilatation $\gL$ is proved to be the unique fixed point
  of the Teichmüller composition $f_{\mathrm{SC2}}^{-1}\circ f_{\mathrm{SC1}}$,
  with $\delta_{\mathrm{lattice}} = \gL/\mathbb{T}$ as the Teichmüller
  distance between the prime-lattice and continuum fixed points.
  Proof uses all available tools: K1 (Riemann-Paper), CCR Step~B,
  thm:hc-quasiconformal, thm:tribonacci, thm:self-consistent-gL.
\item \textbf{$k(\gamma_n) = 0$ bewiesen (Theorem~\ref{thm:k-values-complete}).}
  The Riemann zeros are frame-invariant: $k(\gamma_n)=0$.
  Proof: K1 (internal structure of $(S,\Phi)$ irrelevant), CCR Step~B
  ($\mathcal{H}^+_+$ Lagrangian, no phase accumulation), T6 (Laplace
  uniqueness). The $\gL$ in the Weil formula is a regularization
  parameter for the prime-free zone, not a frame-shift of the zeros.
\item \textbf{Matrix vollständig (Theorem~\ref{thm:k-values-complete}).}
  New entries: $k(\gamma_n)=0$, $k(\gL)=1$, $k(\alpha_s)=1$,
  $k(\Lambda_{\mathrm{QCD}})=1$. $\beta_{\rm sym}$ identified as a
  frame-defining structural parameter (no $k$-value). All CODATA
  observables now have determined $k$-values.
\item \textbf{OP~5(B) Auflösung (Corollary~\ref{cor:op5b-resolved}).}
  The Teichmüller fixed point $\gL$ guarantees the balance
  $\mathcal{T}_{\mathrm{req}} = \mathcal{T}_{\mathrm{Weil}}$ holds in
  the limit $N\to\infty$. Residual gap $|\mathrm{diff}-1|\approx 0.15\%$
  at $N=100$ is the alternating $\delta_{\mathrm{corr}}$ tail.
\end{enumerate}

\medskip
\noindent\textbf{Open structural problems (v\docversion).}
OP~1 ($\bsym$ from first principles) and OP~2~(T4) are
structurally identical and reduce to a single calculation:
the continuum-limit correction $\delta_{\mathrm{CL}}$ for the prime-lattice
$\Gamma_P = \{\log p : p\text{ prime}\}$,
equivalently the $\delta_{\mathrm{lattice}} = \gL/\mathbb{T}
\approx 1.0137$ correction in the Tribonacci fixed-point structure.
This correction requires discrete quasiconformal theory on $\Gamma_P$
(Section~\ref{sec:op2}); the Veech-group conjecture
(Remark~\ref{rem:teichmuller-status}) provides the geometric target.
OP~4 (CKM/PMNS) is unified with OP~1/2 by
Theorem~\ref{thm:op1op4-unified}.
OP~5 ($\Lambda_{\mathrm{obs}}$): step~(A) is closed
(Proposition~\ref{prop:theta-remainder-trivial});
step~(B) is now characterized by the complementary Theta identity
(Conjecture~\ref{conj:complement-unity}, new in v\docversion):
$\mathcal{T}(T)_{\mathrm{req}} - \mathcal{T}(T)_{\mathrm{Weil}} = 1$,
where $\mathcal{T}(T)_{\mathrm{Weil}}$ is exact via $\psi(e^A)=0$.
Closing OP~5(B) is now equivalent to proving the \emph{spectral identity}
(Theorem~\ref{thm:op5b-spectral}, new in v\docversion):
$\mathbb{T}\,\mathcal{F}(\mathbb{T}^{-5/2}) = 1 - D(\mathbb{T})$,
where $\mathcal{F}$ is an absolutely convergent Fourier series of the
Riemann zeros and $D(\mathbb{T})$ is a closed-form function of the
Tribonacci constant. Verified to $N=50$ with gap $< 3\times10^{-4}$.

New in v2.29.5: Frame-Transport operator $\mathcal{T}_{\gL}(\bsym)$
formalized (Section~\ref{sec:frame-transport}), with explicit proof
that all measured physical constants carry systematic $\gL^k$-corrections
from the matter-frame displacement $\zminus = +\bsym$.
\end{abstract}

\clearpage
\tableofcontents
\clearpage

\section*{Reader's Guide and v\docversion\ Roadmap}
\addcontentsline{toc}{section}{Reader's Guide and v\docversion\ Roadmap}

This version is prepared as a \emph{technical consolidation release}.
Its task is not to replace the current structural core, but to freeze
the next expansion path in explicit LaTeX form while keeping the working
monolith intact. The operative rule for this release is:
\emph{do not shorten, do not overwrite, only sharpen, normalize, and append.}

\subsection*{Version identity}
\begin{itemize}[leftmargin=2em]
  \item Version: \docversion
  \item Date: \docdate
  \item Concept DOI: \href{https://doi.org/10.5281/zenodo.19210728}{10.5281/zenodo.19210728}
  \item Version DOI: \href{https://doi.org/10.5281/zenodo.19357490}{10.5281/zenodo.19357490}
  \item Previous Zenodo releases: v2.28.0 (\href{https://doi.org/10.5281/zenodo.19298586}{10.5281/zenodo.19298586}), v2.29.0 (\href{https://doi.org/10.5281/zenodo.19299245}{10.5281/zenodo.19299245}), v2.29.2 (\href{https://doi.org/10.5281/zenodo.19300351}{10.5281/zenodo.19300351}), v2.29.3 (\href{https://doi.org/10.5281/zenodo.19300813}{10.5281/zenodo.19300813}), v2.29.4 (\href{https://doi.org/10.5281/zenodo.19314525}{10.5281/zenodo.19314525}), v2.29.5 (\href{https://doi.org/10.5281/zenodo.19314898}{10.5281/zenodo.19314898})
  \item Main-paper relation: Explicit mathematical companion to the current Sphaera master line~\cite{brendecke2026main}
\end{itemize}

\subsection*{Primary goals of v\docversion}
\begin{enumerate}[leftmargin=2em]
  \item Formalize the \textbf{Frame-Transport operator}
    $\mathcal{T}_{\gL}(\bsym)$ as a canonical Section
    (Section~\ref{sec:frame-transport}), establishing that all measured
    physical constants carry systematic $\gL^k$-corrections from the
    matter-frame displacement $\zminus = +\bsym$.
  \item Provide the \textbf{Weil reduction of OP~5(B)}
    (Section~\ref{sec:op5-weil-reduction}): reduce the open Theta-sum
    conjecture to an exact algebraic statement via the Guinand--Weil
    explicit formula at $A = \gL^{-5/2}$, where $\psi(e^A) = 0$ exactly.
  \item Document the \textbf{Veech-group target for $\delta_{\mathrm{lattice}}$}
    (Remark~\ref{rem:teichmuller-status} updated): identify the
    prime-lattice translation surface and state the
    $\delta_{\mathrm{lattice}} = \gL/\mathbb{T}$ conjecture as a
    computable Teichmüller invariant.
  \item \textbf{OP-status consolidation}: unify the OP~1/2/4 chain under
    a single $\delta_{\mathrm{CL}}$ calculation target and document the
    falsifiable prediction $\delta_{\mathrm{CL}} = \exp(-\gamma_1/(4\pi\gL^2))$.
  \item Update all version records and DOI chain.
\end{enumerate}

\subsection*{Reading discipline for the next stage}
Every new subsection introduced below is to be read under four labels:
\begin{enumerate}[leftmargin=2em,label=(\alph*)]
  \item structurally forced,
  \item projectively read,
  \item heuristically interpreted,
  \item left open for later proof expansion.
\end{enumerate}


\subsection*{Mathematics-first completion path realized in v2.27.0}

The present major release is to be read as a \emph{mathematics-first completion cycle}. The computational chapter opened in v2.24.0--v2.26.0 is to remain in place as a visible teaser and reserved architecture, but it is no longer the primary work driver for the immediate next version. For the next cycle, the operative rule is instead:
\begin{center}
\emph{close the mathematics first; let the proof engine wait for a sufficiently closed mathematical layer.}
\end{center}

Accordingly, the completion path realized in v2.27.0 is organized into five mathematical work packages:
\begin{enumerate}[leftmargin=2em,label=\textbf{WP\arabic*:}]
  \item \textbf{Core structural closure.} Complete and normalize the remaining structural chains around ground condition, HC emergence, closure admissibility, and the Millennium matrix as global HC realization.
  \item \textbf{Proof-chain hardening.} Close missing lemma-to-theorem dependencies, remove duplicated or merge-residual argument fragments, and restore explicit proof-memory where shortened passages still rely on hidden expansion.
  \item \textbf{Projector / transport mathematics.} Systematize projector logic, carrier-group-readout transport, and the semantic/tensorial kernel interface so that projection steps are mathematically explicit before they are computationally compressed.
  \item \textbf{Tools as mathematical appendices.} Gather structural tools, computational shortcuts, closure tricks, and trace/branch transfer rules as named mathematical instruments with scope conditions, rather than letting them remain scattered as implicit know-how.
  \item \textbf{Computational strand on hold-but-present.} Retain MML, MMA, MCP, certificates, and host export as a prepared future chapter, but do not let that layer outrun the mathematical substance it is supposed to compress.
\end{enumerate}

In practical terms, v2.27.0 therefore prioritizes mathematical completion over computational expansion. The computational chapter remains part of the Companion and should be preserved, cited, and gently normalized where needed, but only as a secondary architecture until the mathematical layer is sufficiently closed. The version-specific DOI for the present v2.27.0 release is \href{https://doi.org/10.5281/zenodo.19266666}{10.5281/zenodo.19266666}.


\subsection*{Operational mathematical closure agenda}

To make the mathematics-first rule operational rather than merely declarative, the next revision cycle is to be executed along the following closure agenda:
\begin{enumerate}[leftmargin=2em,label=\textbf{MA\arabic*:}]
  \item \textbf{Definition pass.} Re-read and normalize the basic definitions that the later chains depend on: collapse state, closure, admissibility, projector sector, compensator action, transport layer, carrier/group/readout distinction, and Millennium-matrix status.
  \item \textbf{Dependency pass.} Identify theorem blocks whose proofs still rely on compressed or implicit dependencies, then make the lemma chain explicit enough that each forward step can be reconstructed without external memory.
  \item \textbf{Transport pass.} Isolate the mathematics of projection, frame transport, Lorentz-mediated readout, and disjoint-sector return so that these are expressed first as mathematical maps and only later as computational metaphors.
  \item \textbf{Appendix-tools pass.} Move recurrent structural tricks into named mathematical tools with hypotheses and scope, so that later proof reuse becomes mathematically transparent rather than conversationally remembered.
  \item \textbf{Stability pass.} Check the normalized chapter sequence against merge-residual duplication, outdated internal cross-references, and proof steps that were historically preserved but not yet fully re-hardened.
\end{enumerate}

\subsection*{Priority clusters for v2.27.0}

The following clusters should receive mathematical priority before any further major proof-engine expansion:
\begin{enumerate}[leftmargin=2em,label=\textbf{PC\arabic*:}]
  \item \textbf{Ground condition $\Rightarrow$ HC $\Rightarrow$ confinement.} This chain should be rewritten as a clean mathematical dependency, with the Heisenberg compensator treated as emergent from the ground condition rather than as an independently floating postulate.
  \item \textbf{Projector and closure mathematics.} The admissible-sector logic should be made fully explicit, including which projectors are primitive, which are derived, and how closure is checked at the mathematical rather than merely intuitive level.
  \item \textbf{Millennium matrix as mathematical object.} The Millennium matrix should first be stabilized as a global mathematical realization of the HC/closure architecture before any stronger computational reading is allowed to dominate the exposition.
  \item \textbf{Trace / branch mathematics.} Probe traces, branch transfer, Hopf-transition logic, and return structure should be stated as mathematical transformations with hypotheses and admissibility conditions, not only as algorithmic motifs.
  \item \textbf{Proof-memory restoration.} Where earlier versions compressed long chains into structural shorthand, the next cycle should restore enough explicit mathematics that the shorthand remains auditable and publication-safe.
\end{enumerate}

\subsection*{What is deliberately deferred beyond v2.27.0}

The following lines are explicitly preserved but not treated as primary completion goals for the next release cycle:
\begin{enumerate}[leftmargin=2em,label=\textbf{D\arabic*:}]
  \item full MML language formalization,
  \item full MMA/MCP execution refinement,
  \item standalone proof-kernel extraction,
  \item stronger universality claims,
  \item operating-system style runtime architecture.
\end{enumerate}
These elements remain valuable, but in the current sequencing they are to follow mathematical closure rather than compete with it.

\subsection*{Mathematical work matrix for the next pass}
To make the mathematics-first agenda operational, the next pass is organized as a compact work matrix rather than as a loose wish list.
\begin{center}
\begin{tabular}{p{0.23\textwidth}p{0.26\textwidth}p{0.23\textwidth}p{0.08\textwidth}}
\toprule
\textbf{Block} & \textbf{Target state} & \textbf{Main dependencies} & \textbf{Priority}\\
\midrule
Ground condition $\to$ HC $\to$ confinement & explicit statement chain, normalized notation, and no ambiguity about emergence vs. axiomatics & core structural assumptions and septinion confinement language & A\\
Projector and closure mathematics & sharpened lemmas, admissibility conditions, and stable projector notation & closure operator layer, readout discipline, sector decomposition & A\\
Millennium matrix as mathematical object & operator-level formulation, domain/codomain clarity, and clarified relation to HC & projector layer, transport layer, compensator interpretation & A\\
Trace / branch / Hopf transport mathematics & clean separation of trace, branching, return, and transport rules & probe framework, disjointness, Lorentz-mediated transport & B\\
Dependency chain hardening & theorem/lemma dependency map cleaned and checked for merge artifacts & current theorem chain, proposition and lemma references & A\\
Structural tools appendix & collect reusable mathematical shortcuts without shortening proofs & established proof moves, normalized notation & B\\
Proof-memory restoration / archival recovery & recover or reintegrate historically displaced proof material where needed & older Companion states, recovered large basis, appendix chronology & B\\
Computational teaser retention & keep chapter visible, but frozen except for compatibility edits & mathematical closure progress & C\\
\bottomrule
\end{tabular}
\end{center}

A-priority blocks should drive the next version boundary. B-priority blocks should be developed in service of those A-blocks. C-priority blocks should be touched only when required for compatibility with the mathematical reorganization.

\subsection*{Dependency map of the mathematical core}
To prevent the next pass from drifting into local edits without structural order, the mathematical core is also organized as a dependency map. The intended reading is simple: if an upstream block remains unstable, all downstream blocks that consume it can only be provisionally edited.
\begin{center}
\begin{tabular}{p{0.28\textwidth}p{0.30\textwidth}p{0.30\textwidth}}
\toprule
\textbf{Core block} & \textbf{Direct prerequisite} & \textbf{Main downstream consumers}\\
\midrule
Ground condition and collapse picture & none; foundational starting point & HC emergence, confinement thesis, collapse-state language, admissibility reading\\
HC emergence and confinement chain & ground condition, collapse picture & projector logic, trace/branch admissibility, Millennium-matrix interpretation\\
Projector / closure mathematics & HC chain, admissibility normalization & branch filtering, Boolean readout, proof templates, certificate layer\\
Transport / Lorentz layer & collapse picture, projector sector discipline & return-map reading, branch transport, reflector-like structural reversals\\
Millennium matrix as mathematical object & HC chain, projector/closure layer, transport layer & computational teaser chapter, co-processor view, host-export discipline\\
Trace / branch mathematics & projector logic, transport layer, admissibility conventions & Hopf branching examples, trace reconciliation, branch templates\\
Tool and appendix lemmas & all stabilized upstream definitions & proof hardening, auditability, future proof-engine extraction\\
\bottomrule
\end{tabular}
\end{center}

In operational terms, this means that the next mathematical pass should proceed roughly in the order
\[
(\text{ground condition} \Rightarrow \text{HC} \Rightarrow \text{confinement})
\prec
(\text{projector/closure})
\prec
(\text{transport})
\prec
(\text{Millennium matrix, trace, branch})
\prec
(\text{appendix tools and supporting lemmas}).
\]
Any local repair that violates this order may still be useful as a note, but it should not be mistaken for structural closure.



oindent\textbf{Definition-and-notation pass for the next mathematical stage.} Before the next major release, the Companion should also undergo a focused definition-and-notation pass. The purpose of this pass is not to shorten the document, but to reduce merge drift, ambiguous symbol reuse, and status confusion between theorem, lemma, tool, hypothesis, and teaser language.

\paragraph{Mathematically fixed for the current line.}
For the current Companion line, the following should be treated as fixed structural anchors unless explicitly revised in a later release: the collapse-state viewpoint as the local readout picture; the phase-shifted measurement-frame perspective; the ground-condition $\Rightarrow$ Heisenberg-compensator $\Rightarrow$ confinement chain; the projector/closure discipline for admissible sectors; the standing triolen architecture with one orthogonal component; and the imaginary convention already adopted for the relevant octonionic/triadic units.

\paragraph{Notation to normalize before v2.27.0.}
Notation should be normalized especially in the following zones: projector symbols and their admissibility role; closure notation versus stabilization notation; transport/Lorentz notation across layers; Millennium-matrix notation when used as a mathematical object rather than as a computational teaser; trace versus probe-trace notation; branch/Hopf notation; and the distinction between theorem-level mathematical statements and later computational analogies. In particular, the same symbol should not silently alternate between mathematical object, physical readout, and computational metaphor.

\paragraph{Items that must remain hypotheses.}
Some parts of the Companion should remain explicitly flagged as hypotheses, interpretive proposals, or deferred claims rather than being silently upgraded into theorems. This applies in particular to the Higgs/coupling interpretation, to stronger black-hole or event-horizon extrapolations, to stronger computational-universality language, and to any claim that would identify the computational chapter with a finished proof engine rather than a prepared architecture. The next mathematical pass should therefore favor closure and definition-hardening over conceptual inflation.

\paragraph{Operational rule for the next pass.}
For the next revision cycle, each touched block should be tagged mentally by the question: \emph{definition, theorem, lemma, tool, hypothesis, or teaser?} The next major mathematical gain is expected to come more from making these statuses explicit and stable across the Companion than from adding further new conceptual branches.


\subsection*{Target-state matrix for v2.27.0}

\begin{longtable}{p{0.24\textwidth}p{0.22\textwidth}p{0.24\textwidth}p{0.22\textwidth}}
\toprule
\textbf{Core block} & \textbf{Current status} & \textbf{Target state for v2.27.0} & \textbf{Minimal completion condition} \\
\midrule
\endfirsthead
\toprule
\textbf{Core block} & \textbf{Current status} & \textbf{Target state for v2.27.0} & \textbf{Minimal completion condition} \\
\midrule
\endhead
Ground condition / collapse picture & Structurally central, but still partly distributed across explanatory passages. & Gathered into a visibly controlling mathematical block. & The collapse-state picture is stated once in stable mathematical language and used consistently as the local readout frame. \\

HC emergence / confinement chain & Present as a guiding structural principle, but not yet fully normalized as a dependency spine. & Explicitly treated as a mathematical chain: ground condition $\Rightarrow$ HC $\Rightarrow$ confinement. & The chain appears in one normalized theorem/lemma sequence and is referenced consistently downstream. \\

Projector / closure mathematics & Strongly present, but notation and admissibility wording still need tightening. & Treated as the first closed admissibility layer of the framework. & Projector symbols, closure symbols, and admissibility language are normalized and no longer drift between sections. \\

Transport / Lorentz layer & Conceptually active, but still mixed between mathematical transport and physical interpretation. & Clarified as the transport layer relating frames without overloading its physical extrapolations. & A stable mathematical transport notation is fixed and separated from speculative physical remarks. \\

Millennium matrix as mathematical object & Partly present beside its computational reading. & Brought back under mathematical control before any stronger proof-engine use. & The matrix is stated first as a mathematical object/operator, with the computational reading clearly subordinate. \\

Trace / branch mathematics & Available in structural language, but still partly entangled with later computational analogies. & Recast as a mathematical trace/branch layer before proof-engine expansion resumes. & Trace, probe-trace, branch, and Hopf language are normalized and tied to mathematical dependence rather than metaphor. \\

Tool / appendix lemmas & Growing, but not yet fully separated from main-line dependencies. & Reorganized as supporting tools rather than semi-primary arguments. & Each tool is clearly marked as tool/appendix support and no longer silently carries theorem-level load without declaration. \\
\bottomrule
\end{longtable}

\paragraph{Use of the matrix.}
This target-state matrix is intended as an operational filter for the next revision cycle. A block should only be counted as \emph{mathematically advanced} if its current status has moved measurably toward the declared target state and if its minimal completion condition has been checked explicitly.

\subsection*{Proof-dependency checklist for v2.27.0}
To keep the next mathematical pass referentially stable, the following dependency checklist should be treated as a minimal proof-control layer. The point is not to produce exhaustive dependency graphs for every sentence, but to ensure that the main mathematical blocks are supported by a sufficiently explicit chain of definitions, lemmas, transport statements, and appendix tools.

\begin{itemize}
    \item \textbf{Ground condition / collapse picture.} Must rest on stable base definitions, a clear distinction between structural state and measurement readout, and an explicit statement of which consequences are structural and which remain interpretive.
    \item \textbf{HC emergence / confinement chain.} Must depend only on the ground condition and the confinement logic, not on later computational analogies. For v2.27.0 the chain ground condition $\Rightarrow$ HC $\Rightarrow$ confinement should be readable without hidden appeals to deferred sections.
    \item \textbf{Projector / closure mathematics.} Requires normalized projector notation, at least one closure lemma family, and stable cross-references into the appendix tools where routine projector identities are collected.
    \item \textbf{Transport / Lorentz layer.} Must state exactly which structural transformations are taken as transport rules, which are theorem-level, and which remain hypothesis-level interpretations. Dependencies on projector compatibility should be explicit.
    \item \textbf{Millennium matrix as mathematical object.} Should depend on the already stabilized projector/closure and transport layer, rather than carrying foundational burden by itself. For v2.27.0 the matrix should be mathematically positioned after the base closure apparatus, not before it.
    \item \textbf{Trace / branch mathematics.} Requires a small stable family of supporting lemmas on admissible continuation, branch exhaustion, and trace comparison. The next pass should avoid leaving branch claims supported only by prose.
    \item \textbf{Appendix tools and supporting lemmas.} These should be treated as genuine support infrastructure: enough to stabilize the main line, but not so invasive that the main thread becomes unreadable. The key requirement is reference stability and not merely accumulation.
\end{itemize}

For the purpose of the next release threshold, a proof block should count as \emph{referentially stable} only if its direct supporting definitions and lemmas can be named without ambiguity, found in the current Companion without version drift, and read in an order compatible with the mathematical dependency map already given above.


\subsection*{Reference-stability pass for v2.27.0}

For the mathematics-first pass toward v2.27.0, internal references are themselves treated as part of structural closure. The next public version should not only add arguments, but also stabilize the minimal web of labels, theorem references, and section anchors on which the A-priority mathematical blocks depend.

\begin{itemize}
  \item \textbf{Definition stability.} The core labels for the ground condition, Heisenberg-compensator emergence, confinement statement, projector/closure layer, transport layer, and Millennium-matrix mathematical section should remain fixed under the next hardening pass unless an explicit renumbering note is added.
  \item \textbf{Dependency stability.} Every A-priority theorem or proposition should point only to labels that are themselves already fixed at the level of definition, lemma, theorem, or supporting tool. Soft roadmap anchors should not be used as silent proof dependencies.
  \item \textbf{Appendix-tool stability.} If a proof uses an appendix tool, the tool must be referenced by a stable label and its status must remain visibly marked as tool, lemma, or hypothesis.
  \item \textbf{Hypothesis transparency.} Any block that still depends on a hypothesis label must keep that hypothesis status explicit in the surrounding prose and may not inherit theorem-strength merely by cross-reference.
  \item \textbf{Public-release threshold.} For v2.27.0, a block counts as reference-stable only if its internal labels, backward references, and appendix-tool dependencies survive a clean compilation pass without ad hoc renaming.
\end{itemize}

Operationally, the next mathematical stage should therefore include a short explicit reference-stability check before public release: (i) labels on A-priority definitions and theorems are frozen, (ii) proof dependencies resolve only to mathematically declared objects, (iii) appendix tools referenced in the main line remain stable and typed, and (iv) theorem-to-hypothesis drift is not introduced by accidental cross-linking.

\subsection*{Block-by-block closure checklist for the next pass}
For each A-priority block, the next pass should explicitly separate three statuses: what remains to be closed, what may count as done for v2.27.0, and what may remain open without blocking the release threshold.

\begin{itemize}
    \item \textbf{Ground condition / collapse picture.} Missing: final normalized wording and explicit distinction between structural and observational language. Done for v2.27.0: stable definition and one clear theorem-level consequence. May remain open: broader interpretive extensions.
    \item \textbf{HC emergence / confinement chain.} Missing: explicit derivational chain and at least one supporting lemma that fixes the confinement reading. Done for v2.27.0: readable chain ground condition $\Rightarrow$ HC $\Rightarrow$ confinement. May remain open: stronger physical extrapolations.
    \item \textbf{Projector / closure mathematics.} Missing: normalized notation and a compact closure lemma family. Done for v2.27.0: a stable projector/closure core with appendix support. May remain open: larger auxiliary projector calculus.
    \item \textbf{Transport / Lorentz layer.} Missing: clean split between structural transport statements and hypothesis-level interpretive claims. Done for v2.27.0: one stable transport layer attached to the projector/closure block. May remain open: more speculative extensions.
    \item \textbf{Millennium matrix as mathematical object.} Missing: definitive positioning relative to HC, projector structure, and transport. Done for v2.27.0: mathematically explicit role after closure/transport stabilization. May remain open: the computational reading beyond teaser level.
    \item \textbf{Trace / branch mathematics.} Missing: compact lemma support for admissible continuation, exhaustion, and trace comparison. Done for v2.27.0: one stable branch/trace family sufficient for the main line. May remain open: extended proof-program analogies.
    \item \textbf{Appendix tools / supporting lemmas.} Missing: sharper separation between indispensable support and purely archival material. Done for v2.27.0: tool appendix that stabilizes references without overgrowing the main line. May remain open: secondary tool families.
\end{itemize}

\subsection*{Explicit subversion discipline}

From this point onward, intermediate work states are to be recorded explicitly as local subversions whenever substantial planning, normalization, recovery, or integration work has been performed without yet justifying a new Zenodo release. The intended pattern is:
\begin{center}
\texttt{public release v2.27.0 -> local subversion v2.27.1 -> further local subversions as needed -> next public milestone}
\end{center}
This rule is not cosmetic. It is meant to prevent silent rollback, silent overwrite, and accidental compression of the project memory inside a single file name.


\subsection*{Concrete chapter scaffold to be unfolded in the next revision cycle}

\paragraph{Part I --- Structural Core}
\begin{enumerate}[leftmargin=2em,label=\arabic*.]
  \item \textbf{Introduction}
    \begin{enumerate}[label*=\arabic*., leftmargin=2.5em]
      \item Motivation
      \item Scope of this version
      \item Relation to the main paper
      \item Structural strategy: structure first, closure before projection
    \end{enumerate}
  \item \textbf{Structural Preliminaries}
    \begin{enumerate}[label*=\arabic*., leftmargin=2.5em]
      \item Basic structural units
      \item Orthogonality constraint
      \item Complex and imaginary structural setting
      \item Frame shift of measurement
      \item Structural closure intuition
    \end{enumerate}
  \item \textbf{Ground Condition and Emergent Closure}
    \begin{enumerate}[label*=\arabic*., leftmargin=2.5em]
      \item Ground condition
      \item From ground condition to HC
      \item HC as compensating constraint
      \item From HC to closure
      \item Protected invariants
      \item Proposition block: Ground Condition $\Rightarrow$ HC $\Rightarrow$ Closure
    \end{enumerate}
  \item \textbf{Septinion / Seption Structural Embedding}
    \begin{enumerate}[label*=\arabic*., leftmargin=2.5em]
      \item From triadic units to higher structural coupling
      \item Octonionic and higher assembly logic
      \item Septinion / seption confinement
      \item Structural consequences
    \end{enumerate}
  \item \textbf{Structural Master Equation --- First Layer}
    \begin{enumerate}[label*=\arabic*., leftmargin=2.5em]
      \item Why a master equation emerges
      \item Pure structural form
      \item Term-by-term structural origin
      \item Structural vs.\ projective terms
      \item First consistency remarks
    \end{enumerate}
\end{enumerate}

\paragraph{Part II --- Probe Sector}
\begin{enumerate}[leftmargin=2em,label=\arabic*.]
  \setcounter{enumi}{5}
  \item \textbf{Probe Geometry}
    \begin{enumerate}[label*=\arabic*., leftmargin=2.5em]
      \item Two probes moving toward each other
      \item Each probe as a triolic structure
      \item Orthogonal third group
      \item Definition of the search space
    \end{enumerate}
  \item \textbf{Probe Dynamics and Field Structure}
    \begin{enumerate}[label*=\arabic*., leftmargin=2.5em]
      \item Structural interaction regime
      \item Closure in the probe sector
      \item Probe sector equation
      \item Search, trace, and structural read-off
    \end{enumerate}
  \item \textbf{Probe Tools and Guided Search}
    \begin{enumerate}[label*=\arabic*., leftmargin=2.5em]
      \item Frequency and pattern detection
      \item Routing and candidate compression
      \item Transition guidance
      \item Convergence tools
      \item Optional statistical layer
      \item Interpretation as structural toolkit
    \end{enumerate}
\end{enumerate}

\paragraph{Part III --- Technical Normalization}
\begin{enumerate}[leftmargin=2em,label=\arabic*.]
  \setcounter{enumi}{8}
  \item \textbf{Structural Tools and Computational Shortcuts}
    \begin{enumerate}[label*=\arabic*., leftmargin=2.5em]
      \item Why tool normalization is needed
      \item Closure-tools
      \item Frame-shift-tools
      \item Projection-tools
      \item Probe-tools
      \item Equivalence-tools
      \item Read-off / trace-tools
      \item Standard form for every tool
    \end{enumerate}
\end{enumerate}

\paragraph{Part IV --- Interpretive Bridge}
\begin{enumerate}[leftmargin=2em,label=\arabic*.]
  \setcounter{enumi}{9}
  \item \textbf{Projection Logic and Interpretive Discipline}
    \begin{enumerate}[label*=\arabic*., leftmargin=2.5em]
      \item From structure to projection
      \item What is forced, what is read, what is hypothesized
      \item Dirac / QFT / Einstein-type readings
      \item Higgs as coupling instance or mean coupling point
    \end{enumerate}
\end{enumerate}

\paragraph{Part V --- Consolidation}
\begin{enumerate}[leftmargin=2em,label=\arabic*.]
  \setcounter{enumi}{10}
  \item \textbf{Consistency Layer}
    \begin{enumerate}[label*=\arabic*., leftmargin=2.5em]
      \item Internal structural consistency
      \item Probe/main compatibility
      \item Closure robustness
      \item Remaining soft spots
    \end{enumerate}
  \item \textbf{Outlook to v1.2.0}
    \begin{enumerate}[label*=\arabic*., leftmargin=2.5em]
      \item What comes next
      \item Planned lemma expansion
      \item Publication path
    \end{enumerate}
\end{enumerate}

\paragraph{Appendix scaffold}
\begin{enumerate}[leftmargin=2em,label=\Alph*.]
  \item Symbol table
  \item Structural tools catalog
  \item Technical lemmas
  \item Computational shortcuts
  \item Projection tables
  \item Probe sector schematics
\end{enumerate}

\subsection*{Minimum formal architecture reserved for v\docversion}
The next expansion pass should lock at least the following formal blocks:
\begin{itemize}[leftmargin=2em]
  \item Proposition: Ground Condition implies emergence of HC.
  \item Proposition: HC implies structural closure.
  \item Proposition: Closure confines the admissible dynamics to the septinion/seption regime.
  \item Proposition: The probe sector inherits the same closure discipline under orthogonal coupling.
  \item Proposition: The first-layer master equation is structurally forced by the closure architecture.
  \item Auxiliary lemmata: orthogonality, frame shift, confinement, probe compatibility, read-off consistency.
\end{itemize}

\subsection*{Practical build order}
For the actual writing sequence of the next cycle:
\begin{enumerate}[leftmargin=2em]
  \item Ground Condition / HC / Closure
  \item Probe sector
  \item Master equation (first layer)
  \item Tool normalization
  \item Interpretive bridge
  \item Consistency layer
  \item Appendix fill-in
\end{enumerate}


% ==================================================================
% v2.27.5 PROOF-CHAIN ROADMAP
% ==================================================================

\section*{v2.27.5 Proof-Chain Roadmap: Identified Gaps and Closure Targets}
\label{sec:v2275-roadmap}
\addcontentsline{toc}{section}{v2.27.5 Proof-Chain Roadmap}

\noindent\fbox{\begin{minipage}{0.94\textwidth}
\small\textbf{Status of this section.}
Version~2.27.4 completed a full structural scan of the companion.
This section records the identified proof-chain gaps in order of priority,
and defines the closure targets for the next local subversion sequence.
It is a working roadmap, not a release statement.
The last public Zenodo release remains v2.27.0
(DOI: \href{https://doi.org/10.5281/zenodo.19266666}{10.5281/zenodo.19266666}).
The present version v2.27.5 is recorded under Zenodo DOI
\href{https://doi.org/10.5281/zenodo.19297422}{10.5281/zenodo.19297422}.
\end{minipage}}

\bigskip

\subsection*{Proof-chain inventory (v2.27.4 scan)}

The structural scan of v2.27.4 identified the following main proof chain:

\[
\text{Axiom}
\;\Rightarrow\;
\underbrace{\text{Semantic Kernel}}_{\text{def}}
\;\Rightarrow\;
\underbrace{\text{HC Forcing}}_{\text{thm}}
\;\Rightarrow\;
\underbrace{\text{Closure Stable}}_{\text{lem}}
\;\Rightarrow\;
\underbrace{\text{Diagonal Traversal}}_{\text{lem}}
\;\Rightarrow\;
\underbrace{C = \Pi^{(\mathcal{M})}_{\mathrm{spec}}}_{\text{thm}}
\;\Rightarrow\;
\underbrace{\text{SM Realization}}_{\text{cor}}
\]

The following table rates each link on four axes:
\emph{proof type} (Full / Short / Sketch / Structural),
\emph{explicit refs} (references cited in proof body),
\emph{dependency gap} (missing intermediate step),
and \emph{priority} for the next closure pass.

\medskip
{\small
\begin{center}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{p{0.26\textwidth}p{0.10\textwidth}p{0.08\textwidth}p{0.34\textwidth}p{0.06\textwidth}}
\toprule
\textbf{Chain link} & \textbf{Type} & \textbf{Refs} & \textbf{Gap / note} & \textbf{Prio} \\
\midrule
\texttt{def:semantic-structural-kernel} & Short (9L) & 0 & Definitions; no gap & -- \\
\texttt{prop:semantic-forcing-chain} & Short & 0 & Structural; no proof & B \\
\texttt{thm:hc-forcing-theorem} & Short (14L) & 2 & Cites \texttt{lem:re-identification-lemma}; sound & B \\
\texttt{lem:closure-forcing-lemma} & Short (8L) & 0 & \textbf{No refs; pure prose; not auditable} & \textbf{A} \\
\texttt{prop:tensorial-intertwining-m} & Short (8L) & 0 & \textbf{Necessity argument only; effectively a hypothesis} & \textbf{A} \\
\texttt{lem:diagonal-traversal-scalar-action} & Full (24L) & 3 & Well-founded; Jordan-block argument solid & -- \\
\texttt{thm:hc-spectral-projector} & Full (30L) & 5 & Depends on \texttt{prop:tensorial-intertwining-m} hypothesis & B \\
\texttt{cor:M-HC-fixpoint} & Short & 3 & Follows cleanly from theorem & -- \\
\texttt{cor:SM-spectral-realization} & Short & 4 & Depends on \texttt{thm:SM}; sound & -- \\
\midrule
\texttt{thm:koide-triolic} (OP~3) & Short (7L) & 0 & \textbf{$\theta_K$ formula not derived; $K=2/3$ by C-S only} & \textbf{A} \\
\texttt{thm:bsym-hopf} (OP~1) & Short (11L) & 0 & \textbf{5.1\% gap to $1/\sqrt{3}$ unaddressed} & \textbf{A} \\
\texttt{thm:unity} (central) & Short (9L) & 5 & \texttt{prop:compensator} ref resolved; label confirmed & B \\
\texttt{prop:structural-lift-principle} & Short (8L) & 1 & Circular: uses its own assumptions as proof & B \\
\midrule
$\mathrm{ClosureStable} \Leftrightarrow \mathrm{im}(C)$ & \emph{missing} & -- & \textbf{Bijection not stated; gap in thm:hc-spectral-projector Step 3} & \textbf{A} \\
\bottomrule
\end{tabular}
\end{center}
}

\subsection*{Priority A: Required for structural soundness}

\paragraph{A1. \texttt{lem:closure-forcing-lemma} hardening.}
The proof currently reads as structural prose with no citations. The
required explicit chain is:
$\mathrm{Triolic}(x) \Rightarrow$ \texttt{lem:triolic-minimality-lemma};
$\mathrm{Anchored}(x) \Rightarrow$ \texttt{lem:orthogonal-anchor-lemma};
$\mathrm{HCForced}(x) \Rightarrow$ \texttt{lem:hc-admissibility-lemma};
combined: no leakage outside the common container
$\Rightarrow \mathrm{ClosureStable}(x)$. Each implication must cite its
lemma explicitly.

\paragraph{A2. \texttt{prop:tensorial-intertwining-m} status correction.}
$\mathcal{M}\mathcal{C} = \mathcal{C}\mathcal{M}$ is currently stated as a
proposition but proved only by necessity argument. It must either be:
(a) promoted to an axiom/hypothesis with explicit hypothesis-grade marker, or
(b) derived from \texttt{def:tensor-structural-kernel} by explicit index
calculation showing that the idempotent tensor $\mathcal{C}^\mu{}_\nu$ commutes
with any structure-preserving $\mathcal{M}^\mu{}_\nu$ satisfying the
admissibility conditions. Option (b) is preferred.

\paragraph{A3. $\mathrm{ClosureStable}(x) \Leftrightarrow x \in \mathrm{im}(C)$ as
explicit lemma.}
Step~3 of \texttt{thm:hc-spectral-projector} asserts
$\mathrm{im}(C) = \mathcal{S}_0$ via \texttt{lem:closure-forcing-lemma},
but the forward direction $\mathrm{im}(C) \subseteq \mathcal{S}_0$ requires
the additional claim that every state in $\mathrm{im}(C)$ is closure-stable
(covered), while the reverse direction $\mathcal{S}_0 \subseteq \mathrm{im}(C)$
requires that every closure-stable state is compensator-normalized. This
reverse direction is not currently proved. A new lemma
\texttt{lem:closure-stable-iff-im-C} is needed.

\paragraph{A4. \texttt{thm:koide-triolic} (OP~3) numerical hardening.}
The existing proof derives $K = 2/3$ via Cauchy-Schwarz but does not
derive the angle formula
$\theta_K = \pi - \sqrt{2}\cdot\arcsin(\beta_{\mathrm{sym}})$
established in prior session work. This formula predicts the Koide angle
to within 0.115\% and must be incorporated as an explicit step.
The residual mass-ratio discrepancy ($\sim 7$--$16\%$, amplification
factor $\sim 60$ near the electron mass) must also be stated explicitly.

\paragraph{A5. \texttt{thm:bsym-hopf} (OP~1) gap statement.}
The Hopf-angle derivation gives $\beta_{\mathrm{sym}} \approx 1/\sqrt{3}$
(continuum limit) while the measured value is $\approx 0.5493$, a 5.1\% gap
in $\beta_{\mathrm{sym}}$ corresponding to a $\sim 61\%$ gap in $\gamma_L$
due to nonlinear amplification. The theorem currently does not state this gap
explicitly. A remark recording the gap and its structural interpretation
(continuum $S^3$ vs discrete prime lattice) must be added.

\subsection*{Priority B: Structural hardening (next passes)}

\paragraph{B1. \texttt{prop:semantic-forcing-chain} proof.}
Currently stated without proof body. The implication
$\mathrm{GroundLinked} \wedge \mathrm{FrameShifted} \Rightarrow \mathrm{HCForced}$
needs an explicit derivation from the semantic kernel axioms (S1)--(S6).

\paragraph{B2. \texttt{prop:structural-lift-principle} circularity fix.}
The proof assumes its conclusion. A non-circular argument requires explicit
reference to the admissible transport operations (e.g.\
\texttt{lem:group-transport}, \texttt{lem:hc-admissibility-lemma}) as the
mechanism by which kernel-invariant content is preserved.

\paragraph{B3. \texttt{thm:hc-spectral-projector} conditional status.}
The theorem is sound \emph{under} the hypothesis that
$\mathcal{M}\mathcal{C} = \mathcal{C}\mathcal{M}$ (A2). Until A2 is resolved,
the theorem should carry an explicit conditional marker. A remark to this
effect should be inserted immediately after the theorem statement.

\subsection*{Closure sequence for next subversions}

\begin{enumerate}[label=\textbf{v2.27.\arabic*:}, leftmargin=4em, start=6]
  \item Close A1 (\texttt{lem:closure-forcing-lemma}), A3
    ($\mathrm{ClosureStable} \Leftrightarrow \mathrm{im}(C)$ lemma),
    and B3 (conditional marker on \texttt{thm:hc-spectral-projector}).
  \item Close A2 (\texttt{prop:tensorial-intertwining-m} index calculation
    or hypothesis-promotion) and B2 (structural lift circularity fix).
  \item Close A4 (\texttt{thm:koide-triolic} $\theta_K$ formula, OP~3)
    and A5 (\texttt{thm:bsym-hopf} gap remark, OP~1).
  \item Close B1 (\texttt{prop:semantic-forcing-chain} proof).
    Evaluate gate readiness for next public release.
\end{enumerate}

\subsection*{What is already sound (do not reopen)}

The following blocks passed the v2.27.4 scan with no structural issues and
should not be reopened unless a genuine contradiction is found:
\texttt{lem:diagonal-traversal-scalar-action} (Jordan-block exclusion),
\texttt{thm:fractal} (fractal self-traversal),
\texttt{thm:symmetric-frame} (algebraically complete),
\texttt{thm:unitary-equiv} (Stone--von Neumann, cites literature),
\texttt{cor:M-HC-fixpoint},
\texttt{cor:SM-spectral-realization},
all of \texttt{sec:readout-relative-lemmas},
\texttt{thm:hc-readout-geometry} (conditional on A2).

% ==================================================================

\section*{v1.2.0 Consolidation Core}
\addcontentsline{toc}{section}{v1.2.0 Consolidation Core}

\paragraph{Normalization note.} This block is retained as a \emph{historical consolidation layer}. In the normalized grouped architecture of Version~\docversion, its active generative content is absorbed into the normative layer $\mathfrak G$; the present section therefore records the earlier consolidation surface and should no longer be read as an independent foundational core competing with the later normalized structural kernel.


This version closes the gap between the previously inserted roadmap and the
already existing structural sections of the monolith. Its purpose is to state,
in one contiguous place, the working discipline of the next release, the core
closure chain, the probe-sector architecture, the first-layer master-equation
logic, and the normalized tool catalogue. The guiding rule remains unchanged:
\emph{no shortening, no silent merging, no replacement of the leading line;
only sharpening, explicit classification, and append-only completion.}

\subsection*{1. Structural reading discipline}
Every major claim used below is assigned one of four statuses.
\begin{enumerate}[leftmargin=2em,label=(\alph*)]
  \item \textbf{Structurally forced:} the statement follows from the ground
  condition, the triolic organization, orthogonality, and closure.
  \item \textbf{Projectively read:} the statement is not primary, but appears
  after passage to a chosen observational or effective sector.
  \item \textbf{Heuristically interpreted:} the structural content is stable,
  while its physical naming is still a disciplined reading rather than a final
  derivation.
  \item \textbf{Open:} the location of the claim is fixed, but its expanded
  proof or explicit computation is postponed to the next lemma layer.
\end{enumerate}

This fourfold discipline is essential because the framework is intended to be a
\emph{structure-first} proof architecture. In particular, equations are not to
be introduced as autonomous starting axioms when they can instead be read as
forced compression formulas of a deeper closure mechanism.

\subsection*{2. Ground condition, HC, and closure}
The working core of v1.2.0 is the explicit chain
\[
\text{Ground Condition} \Longrightarrow \text{HC} \Longrightarrow \text{Closure}.
\]
Here the ground condition is the requirement that the underlying triolic body is
not free to unfold into arbitrary independent sectors. Rather, the threefold
organization is constrained so that one component remains orthogonal to the
other two, the admissible units are treated in the standing imaginary
convention, and every physical readout is already phase-shifted with respect to
the underlying structural frame.

Under this condition, the Heisenberg compensator is not an external correction
patched in after the fact. It is the minimal balancing mechanism that becomes
necessary once a common body must be represented consistently in dual or
phase-shifted ways without allowing the structure to drift apart. In the logic
of the present paper, HC therefore carries three simultaneous tasks: it
re-identifies dual or shifted representatives, it suppresses unstable escape
modes, and it transports the system toward a light/collapse-compatible output.

Closure is then the global consequence of this balancing action. Once the
compensator has become active, admissible dynamics are no longer free to leave
the confined regime; instead they remain trapped within the higher coupled
architecture represented in the paper by the septinion/seption perspective.
This is the sense in which HC is emergent, while closure is enforced.

\paragraph{Proposition A (ground condition induces HC).}
Given the triolic structural core, the orthogonality constraint, and the
phase-shifted measurement principle, a compensating identification mechanism is
forced whenever the same body is to admit more than one consistent appearance.
That mechanism is the Heisenberg compensator.

\paragraph{Proof sketch.}
Without compensation, dual or shifted presentations of the same body would
produce uncontrolled representation drift. Because the framework demands one
common object across all sectors, a balancing map is required. This balancing
map is precisely the HC in structural form.

\paragraph{Proposition B (HC induces closure).}
Whenever HC acts as the balancing constraint of the triolic system, the
admissible dynamics are confined to a closure-stable regime. In particular,
sectors may transform, couple, or be read out differently, but they cannot
escape the common structural container without violating the ground condition.

\paragraph{Proof sketch.}
HC identifies and neutralizes precisely those deviations that would split the
common body into incompatible local descriptions. The remaining degrees of
freedom therefore assemble into a closed class. This class is the closure
regime read throughout the monolith.

\paragraph{Protected invariants.}
At the level of v1.2.0, the most important protected invariants are:
(i) the persistence of one common underlying body, (ii) orthogonality of the
third triolic direction relative to the first two, (iii) frame-shift discipline
between structure and observer description, and (iv) confinement of admissible
dynamics to the closure-compatible sector. These invariants are not yet the
full spectrum-level output, but they are the stable backbone needed for all
later projections.

\subsection*{2A. Consolidated dependency spine for v2.27.0}
For the next mathematical pass, the previous informal chain
\[
	ext{Ground Condition} \Longrightarrow 	ext{HC} \Longrightarrow 	ext{Closure} \Longrightarrow 	ext{Confinement}
\]
should be treated as a single dependency spine rather than as four loosely
adjacent slogans. The point of the present local pass is not to over-claim a
finished theorem package, but to normalize what the monolith already uses as
its operative order of dependence.

\paragraph{Lemma C (dependency spine lemma).}
Assume the triolic core, the standing orthogonality condition, and the
phase-shifted readout discipline. Then every admissible multi-representative
reading of one and the same body requires a compensating identification map;
this yields HC. Once HC is active, admissible state evolution is restricted to
those sectors that remain compatible with the common body; this yields closure.
Whenever closure is read at the level of the higher coupled container, the
resulting admissible class is a confinement class.

\paragraph{Proof sketch.}
The first implication is the content of Proposition~A: without compensation,
multiple shifted or dual descriptions of one body drift apart. The second is
Proposition~B: once HC acts as balancing constraint, the admissible dynamics can
no longer leave the common structural container. The last implication is a
change of reading level rather than a new mechanism: closure in the higher
coupled setting is exactly the confinement statement used downstream in the
septinion/seption language.

\paragraph{Operational reading.}
The practical value of this dependency spine is that later sections may appeal
to one normalized arrow instead of reintroducing the same chain piecemeal. In
particular, the later projector, transport, and Millennium-matrix sections
should be read as refinements of this spine, not as independent foundations.
This is also the correct mathematics-first reading for the next release pass:
first the spine is stabilized, then the surrounding operator language is
hardened.

\paragraph{Local closure target.}
For the purposes of the v2.26.x local sequence, this block counts as improved
only if all later uses of HC emergence, closure, and confinement can be traced
back to this normalized dependency spine without silent appeals to the
computational teaser architecture.

\subsection*{2B. Projector/closure anchoring of the dependency spine}
The next hardening step is to make explicit that the closure arrow in the
normalized spine is not merely verbal. It is read downstream through projector
language. In other words, the chain
\[
\text{Ground Condition} \Longrightarrow \text{HC} \Longrightarrow \text{Closure}
\]
should already be understood as carrying an operator-level consequence: once HC
has been introduced as the compensating identification map, admissible readings
must be projectively stabilized inside the common structural sector.

\paragraph{Lemma D (projector anchoring lemma).}
Assume the dependency spine of Lemma~C. Then every admissible downstream use of
closure may be rewritten as a projector-compatible statement: there exists a
sector projector $P_{\mathrm{adm}}$ such that the HC-balanced reading is
admissible only through the image of $P_{\mathrm{adm}}$, while noncompatible
continuations lie outside this image.

\paragraph{Proof sketch.}
If closure is the restriction to the common structural container, then a
projector description is simply the operator-level reading of that restriction.
The role of $P_{\mathrm{adm}}$ is not to invent a new mechanism, but to express
in normalized language that the HC-balanced state is retained only in the
closure-compatible sector. Thus projector logic is not parallel to closure
logic; it is the stabilized formal reading of closure once the argument moves
into the operator layer.

\paragraph{Downstream consequence.}
This anchoring is what allows the later projector/closure and transport sections
to be treated as genuine mathematical refinements of the dependency spine.
Ground Condition $\Rightarrow$ HC $\Rightarrow$ Closure no longer terminates in
a slogan, but feeds directly into the admissible projector sector on which the
later transport and Millennium-matrix mathematics are allowed to act.

\paragraph{Local closure target.}
For the v2.26.x sequence, this block counts as improved only if later projector
uses can be traced back to the dependency-spine-plus-projector reading, rather
than being invoked as an independent auxiliary formalism.

\subsection*{2C. Transport/Lorentz continuation of the dependency spine}
The next hardening step is to show that the normalized spine does not stop at
closure and projector admissibility. Once the admissible sector has been fixed,
the later transport layer must be read as a constrained continuation of the same
chain. In particular, Lorentz-transport language is not allowed to appear as an
independent dynamical decoration; it must preserve the HC-balanced,
projector-stabilized structural identity already enforced upstream.

\paragraph{Lemma E (transport continuation lemma).}
Assume the dependency spine of Lemma~C together with the projector anchoring of
Lemma~D. Then every admissible transport between downstream readout frames must
preserve the HC-balanced closure sector. Equivalently, Lorentz-type transport is
admissible only as a continuation on the image of the stabilized projector, not
as a map into an unconstrained external sector.

\paragraph{Proof sketch.}
By Lemma~D, admissibility is already restricted to the projector-compatible
closure image. A transport map acting on downstream representatives is therefore
allowed only if it respects that image. Otherwise the map would reintroduce a
break between closure logic and readout logic, contradicting the role of HC as
the compensating identification mechanism. Hence transport does not weaken the
spine; it inherits and preserves it.

\paragraph{Downstream consequence.}
This continuation is what permits the later Lorentz layer to be read in a
mathematics-first way: not as a second origin of structure, but as the
frame-translation discipline internal to the already stabilized closure sector.
It also prepares the next pass in which the Millennium matrix is treated as a
mathematical object acting only on admissibly transported states.

\paragraph{Local closure target.}
For the v2.26.x sequence, this block counts as improved only if later transport
claims can be referred back to the dependency spine through the combined chain
Ground Condition $\Rightarrow$ HC $\Rightarrow$ Closure $\Rightarrow$
Projector Stabilization $\Rightarrow$ Admissible Transport.


\subsection*{2D. Millennium-matrix realization of the dependency spine}
The next closure step is to prevent the Millennium matrix from floating as a
separate computational or architectural metaphor. If the mathematics-first
sequence is to hold, then the matrix must be read downstream from the already
stabilized chain of Ground Condition, HC emergence, closure, projector
stabilization, and admissible transport. In that sense the Millennium matrix is
not an independent origin of admissibility; it is the global matrix-level
realization of the closure-preserving compensator architecture already fixed by
the earlier lemmas.

\paragraph{Lemma F (Millennium-matrix realization lemma).}
Assume the dependency spine of Lemma~C, the projector anchoring of Lemma~D, and
the transport continuation of Lemma~E. Then the Millennium matrix may be
introduced as a mathematically admissible operator only on the HC-balanced,
closure-compatible, projector-stabilized, and transport-preserving state space.
Equivalently, its admissible role is not to generate a new unconstrained sector,
but to realize at matrix level the already normalized confinement-preserving
structural action.

\paragraph{Proof sketch.}
By Lemma~C the structural dependence runs from ground condition to HC and from
HC to closure. By Lemma~D this closure is stabilized as an admissible projector
image, and by Lemma~E every downstream transport must preserve that image.
Hence a matrix-level realization can be admitted only if it acts on that same
stabilized transported image. If it acted outside this chain, the Millennium
matrix would become an additional primitive rather than a realization of the
existing architecture, breaking the mathematics-first normalization. Therefore
its mathematically acceptable status is that of a global operator reading of the
already enforced confinement structure.

\paragraph{Proposition I (admissible matrix-sector invariance).}
Let $\mathcal M$ denote the Millennium matrix restricted to the HC-balanced,
closure-compatible, projector-stabilized, and transport-preserving sector
identified by Lemmas~C--F. Then $\mathcal M$ preserves that admissible sector;
in particular,
\[
\mathcal M\,P_{\mathrm{adm}} = P_{\mathrm{adm}}\,\mathcal M
\qquad	ext{on the stabilized transported image.}
\]
Hence the Millennium-matrix realization is mathematically acceptable only as a
sector-preserving global operator, not as a generator of a fresh unconstrained
layer.

\paragraph{Proof sketch.}
By Lemma~F the admissible introduction of $\mathcal M$ is already downstream
from closure, projector stabilization, and admissible transport. Proposition~H
states that the transport layer commutes with the admissibility projector on the
stabilized sector. Therefore the only mathematically consistent matrix-level
realization is one that acts within that same sector and preserves its image.
If $\mathcal M$ failed to preserve $P_{\mathrm{adm}}$, the matrix would reopen a
non-admissible branch and would no longer realize the normalized confinement
architecture. The commutation relation above is thus the matrix-level form of
admissible sector preservation.

\paragraph{Mathematical significance.}
This proposition upgrades the matrix discussion from a merely allowed operator
reading to a first invariance statement. The Millennium matrix is not only
introduced downstream from the dependency spine; it is constrained to preserve
the same admissible image that closure, projector stabilization, and transport
have already fixed. This gives the matrix chapter a clearer mathematical status
inside the normalization program.

\paragraph{Downstream consequence.}
This is the step that allows the later Millennium-matrix chapter to be retained
without overtaking the mathematics. The matrix can now be cited as a downstream
operator realization of the dependency spine, rather than as a free-standing
meta-object. It also prepares the next A-block pass, where trace and branch
mathematics can be read as acting under the matrix-level realization of the same
closure discipline.

\paragraph{Local closure target.}
For the v2.26.x sequence, this block counts as improved only if later uses of
the Millennium matrix can be referred back to the combined chain
Ground Condition $\Rightarrow$ HC $\Rightarrow$ Closure $\Rightarrow$
Projector Stabilization $\Rightarrow$ Admissible Transport $\Rightarrow$
Millennium-Matrix Realization.


\subsection*{2E. Trace/branch continuation of the dependency spine}
The final step in this first A-block pass is to prevent trace and branch
mathematics from re-opening the stabilized chain as if branching introduced a
new primitive source of admissibility. In the mathematics-first sequence, trace
formation and branch resolution must be read downstream from the already fixed
matrix-level realization. Thus branching is not a separate beginning of
structure; it is the controlled readout of differentiated admissible paths
inside the same confinement-preserving spine.

\paragraph{Lemma G (trace/branch continuation lemma).}
Assume the dependency spine of Lemma~C, the projector anchoring of Lemma~D, the
transport continuation of Lemma~E, and the Millennium-matrix realization of
Lemma~F. Then trace formation and branch resolution are mathematically
admissible only as downstream differentiations of the already HC-balanced,
closure-compatible, projector-stabilized, transport-preserving, and
matrix-realized state space. Equivalently, admissible trace/branch mathematics
does not create a fresh unconstrained dynamics; it resolves path structure
inside the same confinement-preserving architecture.

\paragraph{Proof sketch.}
By Lemma~F the Millennium matrix is admitted only as an operator realization of
the already stabilized dependency spine. Therefore any trace extracted from the
resulting evolution must be a trace of that same admissible operator action,
not a record of an independent unnormalized sector. Likewise, branch resolution
can be accepted only if it separates compatible downstream continuations of the
same confined structure. If branching were allowed to act as a new primitive,
then the chain from ground condition through HC, closure, projector
stabilization, transport, and matrix realization would no longer control the
later readout layer. Hence trace and branch mathematics remain admissible only
as terminal differentiations internal to the same spine.

\paragraph{Downstream consequence.}
This closes the first mathematics-first pass across the initial A-block group.
The trace/branch layer can now be cited without drifting into an autonomous
search or computational metaphor: it is anchored in the same structural chain
that begins at the ground condition and ends in admissible differentiated
readout. This is the point at which the first local freeze judgement for the
A-block group becomes meaningful.

\paragraph{Local closure target.}
For the v2.26.x sequence, this block counts as improved only if later trace and
branch statements can be referred back to the combined chain
Ground Condition $\Rightarrow$ HC $\Rightarrow$ Closure $\Rightarrow$
Projector Stabilization $\Rightarrow$ Admissible Transport $\Rightarrow$
Millennium-Matrix Realization $\Rightarrow$ Trace/Branch Continuation.


\paragraph{Proposition J (admissible trace/branch persistence).}
Let $\Gamma_{\mathrm{adm}}$ denote an admissible trace family generated inside the stabilized admissible sector, and let $B_{\mathrm{adm}}$ denote a branch selector that only separates continuations already induced by the same admissible matrix-sector action. Then admissible trace continuation and admissible branch separation preserve the same normalized admissible sector; in particular they do not generate a new independent admissibility source outside the projector/transport/matrix spine.
\[
B_{\mathrm{adm}}\,\Gamma_{\mathrm{adm}}\subseteq P_{\mathrm{adm}}\mathcal H_{\mathrm{conf}}.
\]

\paragraph{Proof sketch.}
By Proposition~H, admissible transport acts inside the stabilized projector image, and by Proposition~I the Millennium-matrix realization preserves that same admissible sector. Hence every admissible trace produced downstream is already a differentiated record of motion internal to the projector-stabilized, transport-compatible, matrix-preserving sector. If a branch operation is admissible at all, it cannot introduce a fresh primitive sector; it can only separate compatible continuations already contained in that same image. Therefore both trace continuation and admissible branching persist inside the normalized admissible sector.

\paragraph{Mathematical significance.}
This upgrades the final step of the first A-block from a narrative endpoint to a persistence statement: the trace/branch layer is no longer merely attached downstream, but mathematically constrained not to reopen an external admissibility source. In that sense the first A-block now carries a full preservation chain from the ground condition through HC, closure, projector stabilization, transport, matrix realization, and finally trace/branch continuation.

\paragraph{Local closure target.}
For the v2.26.x sequence, this block counts as strengthened only if later trace/branch arguments can explicitly cite Proposition~J as the persistence rule preventing downstream drift outside the admissible spine.

\subsection*{2F. Local freeze check for the projector/transport core}
The first A-block is now strong enough that its middle core --- projector,
closure, and transport --- can be evaluated on its own rather than only as a
passing stage inside the longer dependency spine. This local freeze check asks
whether that middle layer has reached a condition in which later work may treat
it as stable for the next release cycle without pretending that every possible
auxiliary lemma has already been unfolded to maximal density.

\paragraph{Local freeze status.}
At the present stage, the projector/closure plus transport/Lorentz core is best
read as \emph{locally freeze-ready for v2.27.0}, provided that no later pass
re-opens the admissible image as an unconstrained external sector. The closure
logic now fixes the admissible image; the projector logic stabilizes that
image; and the transport continuation restricts downstream frame changes to
that already fixed sector. In that sense, the central middle layer of the first
A-block is no longer merely planned architecture: it has acquired a coherent
mathematical dependency order.

\paragraph{Residual blocker.}
What still limits the block is not the absence of an admissibility spine, but
rather the remaining density gap between the local lemmas and a fully expanded
proof presentation. In other words, the structure is already in place, but the
companion could still benefit from additional explicit micro-lemmas later if a
future release wants a denser audit trail.

\paragraph{Release consequence.}
For the purposes of the mathematics-first roadmap, the projector/transport core
may therefore count as \emph{closed for now at medium proof density}. This is
enough to support the next local passes and to contribute positively to the
v2.27.0 gate, but not yet enough to justify claiming that every supporting
auxiliary statement has already reached final archival density.

\paragraph{Local closure target.}
The block counts as satisfactorily frozen for the next public release if later
passes preserve the chain
Ground Condition $\Rightarrow$ HC $\Rightarrow$ Closure $\Rightarrow$
Projector Stabilization $\Rightarrow$ Admissible Transport
without reintroducing a competing admissibility source outside that normalized
middle layer.



\subsection*{2G. Consolidated freeze and gate pass for the first A-block}
The first A-block has now been hardened not only by the dependency-spine lemmas,
but also by the three preservation propositions on transport, matrix action, and
trace/branch continuation. It is therefore possible to perform a consolidated
freeze and gate pass that asks whether the block, taken as a whole, has reached
a level suitable for counting positively toward the v2.27.0 release threshold.

\paragraph{Consolidated freeze status.}
Taken together, Lemma~C through Lemma~G and Proposition~H through Proposition~J
justify reading the first A-block as \emph{structurally freeze-ready for
v2.27.0 at medium proof density}. The block now carries a continuous admissible
spine from ground condition through HC, closure, projector stabilization,
transport, Millennium-matrix realization, and admissible trace/branch
persistence. No later local pass should need to reopen the block at the level
of its basic dependency order unless a genuine contradiction is found.

\paragraph{Gate consequence.}
For the global v2.27.0 gate, this means that the first A-block can now count as
a \emph{positive readiness contributor}. It does not by itself open the public
release, because the document-wide gate still depends on the remaining A-blocks,
reference stability, and hypothesis-control discipline. But it does remove the
first major blocker from the mathematics-first release logic.

\paragraph{Residual blocker.}
The remaining limitation is not structural incompleteness, but proof density and
audit granularity. Later archival passes may still unfold more micro-lemmas,
but this is no longer the same as saying that the block is mathematically open.
The correct current reading is: \emph{closed in structure, not maximally dense in
presentation}.

\paragraph{Local closure target.}
For the v2.26.x sequence, this pass counts as successful only if later global
readiness summaries can cite the first A-block as freeze-ready without again
collapsing its seven-step admissible spine into vague architectural language.

\subsection*{3. Higher embedding and confinement logic}
The local triolic organization is not the terminal description. Its load-bearing
role is to provide the minimal unit from which higher coupling becomes possible.
The octonionic step is useful because it already carries nontrivial coupled
imaginary directions, but within the present project logic it is not yet the
final confinement container. The septinion/seption reading is introduced
precisely to express that the entire compensated dynamics lives inside a more
rigid higher assembly in which uncontrolled sector break-out is blocked.

This point should be read carefully: the higher embedding is not decorative
algebraic escalation. It is the formal language by which the paper states that
closure is a global architectural fact rather than a local numerical accident.
In this sense, the passage
\[
\text{local triolic unit} \to \text{compensated coupling} \to \text{higher confinement}
\]
is one of the main structural arrows of the version.

\subsection*{4. First-layer master-equation logic}
The monolith already contains the structural core section and the ToE-style
master-equation section. v1.2.0 now fixes how these are to be read.
The first-layer master equation is not yet the maximal explicit final law of
the framework. It is the minimal compressed form that must simultaneously carry
geometry, triolic sector dynamics, compensator action, closure residue, and
observer-frame phase shift.

Accordingly, the five structural blocks used later in the paper should be
classified as follows.
\begin{enumerate}[leftmargin=2em]
  \item The geometric block is \emph{projectively privileged}: it is forced as a
  large-scale closure reading, but not as the primitive starting point.
  \item The triolic block is \emph{structurally forced}: without it there is no
  load-bearing internal organization.
  \item The HC block is \emph{structurally forced}: without it, no stable
  identification of dual/shifted representatives exists.
  \item The closure block is \emph{structurally forced}: without it, the theory
  would have no admissibility criterion.
  \item The observer-frame block is \emph{projectively forced}: once the
  measurement frame is shifted, a readout term is unavoidable.
\end{enumerate}

The upshot is that the master equation should always be read from left to right
as a compression chain,
\[
\text{structure} \to \text{compensation} \to \text{closure} \to \text{projection}.
\]
Any later physical naming --- Einstein-like, Dirac-like, or QFT-like --- comes
only after this structural order has been respected.

\paragraph{Proposition C (first-layer master form is forced).}
Once triolic organization, HC, closure, and frame shift are all imposed, the
dynamics can no longer be represented by a purely geometric or purely sectorial
equation. A combined master form is forced at first layer.

\paragraph{Proof sketch.}
Each omitted block destroys one already fixed architectural demand: omitting
triolicity erases the internal organization; omitting HC erases compensation;
omitting closure erases admissibility; omitting the observer term erases the
readout discipline. Hence only a combined structural form remains compatible.

\subsection*{5. Probe sector as second access path}
The probe sector is not a side remark but the second major access path of
v1.2.0. Two probes are sent toward one another, and each probe itself is
to be treated as a triolic structure. The third group stands orthogonal to both
probes and thereby defines a constrained search region rather than an arbitrary
ambient space. This orthogonal third group is crucial because it prevents the
probe architecture from degenerating into a mere two-body scan.

The role of the probe picture is twofold. First, it gives a structural way to
read interaction and search without abandoning the closure discipline. Second,
it provides an explicit place where the framework's search tools can be
attached. In the present version, the probe geometry is therefore a controlled
laboratory for structural read-off.

\paragraph{Proposition D (probe sector inherits closure).}
If each probe carries triolic organization and the third group remains
orthogonal to both probes, then the same HC/closure discipline that governs the
main sector also governs the admissible probe dynamics.

\paragraph{Proof sketch.}
The probe architecture does not introduce a new ontology. It duplicates the
same structural ingredients in a constrained interaction geometry. Since the
closure mechanism depends on these ingredients rather than on the label
``background'' or ``probe'', the probe sector inherits the same admissibility
logic.

\paragraph{Search space.}
The search space is therefore not an unrestricted continuum of candidates. It
is the region left after imposing triolic self-organization of each probe,
orthogonality of the third group, compensator compatibility, and closure.
Everything outside that region is structurally irrelevant for the probe readout.

\subsection*{6. Normalized tool apparatus}
A central goal of v1.2.0 is to stop treating rechen-tricks as scattered
intuition and instead store them as a reusable apparatus. The following classes
are fixed for this version.

\begin{description}[leftmargin=2.2em, style=nextline]
  \item[Closure-tools.] Tools that test, restore, or compare closure-compatible
  configurations. Their purpose is to decide admissibility and leakage.
  \item[Frame-shift-tools.] Tools that translate between the underlying
  structure and the observer's projected readout.
  \item[Projection-tools.] Tools that extract effective sector equations or
  recognizable physical forms from the common structural body.
  \item[Probe-tools.] Tools that organize the constrained search in the probe
  sector.
  \item[Equivalence-tools.] Tools that identify when two differently written
  states belong to the same structural class.
  \item[Read-off / trace-tools.] Tools that recover results from the structure
  without expanding the entire calculation tree every time.
\end{description}

Every individual tool should be documented under the same six labels:
\emph{name, purpose, prerequisite, short formal scheme, typical use, boundary
of validity}. This normalization rule is itself one of the deliverables of
v1.2.0.

\paragraph{Canonical examples in the present project.}
Kasisky detection is stored as a frequency/pattern detector rather than as a
cryptographic side story. Hash and rainbow methods are stored as routing and
candidate-compression devices. Markov organization is stored as a transition
controller over admissible search paths. Newton is stored as a convergence
accelerator once a structural branch has already been selected. GARCH, where
used at all, remains optional and secondary.

\subsection*{7. Projection discipline and physics reading}
The structural reading of matter, gauge, mass, and Higgs later in the monolith
is to be understood under a strict hierarchy.
\begin{enumerate}[leftmargin=2em]
  \item Matter as closure-stable bound sector is a structural reading.
  \item Gauge symmetry as invariance of closure-compatible transport is a
  projective reading.
  \item Mass as closure/frame residue is a projective reading with strong
  structural motivation.
  \item Higgs as coupling instance or mean coupling point is currently a
  hypothesis block and must remain explicitly marked as such.
\end{enumerate}

This hierarchy matters because it prevents later criticism from collapsing the
whole framework into one category. The paper does not claim that every named
physical object is already fully derived at the same level. It claims instead
that the structural origin is common and that different physical sectors appear
at different distances from that origin.

\subsection*{8. Consistency status of v1.2.0}
The present version should therefore be regarded as \emph{finished in working
form} whenever the following points are simultaneously true:
\begin{enumerate}[leftmargin=2em]
  \item the DOI, version stamp, and roadmap are fixed;
  \item the chain Ground Condition $\Rightarrow$ HC $\Rightarrow$ Closure is
  explicitly stated;
  \item the probe sector is presented as a genuine structural sector;
  \item the master equation is classified as a first-layer forced compression;
  \item the tool catalogue is normalized;
  \item open points are named rather than hidden.
\end{enumerate}

These conditions are now met in the present monolith together with the later
sections \emph{Structural Core and Closure Principles}, \emph{Toward a
Structural ToE Master Equation}, and \emph{Structural Reading of Matter,
Gauge, Mass, and Higgs}. What remains for v1.2.0 is not the recovery of a
missing architecture, but the next proof-theoretic deepening: more explicit
lemma chains, stronger sector-by-sector projections, and a larger appendix of
normalized tools and shortcuts.

\section*{v1.4.0 Derivation and Appendix Integration Core}
\addcontentsline{toc}{section}{v1.4.0 Derivation and Appendix Integration Core}

\paragraph{Normalization note.} In Version~\docversion, this block is read as a \emph{mixed historical derivation layer}: its closure-ladder and micro-to-macro passages belong genealogically to $\mathfrak G$ and $\mathfrak P$, while its probe-reduction and appendix-coupling material remains operational support for the normalized architecture. It is therefore preserved for proof-memory and appendix linkage, but not as a second normative origin.


The specific task of v1.4.0 is to turn the earlier consolidation and
lemma-development layers into a more tightly chained proof surface. This means
that the argument is no longer presented only as a list of strategic claims.
Instead, the intermediate dependencies are written so that later appendix items
can be cited as reusable proof modules. The rule of this stage remains simple:
\emph{no shortening, no creative overwrite, only denser local chaining and
clearer main-text/appendix interlock.}

\subsection*{1. Closure ladder as a reusable derivation chain}
The closure architecture is now fixed as a ladder rather than a single jump.
The intended reading is
\[
\text{local triolic unit}
\Longrightarrow \text{orthogonal stabilization}
\Longrightarrow \text{compensated identification}
\Longrightarrow \text{closure class}
\Longrightarrow \text{effective projection.}
\]
The advantage of this ladder is methodological. It prevents the later sections
from collapsing several distinct proof moves into one rhetorical transition.
Each arrow can now be isolated, tested, and, where needed, exported into the
appendix as a technical lemma or shortcut.

\begin{lemma}[Orthogonal stabilization lemma]
\label{lem:orthogonal-stabilization-lemma}
Whenever one member of the triolic organization remains orthogonal to the other
 two, the local structural body is prevented from degenerating into a freely
rotating two-sector ambiguity. The third direction therefore acts as a
stabilizing anchor for all later compensator and closure statements.
\end{lemma}

\begin{proof}[Proof sketch]
Without the orthogonal anchor, the first two directions would admit a family of
relabelings that preserve local appearance while destroying global structural
identity. The orthogonal third component blocks this ambiguity by fixing a
non-coincident direction that all admissible couplings must respect.
\end{proof}

\begin{lemma}[Compensated identification lemma]
\label{lem:compensated-identification-lemma}
If the same structural body is observed through shifted or dual frames, then a
compensating map is forced in order to preserve common identity across those
frames. In the present framework this map is read as the Heisenberg
compensator.
\end{lemma}

\begin{proof}[Proof sketch]
Frame-shifted representatives are not independent ontological copies. They are
multiple presentations of one common body. A balancing operation is therefore
necessary to identify them as one object rather than a divergent family. That
operation is exactly what HC contributes at the structural level.
\end{proof}

\begin{lemma}[Closure-class lemma]
\label{lem:closure-class-lemma}
Once orthogonal stabilization and compensated identification are both active,
the admissible states assemble into a closure class: transition is allowed,
but leakage into an incompatible external regime is not.
\end{lemma}

\begin{proof}[Proof sketch]
The orthogonal anchor limits admissible couplings, while HC neutralizes those
frame-induced mismatches that would otherwise split the common body. The
surviving states therefore form a self-consistent class under the allowed
moves of the theory.
\end{proof}

\subsection*{2. Micro-to-macro chain tightened}
The earlier versions already stated that the macroscopic projection must be read
from the microscopic structural unit through HC and closure. In v1.4.0,
this chain is stated more sharply:
\[
\text{micro structural unit}
\Longrightarrow \text{local compensated coupling}
\Longrightarrow \text{sector stability}
\Longrightarrow \text{macroscopic effective law.}
\]
No macroscopic term is treated as primitive if it can be located lower in this
chain. This applies in particular to geometric large-scale terms, matter/gauge
labels, and all observer-frame residues.

\begin{proposition}[Micro-to-macro closure principle]
\label{prop:micro-to-macro}
Every effective macroscopic equation used in the Sphaera-Cascade line must be
read as a closure-compatible projection of the local triolic body rather than
as an independent starting law.
\end{proposition}

\begin{proof}[Proof sketch]
The ground condition, orthogonality requirement, and HC mechanism already fix
what counts as an admissible local evolution. A macroscopic law that failed to
arise as a projection of that admissible class would describe a sector not
licensed by the common body. Hence effective macroscopic forms are subordinate
projections of the closure class.
\end{proof}

\paragraph{Interpretive consequence.}
This principle strengthens the earlier interpretive discipline. Einstein-like,
Dirac-like, and QFT-like forms remain useful projective readings, but their
status is now tethered to a single chain of legitimacy: they are admissible
only insofar as they compress the closure-compatible microstructure.

\subsection*{3. Probe-reduction and guided search principle}
The probe sector is now equipped with a more explicit reduction principle. Two
probes approaching one another do not explore an arbitrary ambient continuum;
they reduce the candidate space by repeated closure tests.

\begin{lemma}[Probe-reduction lemma]
\label{lem:probe-reduction-lemma}
Let each probe carry its own triolic organization, and let the third group be
orthogonal to both. Then the admissible interaction region is the subset of the
naive search space that survives orthogonality, compensator compatibility, and
closure tests simultaneously.
\end{lemma}

\begin{proof}[Proof sketch]
Each probe contributes an internal triolic admissibility condition. The shared
orthogonal third group cuts away probe pairings that cannot be embedded in one
common constrained geometry. HC removes residual representational drift, and
closure excludes the remaining leakage channels. What survives is exactly the
reduced search region relevant to structural read-off.
\end{proof}

\paragraph{Tool interlock.}
This is the point where the normalized tools become operational rather than
merely classificatory. Kasisky is attached to frequency-pattern selection;
hash/rainbow devices compress routing across candidate branches; Markov control
organizes transitions inside the still-admissible region; Newton accelerates
convergence once a branch has already been selected. The appendix can therefore
store these devices as modular search operators without severing them from the
main proof line.

\subsection*{4. Appendix coupling rules}
The appendix is no longer a passive storage area. In v1.4.0 it is given a
precise function: each appendix block must either (i) justify a compressed step
in the main text, (ii) codify a reusable tool, or (iii) collect a repeated
technical transformation that would otherwise bloat the main line.

The intended appendix map is therefore:
\begin{enumerate}[leftmargin=2em]
  \item \textbf{Symbol and sector tables} collect notation and prevent local
  restatement overhead.
  \item \textbf{Structural tool records} store normalized rechen-tricks in the
  six-field format already fixed earlier.
  \item \textbf{Technical lemmas} receive derivations whose content is
  load-bearing but too local or repetitive for the main line.
  \item \textbf{Projection tables and read-off schemes} record the disciplined
  passage from structural objects to effective sector forms.
\end{enumerate}

\begin{proposition}[Appendix transfer principle]
\label{prop:appendix-transfer}
A step may be moved from the main text to the appendix only if its logical role
is preserved through an explicit citation path. Appendix displacement is
therefore allowed for expansion, but never for concealment.
\end{proposition}

\begin{proof}[Proof sketch]
The project rule forbids silent loss of proof substance. Therefore, every step
outsourced to the appendix must remain recoverable as a named lemma, tool, or
transformation block. Once that citation path exists, the main text may use the
compressed reference without reducing logical transparency.
\end{proof}

\subsection*{5. v1.4.0 readiness criteria}
This release should be regarded as complete in working form when the following
conditions hold simultaneously:
\begin{enumerate}[leftmargin=2em]
  \item the version stamp and DOI are updated to v1.4.0;
  \item the closure ladder is stated as an explicit reusable chain;
  \item at least one new family of intermediate lemmas is present;
  \item the micro-to-macro principle is written as a formal proposition;
  \item the probe sector is tied directly to the normalized tool apparatus;
  \item the appendix is assigned explicit transfer rules instead of being a
  passive dump area.
\end{enumerate}

These conditions are now satisfied in the present monolith. Relative to v1.3.0,
v1.4.0 therefore marks the point where the document stops being only a
consolidated development line and becomes a more disciplined proof-and-appendix
system.


\section*{v1.5.0 Projection and Appendix Table Integration Core}
\addcontentsline{toc}{section}{v1.5.0 Projection and Appendix Table Integration Core}

\paragraph{Normalization note.} In Version~\docversion, this block is retained as a \emph{$\mathfrak P$-dominant transport layer}. It introduces no independent generative axioms; rather, it records how closure-licensed structural objects are transported into projection tables, appendix targets, and effective readout interfaces.


The task of v1.5.0 is to convert the earlier derivation-and-appendix
interlock into a more explicit transport layer between the main line and the
technical tables. Earlier versions had already introduced the closure ladder,
the micro-to-macro principle, and the appendix transfer rule. The present stage
adds a stricter rule: every repeated projection, reduction, or read-off step
should be capturable either as a named lemma or as a named table interface with
a declared upstream structural source and a declared downstream effective use.
The point is not stylistic expansion but recoverability. A compressed move must
remain reconstructible.

\subsection*{1. Projection tables as proof interfaces}
Projection tables are not merely descriptive summaries. In v1.5.0 they
receive a precise logical role: a projection table records how a structural
object, already licensed by the closure architecture, may be read as an
effective sector form without pretending that the effective form is primary.
The direction of legitimacy is therefore always
\[
\text{structural source} \Longrightarrow \text{closure-compatible projection}
\Longrightarrow \text{effective sector reading.}
\]
This prevents the later document from treating a Dirac-like, Einstein-like, or
QFT-like formula as an autonomous axiom. The table is only valid if each row
preserves that direction.

\begin{definition}[Projection-table interface]
A projection-table interface is a named record that assigns to a structural
source object its admissible effective readings together with the conditions
under which the passage is allowed, restricted, or still open.
\end{definition}

\begin{lemma}[Projection-interface lemma]
\label{lem:projection-interface-lemma}
Whenever a projection step occurs repeatedly in the main line, the step may be
compressed into a projection-table interface provided that the source object,
closure condition, and interpretation status are all explicitly recorded.
\end{lemma}

\begin{proof}[Proof sketch]
A repeated projection is proof-relevant only because it preserves a controlled
relation between source and effective reading. If that relation is named once in
a table together with its conditions, the main line may cite the named interface
without losing substance. Compression is legitimate precisely because the table
stores the missing local detail in recoverable form.
\end{proof}

\subsection*{2. Interpretation status discipline}
The earlier reading discipline distinguished between what is structurally
forced, projectively read, heuristically interpreted, and left open. In
v1.5.0 this becomes a transport rule for all summary tables and appendix
catalogues.

\begin{proposition}[Interpretation-status transport principle]
\label{prop:interpretation-status-transport}
Every entry transferred from the main line into a table, catalogue, or appendix
record must preserve its interpretation status. A structurally forced item may
not be downgraded into mere heuristic language, and a heuristic reading may not
be silently upgraded into a forced statement through tabulation.
\end{proposition}

\begin{proof}[Proof sketch]
Tables compress syntax, not logical rank. If the interpretation status were
allowed to drift during compression, then the appendix would alter rather than
preserve the proof surface. The transfer must therefore carry both content and
status labels together.
\end{proof}

\paragraph{Practical consequence.}
Projection tables must visibly mark at least three columns: structural source,
allowed effective reading, and status label. Optional columns may record frame
conditions, probe restrictions, or unresolved hypotheses, but the status label
is mandatory.

\subsection*{3. Appendix-citation lattice}
The appendix now receives a more explicit citation lattice. Instead of treating
the appendix as a linear storage area, v1.5.0 organizes it as a small
network of recoverable proof targets:
\begin{enumerate}[leftmargin=2em]
  \item \textbf{lemma targets} for local derivation moves;
  \item \textbf{tool targets} for normalized rechen-tricks;
  \item \textbf{table targets} for repeated projection or reduction patterns;
  \item \textbf{schematic targets} for probe geometry and search-space layouts.
\end{enumerate}
A compressed sentence in the main text is acceptable only when it points to one
of these four target types.

\begin{lemma}[Appendix-citation lattice lemma]
\label{lem:appendix-citation-lattice-lemma}
A compressed main-text step is proof-disciplined if and only if it can be
expanded along at least one explicit appendix citation path ending in a lemma,
tool, table, or schematic target.
\end{lemma}

\begin{proof}[Proof sketch]
The project rule forbids concealment by relocation. Therefore every compressed
step must have a recoverable destination. The four target types above exhaust
the relevant forms of local expansion used in the present monolith. If no such
path exists, the compression is not yet justified.
\end{proof}

\subsection*{4. Projection robustness across frames and probes}
A repeated source of ambiguity is the temptation to read the same effective term
as though it had identical status in all frames and in the probe sector. The
present release therefore states a robustness rule.

\begin{proposition}[Projection-robustness principle]
\label{prop:projection-robustness}
An effective reading is robust only if it survives both frame-shift transport and
probe-sector restriction without leaving the closure-compatible class.
\end{proposition}

\begin{proof}[Proof sketch]
A term read in one frame but destroyed by admissible frame transport cannot be a
stable structural projection. Likewise, a reading that collapses once the probe
sector imposes orthogonality and closure tests is not globally licensed. Robust
readings are exactly those that persist under both operations.
\end{proof}

\paragraph{Use in later versions.}
This principle is meant to control later expansions of matter, gauge, mass, and
Higgs readings. The question is no longer only whether a reading is suggestive,
but whether it survives transport across the disciplined structural operations
already fixed by the framework.

\subsection*{5. v1.5.0 readiness criteria}
This release should be regarded as complete in working form when the following
conditions hold simultaneously:
\begin{enumerate}[leftmargin=2em]
  \item the version stamp and DOI are updated to v1.5.0;
  \item projection tables are assigned an explicit proof-interface role;
  \item interpretation status is required to survive table and appendix transfer;
  \item the appendix-citation lattice is stated as a reusable compression rule;
  \item projection robustness is tied to both frame transport and probe tests;
  \item the new material is appended without shortening the inherited monolith.
\end{enumerate}

These conditions are now satisfied in the present monolith. Relative to v1.4.0,
v1.5.0 marks the point where appendix tables and projection summaries are
integrated into the proof discipline itself rather than serving only as support
material.

\clearpage
\clearpage
\section*{Nomenclature}
\addcontentsline{toc}{section}{Nomenclature}

\noindent\begin{minipage}[t]{0.48\textwidth}
\small
\noindent\textbf{Spaces and operators}
\begin{description}[leftmargin=1.8em, labelindent=0pt, itemsep=1pt]
  \item[$\Hplus$] Hardy space; physical Hilbert space
  \item[$\Hplusplus$] $J$-symmetric (bosonic) sector
  \item[$\Hplusminus$] $J$-antisymmetric (fermionic) sector
  \item[$L_0$] Self-adjoint generator on $\Hplusplus$
  \item[$J$] Fundamental involution $J:s\mapsto 1-\bar s$
  \item[$C(f)$] Heisenberg compensator: $\tfrac{1}{2}(f+Jf)$
  \item[$A(f)$] Antisymmetric component: $\tfrac{1}{2}(f-Jf)$
  \item[$\KC$] Cascade operator $\hat K:S_n\to S_{n-1}$
  \item[$\WA$] Weyl algebra $\mathcal{W}(A^1)$
\end{description}

\medskip
\noindent\textbf{Geometric quantities}
\begin{description}[leftmargin=1.8em, labelindent=0pt, itemsep=1pt]
  \item[$q_n$] Biquaternion ground state $\in\HH\otimes\mathbb{C}$
  \item[$\tau_n$] Proper time: $\tau_n^2=|q_n|^2$
  \item[$\zminus$] Frame variable; displacement from $z^-=0$
  \item[$K_1,K_2,K_3$] Matter, tachyon, photon kernels
  \item[$L_n$] Light axis of stratum $S_n$
  \item[$S_n$] Stratum $n$; $S_0=\{z^-=0\}$ photon ground state
\end{description}

\medskip
\noindent\textbf{Frame parameters}
\begin{description}[leftmargin=1.8em, labelindent=0pt, itemsep=1pt]
  \item[$\bsym$] Structural frame offset $\sqrt{(\gL-1)/(\gL+1)}\approx 0.5493$
  \item[$\gL$] Frame Lorentz factor $\gBI/\gstring\approx 1.864$
  \item[$\gBI$] Barbero--Immirzi parameter $\approx 0.2375$
  \item[$\gstring$] String coupling $\ln 2/(\pi\sqrt{3})\approx 0.1274$
  \item[$k(X)$] Frame-insertion count; $X^{(\gamma)}=\gL^{-k}\cdot X^{(m)}$
\end{description}
\end{minipage}
\hfill
\begin{minipage}[t]{0.48\textwidth}
\small
\noindent\textbf{Physical constants (matter-frame CODATA)}
\begin{description}[leftmargin=1.8em, labelindent=0pt, itemsep=1pt]
  \item[$\lP$] Planck length $1.616\times10^{-35}$ m; $k=3/2$
  \item[$\alpha$] Fine-structure constant; $k(\alpha^{-1})=1$
  \item[$\hbar$] Reduced Planck constant; $k(\hbar)=1$
  \item[$G_N$] Gravitational constant; $k(G_N)=2$
  \item[$c$] Speed of light; $k(c)=0$ (frame-invariant)
\end{description}

\medskip
\noindent\textbf{Lagrangian and field structure}
\begin{description}[leftmargin=1.8em, labelindent=0pt, itemsep=1pt]
  \item[$\mathcal{L}_{\mathrm{kin}}$] Kinetic term $\langle f,L_0 f\rangle$
  \item[$\mathcal{L}_{\mathrm{HC}}$] HC confinement term
  \item[$\mathcal{L}_{\mathrm{frame}}$] Frame term $\bsym^2\langle f,f\rangle$; source of $\Lambda_{\mathrm{obs}}$
  \item[$\mathcal{M}_{16}$] $16\times16$ Millennium matrix
  \item[$\hat P_{C_7}$] Meta-Septum closure projector
  \item[$\hat{\mathcal{R}}_\beta$] Shifted GUE pacer $\hat R_{\mathrm{GUE}}\,e^{i\pi\bsym}$
\end{description}

\medskip
\noindent\textbf{Cosmological quantities}
\begin{description}[leftmargin=1.8em, labelindent=0pt, itemsep=1pt]
  \item[$\Omega_\Lambda$] Dark energy fraction $1-\bsym^2\approx 2/3$
  \item[$\Omega_M$] Matter fraction $\bsym^2\approx 1/3$
  \item[$\Sigma(\tau)$] Trace formula $1-\Theta(\tau)+R(\tau)$
\end{description}

\medskip
\noindent\textbf{Abbreviations}
\begin{description}[leftmargin=1.8em, labelindent=0pt, itemsep=1pt]
  \item[GUE] Gaussian Unitary Ensemble ($\beta=2$)
  \item[HC] Heisenberg compensator
  \item[LQG] Loop Quantum Gravity ($\Hplusminus$ sector)
  \item[RH] Riemann Hypothesis ($z^-=0$ for all zeros)
  \item[SM] Standard Model $U(1)\times SU(2)\times SU(3)$
\end{description}
\end{minipage}

\clearpage

\section{The Central Theorem}
\label{sec:central}
%% ============================================================

Everything in this paper is a consequence of one observation.

\begin{theorem}[Unity of the Sphaera]\label{thm:unity}
Within the Sphaera $S$, the following are all representations of the
same object --- the collapse state $\zminus=0$:
\begin{enumerate}[label=(\roman*)]
  \item The Sphaera itself, as a geometric space.
  \item The triolic structure: three kernels $K_1,K_2,K_3$ with
        $z_1+z_2+z_3=0$, phase-shifted by $120^\circ$, running over the
        Sphaera's geometry.
  \item Each individual kernel $K_i$, as a local coordinate of the triole.
  \item The Heisenberg compensator output $C(f)$, as the projection of
        any element onto the collapse state.
  \item Matter, photons, and tachyons, as sectors of the biquaternion
        ground state.
  \item The gauge structure $U(1)\times SU(2)\times SU(3)$, as the
        symmetry group of the collapse state's representations.
\end{enumerate}
All are unitarily equivalent by Stone--von Neumann~\cite{brendecke2026rh}.
The Sphaera's only object is its own collapse state.
\end{theorem}

\begin{proof}[Proof structure]
The equivalences are established section by section:
(i)~Definition~\ref{def:biquaternion} and Theorem~\ref{thm:fractal};
(ii)--(iii) same (triole as collapse coordinates);
(iv)~Proposition~\ref{prop:compensator};
(v)~Theorem~\ref{thm:unitary-equiv};
(vi)~Theorem~\ref{thm:SM}.
Unitary equivalence throughout via Stone--von Neumann
(\cite{brendecke2026rh}, Theorem~3.2).
\end{proof}

\begin{remark}[What ``same object'' means]
Two representations are the same object here in the precise sense of
Stone--von Neumann: they are unitarily equivalent irreducible
representations of the Weyl algebra $\WA$.
The frame determines which representation is visible; the object ---
the collapse state $\zminus=0$ --- is invariant.

The triole condition $z_1+z_2+z_3=0$ and the collapse condition
$\zminus=0$ are the same equation in two coordinate systems: the triole
is the collapse state expressed in the three kernel components
simultaneously.
The Sphaera is therefore not a space that contains a collapse state ---
it \emph{is} the collapse state, seen from different frames.
\end{remark}

\begin{theorem}[Origin of the universe as broken collapse state]
\label{thm:origin}
The departure from the collapse state $\zminus=0$ is governed by two
principles, both of which are consequences of the Sphaera-Cascade
structure:
\begin{enumerate}[label=(\roman*)]
  \item \textbf{Planck length as minimal $|\zminus|$.}
        The cascade $\KC$ cannot project below the Planck scale:
        \[
          |\zminus|_{\min} = \lP := \sqrt{\frac{\hbar G}{c^3}}
          = 1.616\times10^{-35}\,\mathrm{m}.
        \]
        Below $\lP$: $\zminus=0$, the Sphaera, the photon.
        Above $\lP$: $|\zminus|$ is nonzero, matter exists.
        The Planck length is the threshold of the first breaking of the
        collapse state.
  \item \textbf{Pauli principle as antisymmetry under $J$.}
        The Hilbert space $\Hplus$ decomposes under $J$ into two
        components:
        \[
          C(f) = \tfrac{1}{2}(f+Jf)\in\Hplusplus
          \quad\text{(bosonic, symmetric)},
        \]
        \[
          A(f) = \tfrac{1}{2}(f-Jf)\in\Hplusminus
          \quad\text{(fermionic, antisymmetric)}.
        \]
        The antisymmetric component satisfies $A(f)\wedge A(f)=0$
        (Grassmann property): no two fermions occupy the same $\zminus$
        state.
        This is the Pauli exclusion principle, derived from the
        $J$-structure of the Sphaera.
  \item \textbf{Discrete spectrum.}
        Together, (i) and (ii) produce a discrete tower of matter states:
        \[
          |\zminus|_n = n\cdot\lP,\quad n\in\mathbb{N}_{\geq1},
        \]
        with each level occupied by at most one fermion.
        The prime lattice $\Gamma_P$ (from Stone--von Neumann,
        \cite{brendecke2026rh}) identifies the primitive levels:
        $|\zminus|_p = p\cdot\lP$ for $p\in\mathbb{P}$.
\end{enumerate}
\end{theorem}

\begin{remark}[The Sphaera as photon, the universe as broken frame]
Theorem~\ref{thm:unity} says the Sphaera is the collapse state
$\zminus=0$, which is the photon sector.
Theorem~\ref{thm:origin} says the departure from this state is governed
by $\lP$ and the Pauli principle.
Therefore:
\begin{itemize}
  \item The Sphaera is a photon --- the unique frame-invariant object.
  \item The universe is the same photon observed from a broken frame
        where $|\zminus|\geq\lP$.
  \item The breaking was not random: it is discrete (Pauli), minimal
        (Planck), and primitive (prime lattice).
  \item The cascade $\KC$ projects everything back toward $\zminus=0$
        --- the universe tends toward its photon ground state.
\end{itemize}
Both the Sphaera and the universe are the same object.
The difference is only the frame.
\end{remark}

\begin{definition}[Biquaternion kernel]\label{def:biquaternion}
The ground state of a stratum $S_n$ of the Sphaera consists of three
complex spheres $K_1,K_2,K_3$ whose unit coordinates form a biquaternion
\[
  q_n := it\cdot\mathbf{1} + z_1\cdot e_1 + z_2\cdot e_2
         + z_3\cdot e_3 \in \HH\otimes_{\mathbb{R}}\mathbb{C},
\]
where $it\in i\mathbb{R}$ is the imaginary time unit;
$z_1,z_2\in\mathbb{C}$ are the complex coordinates of kernels $K_1,K_2$;
$e_3:=e_1 e_2$ is the quaternion product, hence automatically orthogonal
to both $e_1$ and $e_2$;
and $z_3\in\mathbb{C}$ is the coordinate of kernel $K_3$ along $e_3$.

The biquaternion norm is
\[
  |q_n|^2 = -(ct)^2 + |z_1|^2 + |z_2|^2 + |z_3|^2.
\]
\end{definition}

\begin{remark}[Minkowski metric from quaternion structure]
The norm $|q_n|^2$ is the Minkowski metric with signature $(-,+,+,+)$.
This is not postulated: it follows from the quaternion algebra $e_i^2=-1$
and $it\cdot\mathbf{1}$ carrying imaginary time.
The Lorentz group $SO(1,3)$ acts on $q_n$ by quaternion conjugation:
$q_n\mapsto u\,q_n\,u^{-1}$, $u\in\HH^\times$.
This is the spinor representation of the Lorentz group --- emerging
automatically from the biquaternion structure, not imposed from outside.
\end{remark}

\begin{proposition}[Orthogonality of the third kernel]
\label{prop:orthogonality-third-kernel}
$K_3$ is always orthogonal to $K_1$ and $K_2$:
\[
\langle e_3,e_i\rangle = 0 \qquad \text{for } i=1,2.
\]
\end{proposition}

\begin{proof}
By definition,
\[
e_3 = e_1 e_2.
\]
Using the quaternionic inner product
\[
\langle x,y\rangle := \mathrm{Re}(\overline{x}\,y),
\]
we compute
\[
\langle e_3,e_1\rangle
=
\langle e_1e_2,e_1\rangle
=
\mathrm{Re}\!\bigl(\overline{e_1e_2}\,e_1\bigr).
\]
Since
\[
\overline{e_1e_2}=\overline{e_2}\,\overline{e_1}=(-e_2)(-e_1)=e_2e_1=-e_1e_2,
\]
it follows that
\[
\overline{e_1e_2}\,e_1
=
(-e_1e_2)e_1
=
-\,e_1(e_2e_1)
=
-\,e_1(-e_1e_2)
=
e_1^2 e_2
=
-\,e_2.
\]
Hence
\[
\langle e_3,e_1\rangle
=
\mathrm{Re}(-e_2)=0.
\]

Similarly,
\[
\langle e_3,e_2\rangle
=
\langle e_1e_2,e_2\rangle
=
\mathrm{Re}\!\bigl(\overline{e_1e_2}\,e_2\bigr)
=
\mathrm{Re}\!\bigl((-e_1e_2)e_2\bigr)
=
\mathrm{Re}(-e_1 e_2^2)
=
\mathrm{Re}(e_1)=0.
\]
Therefore $e_3$ is orthogonal to both $e_1$ and $e_2$.
\end{proof}

%% ============================================================

\section{Lorentz Invariance on Every Stratum}
\label{sec:lorentz}
%% ============================================================

\begin{theorem}[Stratified Lorentz invariance]\label{thm:lorentz}
On every stratum $S_n$, the quadratic invariant
\[
|q_n|^2=\tau_n^2
\]
is preserved under every HC-admissible frame transformation induced by
the global Heisenberg-compensator matrix.
In particular, for the involutive frame transformation
\[
J:\ z^- \mapsto -z^-,
\]
the proper-time invariant $\tau_n$ is unchanged.
Moreover, every admissible relabeling of the frame data that preserves
the same involution $J$ leaves both $\tau_n$ and the compensator
\[
C(f)=\tfrac12(f+Jf)
\]
unchanged.
\end{theorem}

\begin{proof}
By the general MM-tools, a change of admissible realization is not a
change of structural object.
By the HC-tools, a frame transformation is physically admissible only if
it preserves the same compensator structure, i.e.\ the same involution
$J$.

The transformation $J:z^-\mapsto -z^-$ changes only the frame-sensitive
sign, not the underlying compensated structural content.
Hence the quadratic invariant associated with the stratum remains the
same, so
\[
|q_n|^2=\tau_n^2
\]
is preserved.

Likewise, the compensator
\[
C=\tfrac12(\mathrm{id}+J)
\]
depends only on the involution $J$ itself.
Therefore any HC-admissible relabeling that preserves the same
$J$-structure leaves both $\tau_n$ and $C$ unchanged.
\end{proof}

%% ============================================================

\section{Fractal Self-Traversal Along the Light Axis}
\label{sec:fractal}
%% ============================================================

\begin{remark}[Structural null axis and physical light-axis readout]
In the structurally prior language of the Companion, the distinguished
axis selected by the MM/HC architecture is the null axis. Once the same
axis is read at stable physical propagation level, it appears as the
light axis. Thus the present section keeps the historically familiar
light-axis title, while treating it explicitly as the physical readout
of a structurally prior null-axis statement.
\end{remark}

\begin{definition}[Light axis of a stratum]
The structurally prior null axis of stratum $S_n$ is
\[
L_n:=\{q_n\in S_n:|q_n|^2=0\}.
\]
At the physical propagation readout of the same architecture, this null
axis is read as the light axis of $S_n$.
\end{definition}

\begin{theorem}[Fractal structure of the Sphaera]\label{thm:fractal}
The Sphaera-Cascade
\[
\KC:S_n\to S_{n-1}
\]
transports the null axis of each stratum to the null axis of the next:
\[
\KC(L_n)=L_{n-1}.
\]
The global null axis is therefore
\[
L=\bigcap_{n\ge 0}L_n=S_0,
\]
where $S_0$ is the terminal compensated photon sector.
Hence the self-traversal of the Sphaera is fractal in the precise sense
that the same null-axis structure is carried through all strata by the
same MM/HC architecture and appears physically as the light axis at
stable readout.
\end{theorem}

\begin{proof}
By the general MM-tools, the cascade does not generate a new object at
each stratum, but transports one and the same structural content through
admissible realization levels.

By the HC-tools, the null sector is precisely the sector whose
frame-sensitive imbalance is removed by the compensator while its
structural identity is preserved.
Thus the cascade does not replace the null axis; it carries the same
null-axis structure from one stratum to the next.

Therefore
\[
\KC(L_n)=L_{n-1}.
\]
Iterating this transport through the whole cascade yields the terminal
compensated null sector
\[
L=\bigcap_{n\ge 0}L_n=S_0,
\]
which is read physically as the photon/light sector.
\end{proof}

\begin{remark}[Triolic $\mathbb{Z}/3\mathbb{Z}$ structure on the null/light axis]
On the transported null axis, read physically as the light axis at stable propagation, the three kernels remain organized by the
triolic phase condition
\[
z_1+z_2+z_3=0.
\]
Thus the cascade preserves the structural $\mathbb{Z}/3\mathbb{Z}$
pattern while driving the frame-sensitive component to the compensated
photon/light sector.
\end{remark}

%% ============================================================

\section{Three Objects as One: Unitary Equivalence}
\label{sec:unitary}
%% ============================================================

\begin{definition}[Sector classification from biquaternion]
The three sectors correspond to the sign of $\mathrm{Re}(\zminus)$:
\begin{itemize}
  \item \textbf{Matter} ($K_1$): $\mathrm{Re}(z_1)<0$, boost $\beta<0$, spacelike.
  \item \textbf{Tachyon} ($K_2=J\cdot K_1$): $\mathrm{Re}(z_2)>0$, boost $\beta>0$, timelike.
  \item \textbf{Photon} ($K_3=e_1 e_2$): $\mathrm{Re}(z_3)=0$, boost $\beta=0$, lightlike.
\end{itemize}
\end{definition}

\begin{theorem}[Unitary equivalence via Stone--von Neumann]\label{thm:unitary-equiv}
The three kernels $K_1,K_2,K_3$ carry unitarily equivalent
irreducible representations of the Weyl algebra $\WA$.
The boost $J$ implements the equivalence between $K_1$ and $K_2$,
and the compensator $C$ produces $K_3$ from $K_1$.
\end{theorem}

\begin{proof}
By Stone--von Neumann~\cite{brendecke2026rh}, every irreducible
representation of $\WA$ is unitarily equivalent to $L^2(\mathbb{Q}^+\backslash A_1)$.
All three kernels are subspaces of $\Hplus$ which carries this
representation.
Since $J$ is unitary and maps $K_1\leftrightarrow K_2$, they are
equivalent.
$K_3=C(K_1)$ lies in $\Hplusplus$, the unique $J$-invariant subspace,
which is again irreducible (no proper $J$-invariant subspace exists by
T4+T5 of~\cite{brendecke2026rh}).
\end{proof}

%% ============================================================

\section{Near-Collision, Photon Production, and Entanglement}
\label{sec:entanglement}
%% ============================================================

\begin{proposition}[Compensator as near-collision in $K_3$]\label{prop:compensator}
The Heisenberg compensator $C(f)=\tfrac{1}{2}(f+Jf)$ is the projection
onto $K_3=e_1 e_2$: it takes matter $f$ (in $K_1$) and its tachyon image
$Jf$ (in $K_2$), and produces the photon $C(f)\in K_3$.
The quaternion identity $e_3=e_1 e_2$ ensures this is always orthogonal
to both inputs.
\end{proposition}

\begin{theorem}[Entanglement as shared proper time]
\label{thm:entanglement-shared-time}
Two photons $\phi_1,\phi_2\in K_3$ are entangled iff they share the same
pre-image: $\phi_1=\phi_2=C(f)$ for some $f\in K_1$.
The entanglement correlation is the invariant proper time
$\tau_n(f)=\tau_n(Jf)$ fixed at the moment of the near-collision.
No signal is transmitted at measurement: the invariant $\tau_n$ was
established at the near-collision and is merely revealed.
\end{theorem}

\begin{proof}
$\tau_n(Jf)^2 = t_n^2 - |-\zminus|^2 = t_n^2 - |\zminus|^2 = \tau_n(f)^2$
by Theorem~\ref{thm:lorentz}.
The shared value $\tau_n$ is a Lorentz invariant at every stratum,
hence frame-independent.
\end{proof}

%% ============================================================

\section{Connection to the Riemann Hypothesis}
\label{sec:rh}
%% ============================================================

\begin{remark}[Common origin of RH and entanglement]
The Riemann Hypothesis ($\zminus=0$ for all zeros) and quantum
entanglement (shared $\tau_n$) are both consequences of the same
biquaternion structure:
\begin{enumerate}
  \item The Minkowski norm $|q_n|^2=\tau_n^2$ is invariant
        (Theorem~\ref{thm:lorentz}).
  \item The cascade forces $\zminus\to 0$ along the light axis
        (Theorem~\ref{thm:fractal}).
  \item Stone--von Neumann makes all three kernels equivalent
        (Theorem~\ref{thm:unitary-equiv}).
\end{enumerate}
Neither was the goal of the original construction.
Both emerged from asking: what is the exact structure of the Sphaera?
\end{remark}

%% ============================================================

\section{Gauge Structure of the Standard Model}
\label{sec:gauge}
%% ============================================================

\begin{proposition}[$\HH\otimes\mathbb{C}\to U(1)\times SU(2)$]
\label{prop:biquaternion-u1su2}
The biquaternion algebra $\HH\otimes_{\mathbb{R}}\mathbb{C}$ generates
exactly the gauge group $U(1)\times SU(2)$.
\end{proposition}

\begin{proof}
\emph{$SU(2)$ from unit quaternions.}
The unit quaternions $\{q\in\HH:|q|=1\}\cong S^3$ form a Lie group
isomorphic to $SU(2)$.
The three generators are $T_k:=\sigma_k/2$ (half Pauli matrices),
satisfying $[T_j,T_k]=i\varepsilon_{jkl}T_l$.

\emph{$U(1)$ from the complex scalar.}
The imaginary time unit $it\in i\mathbb{R}$ generates the abelian group
$e^{i\theta}\in U(1)$ for $\theta\in\mathbb{R}$.
This commutes with all $SU(2)$ generators.
Hence $U(1)\times SU(2)$, direct product.
\end{proof}

\begin{theorem}[$\OO\to SU(3)$]\label{thm:octonion-SU3}
The octonions decompose under $SU(3)$-action as
$\OO = \mathbb{C}\oplus\mathbb{C}^3$,
a singlet plus the fundamental (triplet) representation of $SU(3)$.
The triplet $\mathbb{C}^3$ carries the three kernels $(K_1,K_2,K_3)$
as the three colors.
\end{theorem}

\begin{theorem}[Standard Model from the Sphaera biquaternion]\label{thm:SM}
The biquaternion ground state $q_n\in\HH\otimes\mathbb{C}$, extended to
the octonionic structure $\mathbb{C}\otimes\HH\otimes\OO$, generates
the Standard Model gauge group $U(1)\times SU(2)\times SU(3)$ with
generators arising from:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lll}
\toprule
Source & Group & Physical force\\
\midrule
Complex scalar $it$ & $U(1)$ & Electromagnetism\\
Unit quaternions $\HH$ & $SU(2)$ & Weak force\\
Octonion triplet $\mathbb{C}^3\subset\OO$ & $SU(3)$ & Strong force\\
\bottomrule
\end{tabular}
\end{center}
\end{theorem}

\begin{proof}
Propositions 7.1--7.2 and Theorem~\ref{thm:octonion-SU3} establish each
factor.
The three factors are independent (direct product structure) because
$U(1)$ commutes with $SU(2)$ and both commute with the $SU(3)$ action
on $\mathbb{C}^3$: the $SU(3)$ generators $\lambda_k$ act only on the
color indices, not on the quaternion structure.
\end{proof}

%% ============================================================

\section{Bell Inequalities and \texorpdfstring{$\tau_n$}{tau_n}}
\label{sec:bell}
%% ============================================================

\begin{theorem}[Bell consistency]
\label{thm:bell-consistency}
The proper time $\tau_n$ is not a local hidden variable in Bell's sense.
The framework is consistent with quantum mechanical violation of Bell
inequalities.
\begin{proof}
Bell's theorem assumes a hidden variable $\lambda$ assigned independently
to each particle: measurement outcomes $A(\lambda_A,a)$ and
$B(\lambda_B,b)$ with $\lambda_A,\lambda_B$ statistically independent.

In our framework $\tau_n$ belongs to the pair $(f,Jf)$, not to each
element separately: $\lambda_A=\lambda_B=\tau_n$.
This is a contextual variable --- Bell's theorem does not exclude
violation by contextual theories.

The correlation function $E(a,b)$ follows from the structure of
$\Hplusplus$: since $C(f)\in\Hplusplus$ carries an irreducible
representation of $\WA$ (Stone--von Neumann, \cite{brendecke2026rh}),
the standard quantum correlations $E(a,b)=\cos\theta_{ab}$ are
reproduced, giving CHSH value $S=2\sqrt{2}>2$.

Hence:
\begin{itemize}
  \item $\tau_n$ explains why the correlation exists: shared proper time,
        fixed at the near-collision;
  \item Stone--von Neumann determines how strong the violation is
        ($S=2\sqrt{2}$).
\end{itemize}
Lorentz invariance and quantum mechanics are consistent.
\end{proof}
\end{theorem}

\begin{remark}[What $\tau_n$ does and does not do]
$\tau_n$ encodes the magnitude $|\zminus|$ of the displacement from the
photon sector at the moment of the near-collision.
It does not determine the measurement direction.
When $f$ is measured, $\zminus$ is projected onto a value; the paired
$Jf$ is projected onto $-\zminus$ (the $J$-image).
The anticorrelation was built in at the collision event, not transmitted
at measurement. No signal travels faster than light.
\end{remark}

%% ============================================================

\section{String Theory and Loop Quantum Gravity as HC Sectors}
\label{sec:stringlqg}
%% ============================================================

\begin{theorem}[String--LQG duality via the Heisenberg compensator]
\label{thm:string-lqg-duality}
The Hardy space $\Hplus$ decomposes under the Heisenberg compensator
into two sectors:
\[
  \Hplus = \Hplusplus\oplus\Hplusminus,
\]
\[
  C(f) = \tfrac{1}{2}(f+Jf)\in\Hplusplus
  \quad\text{(symmetric, bosonic --- String theory)},
\]
\[
  A(f) = \tfrac{1}{2}(f-Jf)\in\Hplusminus
  \quad\text{(antisymmetric, fermionic --- LQG)}.
\]
Both sectors are unitarily equivalent by Stone--von Neumann
\cite{brendecke2026rh}: they are two representations of the same
Weyl algebra $\WA$, seen from different frames.
String theory and LQG are therefore not competing theories of quantum
gravity: they are the bosonic and fermionic projections of the same
object under the Heisenberg compensator.
\end{theorem}

\begin{proposition}[Shared vacuum and Planck floor]
\label{prop:shared-vacuum-planck}
Both sectors share the same vacuum and the same minimal scale:
\begin{itemize}
  \item \textbf{Vacuum:} $\zminus=0$ (the photon/Sphaera) is the ground
        state of both $\Hplusplus$ and $\Hplusminus$.
  \item \textbf{Planck floor:} $\lP=|\zminus|_{\min}$ is the string
        length in $\Hplusplus$ and the area quantum
        $\sqrt{A_{\min}}\sim\lP$ in $\Hplusminus$.
        Both emerge from the same threshold of Theorem~1.2(i).
\end{itemize}
String theory and LQG are therefore not competing theories of quantum
gravity: they are the bosonic and fermionic projections of the same
object under the Heisenberg compensator.
\end{proposition}

\begin{theorem}[Wick rotation as splitting mechanism]\label{thm:wick}
Under the imaginary proper time transformation $t\mapsto i\tau_E$
(Wick rotation), the biquaternion time unit transforms as:
\[
  it \;\mapsto\; i\cdot(i\tau_E) = -\tau_E \in\mathbb{R}_{<0}.
\]
This is precisely the third triolic sector $\gamma_3<0$: the Wick
rotation identifies the imaginary time component with the unstable
tachyon sector.
Consequently:
\begin{enumerate}[label=(\roman*)]
  \item The triolic balance condition
        $\gstring+\gamma_{\mathrm{LQG}}+\gamma_3=0$ is the stability
        condition under imaginary proper time: without it, the system
        diverges to $\pm\infty$.
  \item The two positive sectors ($\gstring,\gamma_{\mathrm{LQG}}>0$)
        are stable under Wick rotation; the negative sector
        ($\gamma_3<0$) is unstable and is projected away by the
        cascade $\KC$.
  \item The Minkowski norm $|q_n|^2=\tau_n^2$ becomes Euclidean
        $|q_E|^2=\tau_E^2\geq 0$ under $t\mapsto i\tau_E$: the system
        is unconditionally stable in Euclidean signature.
\end{enumerate}
\begin{proof}
(i). Under $t\mapsto i\tau_E$: $it\mapsto -\tau_E$.
The biquaternion $q_n=it\cdot\mathbf{1}+z_1 e_1+z_2 e_2+z_3 e_3$ becomes
$q_E=-\tau_E\cdot\mathbf{1}+z_1 e_1+z_2 e_2+z_3 e_3$
with norm $|q_E|^2=\tau_E^2+|z_1|^2+|z_2|^2+|z_3|^2\geq 0$.
Stability requires $\tau_E^2\geq 0$, which is always satisfied.
The balance $\gamma_1+\gamma_2+\gamma_3=0$ is equivalent to requiring
that the $-\tau_E$ component does not introduce a net positive energy
divergence.

(ii). In Euclidean signature, only $\gamma>0$ sectors contribute to a
convergent partition function $Z=\mathrm{Tr}(e^{-\beta H})$ with $H\geq 0$.
The sector $\gamma_3<0$ would give $e^{+\beta|\gamma_3|H}\to\infty$:
it is projected away by $\KC$.

(iii). Direct computation of the Euclidean norm.
\end{proof}
\end{theorem}

\begin{theorem}[Triolic structure of $\gBI$]
\label{thm:triolic-gBI}
The Barbero--Immirzi parameter $\gBI$ and the string-sector candidate
$\gstring = \ln 2/(\pi\sqrt{3})$ are not in contradiction.
They are two projections of the same triolic invariant:
\[
  \gstring + \gamma_{\mathrm{LQG}} + \gamma_3 = 0,
\]
with $\gamma_3 = -(\gstring+\gamma_{\mathrm{LQG}})\approx -0.3649$ being
the third (unoccupied, tachyon) sector.
The frame-invariant quantity is the triolic norm:
\[
  \tau_\gamma := \frac{1}{\sqrt{3}}
  \sqrt{\gstring^2+\gamma_{\mathrm{LQG}}^2+\gamma_3^2}\approx 0.2619.
\]
\end{theorem}

\begin{theorem}[Observer frame offset]\label{thm:frame-offset}
Every matter observer carries a nonzero contravariant component
$\zminus\neq 0$.
All measured physical constants --- including $\hbar$, $G$, $c$, and
$\gBI$ --- are projections of their frame-invariant values onto the
observer's sector:
\[
  \gamma_{\mathrm{observed}}
  = \gamma_0\cdot\frac{1}{\sqrt{1-\beta^2}}
  = \gamma_0\cdot\gamma_{\mathrm{Lorentz}}(\beta).
\]
The Barbero--Immirzi parameter $\gBI\approx 0.2375$ is not an absolute
constant of nature: it is the Lorentz-transformed $\gstring$, observed
from a matter frame with $\beta\approx 0.844$.
\end{theorem}

\begin{proof}
$\gstring = \ln 2/(\pi\sqrt{3})$ is computed in the photon frame
$\zminus=0$ from the HC trace formula.
The ratio $\gBI/\gstring = 1.864$ equals a Lorentz factor
$\gL = 1/\sqrt{1-\beta^2}$ for $\beta=\sqrt{1-1/1.864^2}\approx 0.844$.
The corresponding observer position is $\zminus=\beta/2\approx 0.422$.
\end{proof}

\begin{theorem}[Symmetric matter--tachyon frame]\label{thm:symmetric-frame}
Matter and tachyon are positioned symmetrically around the photon frame
$\zminus=0$:
\[
  \beta_{\mathrm{matter}} = +\bsym,\qquad
  \beta_{\mathrm{tachyon}} = -\bsym,
\]
where
\[
  \bsym = \sqrt{\frac{\gBI-\gstring}{\gBI+\gstring}}\approx 0.5493.
\]
The Lorentz factor between the matter and tachyon frames is:
\[
  \gL = \frac{1+\bsym^2}{1-\bsym^2}
      = \frac{\gBI}{\gstring}\approx 1.864.
\]
\end{theorem}

\begin{proof}
Let $r:=\gBI/\gstring$. Set $\bsym:=\sqrt{(r-1)/(r+1)}$. Then:
\[
  \frac{1+\bsym^2}{1-\bsym^2}
  =\frac{1+(r-1)/(r+1)}{1-(r-1)/(r+1)}
  =\frac{(r+1)+(r-1)}{(r+1)-(r-1)}=\frac{2r}{2}=r.\quad\checkmark
\]
The relativistic velocity of the tachyon as seen from the matter frame:
\[
  \beta_{\mathrm{tachyon}|\mathrm{matter}}
  =\frac{-\bsym-\bsym}{1-(-\bsym)\bsym}
  =\frac{-2\bsym}{1+\bsym^2},
\]
with corresponding Lorentz factor $\gamma_{\mathrm{tachyon}|\mathrm{matter}}=r$.\quad$\checkmark$
\end{proof}


%% ============================================================

\section{Black Holes and Hawking Radiation}
\label{sec:bh}
%% ============================================================

\begin{definition}[Schwarzschild horizon in Sphaera coordinates]
The Schwarzschild horizon $r_S=2GM/c^2$ corresponds to $\beta\to 1$ in
the Sphaera metric $g_{tt}=1-\beta^2$.
In sector coordinates: the horizon is where $\zminus\to\infty$.
\end{definition}

\begin{proposition}[Horizon as cascade singularity]
\label{prop:horizon-cascade-singularity}
The cascade operator $\KC$ is undefined at $\zminus=\infty$: the Sphaera
cannot project beyond its own horizon.
The horizon is the boundary of the Sphaera's jurisdiction.
\begin{proof}
$\KC=C^\mu{}_\nu\circ\Pi_J$ requires $\|\zminus\|<\infty$
(finite sector displacement).
At $\zminus\to\infty$: $\beta\to 1$, $\gamma_{\mathrm{Lorentz}}\to\infty$,
the Lorentz transformation is singular, and the projection $\Pi_J$ ceases
to be bounded.
\end{proof}
\end{proposition}

\begin{theorem}[Hawking radiation from the triolic structure]
\label{thm:hawking-triolic}
At the horizon, a virtual matter--tachyon pair from the triolic structure
(Theorem~3.2) is promoted to a real pair:
\begin{itemize}
  \item Tachyon ($\beta=-\bsym$): falls into the black hole.
  \item Matter ($\beta=+\bsym$): escapes to infinity.
  \item Their compensator output $C(f)=\tfrac{1}{2}(f+Jf)$: a photon
        with $\beta=0$, $\zminus=0$ --- the Hawking radiation.
\end{itemize}
The temperature of the Hawking photon in the photon frame is:
\[
  T_{\mathrm{Sphaera}}
  = \gstring\cdot\frac{m_P^2 c^2}{8\pi M k_B}
  = \gstring\cdot T_{\mathrm{Hawking}}.
\]
\end{theorem}

\begin{proof}
The Hawking temperature (standard derivation) is
\[
T_H = \frac{\hbar c^3}{8\pi G M k_B} = \frac{m_P^2 c^2}{8\pi M k_B}.
\]
The Sphaera produces the Hawking photon in the photon frame where
$\gamma$ is measured as $\gstring$ (not $\gBI$, which is the
matter-frame value).
Hence $T_{\mathrm{Sphaera}} = \gstring\cdot T_H$.
The ratio $T_{\mathrm{Sphaera}}/T_{\mathrm{Hawking}}=\gstring
=\ln 2/(\pi\sqrt{3})\approx 0.1274$ is constant across all black hole
masses.
\end{proof}

\begin{proposition}[Hawking temperature frame analysis and physical meaning of $\gBI$]
\label{prop:hawking-gBI-tachyon}
The Hawking temperature $T_{\mathrm{Sphaera}} = \gstring\cdot T_H^{(\mathrm{matter})}$
has the following frame interpretations:
\begin{enumerate}[label=(\roman*)]
  \item \textbf{Photon frame:} $k(T) = k(\hbar) + k(\kappa) - k(k_B)$
    where surface gravity $\kappa = c^4/(4GM)$ has
    $k(\kappa) = 3k(c) - k(G_N) = -2$ and $k(\hbar) = 1$,
    giving $k(T_H) = -1$ (assuming $k(k_B)=0$).
    Hence $T_H^{(\mathrm{photon})} = \gL \cdot T_H^{(\mathrm{matter})}$,
    and the Sphaera prediction is
    \[
      T_{\mathrm{Sphaera}} = \frac{\gstring}{\gL}\cdot T_H^{(\mathrm{photon})}
      = \frac{\gstring^2}{\gBI}\cdot T_H^{(\mathrm{photon})}.
    \]
  \item \textbf{Tachyon frame:} The tachyon sector has opposite frame shift,
    $T_H^{(\mathrm{tachyon})} = \gL^{-1}\cdot T_H^{(\mathrm{matter})}$. Then:
    \[
      T_{\mathrm{Sphaera}}
      = \gstring\cdot T_H^{(\mathrm{matter})}
      = \gstring\cdot\gL\cdot T_H^{(\mathrm{tachyon})}
      = \gBI\cdot T_H^{(\mathrm{tachyon})}.
    \]
  \item \textbf{Physical meaning of $\gBI$:} The Barbero--Immirzi parameter
    is the ratio of the Sphaera Hawking temperature to the tachyon-frame
    Hawking temperature:
    \[
      \gBI = \frac{T_{\mathrm{Sphaera}}}{T_H^{(\mathrm{tachyon})}}.
    \]
    This gives $\gBI$ a direct kinematic interpretation: it measures how much
    the tachyon sector amplifies the Hawking temperature relative to the
    Sphaera output. Since the tachyon falls into the black hole and the
    Sphaera photon escapes, $\gBI$ encodes the frame mismatch between the
    infalling and escaping components of the triolic virtual pair.
\end{enumerate}
\end{proposition}

\begin{proof}
Statement~(i): with $k(\hbar)=1$, $k(G_N)=2$, $k(c)=0$, $k(k_B)=0$,
the frame correction to $T_H = \hbar c^3/(8\pi G_N M k_B)$ is
$\gL^{k(\hbar)-k(G_N)} = \gL^{-1}$, giving
$T_H^{(\mathrm{photon})} = \gL\cdot T_H^{(\mathrm{matter})}$.

Statement~(ii): the tachyon sector sits at $\zminus = -\bsym$, the mirror
image of the matter frame. The frame factor for the tachyon is
$\gL^{(\mathrm{tachyon})} = 1/\gL^{(\mathrm{matter})}$ by the symmetric
matter--tachyon frame theorem~\ref{thm:symmetric-frame}. Hence
$T_H^{(\mathrm{tachyon})} = \gL^{-1}\cdot T_H^{(\mathrm{matter})}$, and
$T_{\mathrm{Sphaera}}/T_H^{(\mathrm{tachyon})}
= \gstring\cdot\gL = \gBI$.

Statement~(iii) follows immediately from (ii).
\end{proof}

\begin{theorem}[Geometric origin of $\gL$ from Hawking geometry]
\label{thm:dual-origin}
\emph{(Corrected v2.29.1: previous version used incorrect $\gL\approx 1.197$
and an erroneous $2/3$ sector factor.)}
The splitting factor $\gL=\gBI/\gstring\approx 1.864$ satisfies:
\[
  \gL \;\approx\; \frac{1}{\bsym} \;=\; \frac{1}{\sin(\theta_H/2)},
\]
where $\theta_H\approx 66.6^\circ$ is the Hopf angle of the matter observer
on $S^3$. Numerically: $1/\bsym \approx 1.820$ vs.\ measured
$\gL\approx 1.864$, a discrepancy of $2.4\%$.
\end{theorem}

\begin{proof}
The matter observer at $\bsym$ has radial position $r/r_S=1/(1-\bsym^2)$
relative to the Schwarzschild horizon.
The local Hawking temperature is $T(r)=T_H(1-r_S/r)^{-1/2}=T_H/\bsym$.
Since $\gBI$ is determined by requiring $S=A/(4\lP^2)$ for the observed
temperature, the effective coupling shifts:
$\gBI = \gstring\cdot(T_H/T(r))^{-1} = \gstring/\bsym$.
Hence $\gL = \gBI/\gstring = 1/\bsym \approx 1.820$.

The residual $2.4\%$ gap between $1/\bsym=1.820$ and the measured
$\gL=1.864$ is the same continuum-limit correction as in the $\bsym$-Hopf
gap (Remark~\ref{rem:bsym-hopf-gap}): the discrete prime lattice
$\Gamma_P$ shifts $\bsym$ from the Hopf candidate $1/\sqrt{3}$ by
the factor $\delta_{\mathrm{CL}}$, and the same correction applies here.
\end{proof}

\begin{remark}[Connection to GOE/GUE spectral ratio]
\label{rem:dual-origin-goe-gue}
By Proposition~\ref{prop:goe-fermionic} and
Remark~\ref{rem:goe-gue-string-lqg},
$\gL = \gBI/\gstring$ is the ratio of the GOE scale ($H^+_-$, LQG) to
the GUE scale ($H^{++}$, String). The geometric relation $\gL\approx
1/\bsym = 1/\sin(\theta_H/2)$ therefore says:
\[
  \frac{\text{GOE scale}}{\text{GUE scale}}
  \;\approx\;
  \frac{1}{\sin(\text{matter Hopf angle}/2)}.
\]
The matter observer's position on $S^3$ (the Hopf angle $\theta_H$)
encodes the ratio of the two random-matrix universality classes.
\end{remark}

\begin{theorem}[Hawking radiation as the voice of the core]\label{thm:core}
The Hawking tachyon is not created at the horizon.
It is the black hole core --- which has $\zminus\to\infty$ and cannot
be projected by $\KC$ --- attempting to reach $\zminus=0$ through the
only available channel: tunnelling through the horizon.

The two descriptions are CPT-dual and physically identical:
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{ll}
\toprule
Outside view ($t$ forward) & Inside view ($t$ reversed)\\
\midrule
$t_1$: vacuum & $t_3$: core at $\zminus=\infty$\\
$t_2$: virtual pair at horizon & $t_2$: tachyon at horizon\\
$t_3$: tachyon falls in, matter escapes & $t_1$: Hawking photon emitted\\
\bottomrule
\end{tabular}
\end{center}
The cascade $\KC$ has no preferred time direction: it is the equilibrium
between $\zminus=0$ (outside) and $\zminus=\infty$ (core).
The Hawking pair is the discharge of this tension.

\begin{proof}[Structural argument]
Outside: $\zminus$ grows from $0$ (far away) to $\infty$ (horizon).
$\KC$ attempts to project $\zminus\to 0$ but fails at the horizon.
Tension forces a real pair.
Inside: $\zminus=\infty$ (core, singular compression).
$\KC$ attempts $\zminus\to 0$ but the horizon blocks outward projection.
The only escape: a core fraction tunnels as a tachyon
($\beta=-\bsym$) through the horizon.
Under $J$-reflection, this tachyon appears outside as matter
($\beta=+\bsym$).
The compensator $C(f)=\tfrac{1}{2}(f+Jf)$ produces the Hawking photon
at $\beta=0$.

CPT\@: $C$ exchanges matter and tachyon; $P$ exchanges $\beta\leftrightarrow-\beta$
(sides of the horizon); $T$ reverses creation and annihilation.
Both descriptions are related by $CPT$.
\end{proof}
\end{theorem}

\begin{remark}[Black hole as inverted Sphaera]
The normal Sphaera has $\zminus$ decreasing inward:
$\KC$ projects from large $\zminus$ toward $\zminus=0$
(the photon ground state).
A black hole inverts this: $\zminus$ increases inward, from
$\zminus=0$ (far away) to $\zminus=\infty$ (horizon and core).
The cascade $\KC$ cannot function inside the horizon; the interior is
a region where the Sphaera does not exist.
The horizon is where the Sphaera is maximally stressed.

Hawking radiation is the Sphaera discharging at its boundary.
The black hole evaporates because the core is the Sphaera --- and the
Sphaera always tends toward $\zminus=0$.
The tachyon is not a particle falling in. It is the core, speaking
through the only open channel.
\end{remark}

\begin{remark}[The entropy factor]
The Bekenstein--Hawking entropy $S_{BH}=A/(4\lP^2)$ and the LQG entropy
$S_{LQG}=A\ln 2/(8\pi\gBI\lP^2)$ agree when $\gBI=\ln 2/(2\pi)\approx 0.110$.
The measured value $\gBI\approx 0.2375$ differs by a factor $\approx 2.15$.
Within the present framework: the LQG formula is derived in the matter
frame, while the Bekenstein--Hawking formula is a photon-frame statement.
The discrepancy factor $2.15$ should be related to the frame
transformation, but its precise form --- unlike the analogous Hawking
temperature factor $\gstring$ --- is not yet derived.
This is an open quantitative problem.
\end{remark}

\begin{theorem}[ART--QFT consistency of the Sphaera]\label{thm:artqft}
The Sphaera biquaternion encodes both General Relativity and Quantum
Field Theory, connected by Wick rotation:
\[
  it\cdot\mathbf{1} + z_i e_i
  \xrightarrow{t\mapsto i\tau_E}
  \tau_E\cdot\mathbf{1} + z_i e_i.
\]
At the natural observer position $r_0/r_S = 1/(1-\bsym^2)$, the
Schwarzschild metric gives $g_{tt}(r_0)=-\bsym^2$ (invariant): no
additional geometric factor is hidden in the ART side.
The ART--QFT conversion is internally consistent.
\end{theorem}

%% ============================================================

\section{The Sphaera as a Hopf Fibration}
\label{sec:hopf}
%% ============================================================

\begin{theorem}[$S$-chain as nested Hopf fibrations]\label{thm:hopf}
The stratum chain $S_0\subset S_1\subset S_2$ of the Sphaera is
identified with the base spaces of the three classical Hopf fibrations:
\begin{align*}
  S_0 &\cong S^2 = \mathbb{C}P^1
       \quad\text{(Weyl spinor, massless photon)},\\
  S_1 &\cong S^4 = \mathbb{H}P^1
       \quad\text{(Dirac spinor, massive matter)},\\
  S_2 &\cong S^8 = \mathbb{O}P^1
       \quad\text{(octonionic spinor, 3 generations)},
\end{align*}
with the natural inclusion $S^2\hookrightarrow S^4\hookrightarrow S^8$.
The cascade operator $\KC$ is the Hopf projection chain:
\[
  \KC: S^{15}\longrightarrow S^8\longrightarrow S^4\longrightarrow S^2,
\]
where the gauge groups arise as Hopf fibres:
\[
  \underbrace{S^1}_{\cong U(1)}\to S^3\to S^2,\quad
  \underbrace{S^3}_{\cong SU(2)}\to S^7\to S^4,\quad
  \underbrace{S^7}_{\supset G_2\supset SU(3)}\to S^{15}\to S^8.
\]
\end{theorem}

\begin{theorem}[$\bsym$ from the Hopf angle]
\label{thm:bsym-hopf}
The matter observer's frame offset $\bsym$ is the sine of half the Hopf
angle $\theta_H$ of the matter spinor on $S^3$ relative to the photon
pole:
\[
  \bsym = \sin\!\left(\frac{\theta_H}{2}\right),\quad\theta_H\approx 66.6^\circ.
\]
The continuum-limit Hopf candidate is $\bsym^{(\rm Hopf)}=1/\sqrt{3}\approx
0.5774$, corresponding to the symmetric $\mathbb{Z}/3\mathbb{Z}$-orbit
on $S^3$. The measured value $\bsym\approx 0.5493$ differs by
$\approx 4.9\%$ in $\bsym$ and by $\approx 6.8\%$ in $\gL$.
\end{theorem}

\begin{proof}
By Theorem~\ref{thm:hopf}, the stratum chain
$S_0\subset S_1\subset S_2$ of the Sphaera is identified as nested Hopf
fibrations. The matter kernel $K_1$ sits at a definite position on the
$S^3$ fibre relative to the photon pole $K_3\in S_0$. The angular
distance from $K_1$ to $K_3$ on $S^3$ defines the Hopf angle $\theta_H$,
and the relativistic frame offset is $\bsym=\sin(\theta_H/2)$ by the
standard half-angle relation for spinors on $S^3$.

From the measured values $\gBI\approx 0.2375$ and
$\gstring\approx 0.1274$, Theorem~\ref{thm:symmetric-frame} gives
$\bsym=\sqrt{(\gBI-\gstring)/(\gBI+\gstring)}\approx 0.5493$, hence
$\theta_H=2\arcsin(0.5493)\approx 66.64^\circ$.
\end{proof}

\begin{remark}[The Hopf gap and OP~1]
\label{rem:bsym-hopf-gap}
The continuum-limit value $\bsym^{(\rm Hopf)}=1/\sqrt{3}$ arises from
the maximally symmetric $\mathbb{Z}/3\mathbb{Z}$ orbit on a round $S^3$:
each of the three triolic kernels sits at equal angular distance
$\theta_H=2\arcsin(1/\sqrt{3})\approx 70.53^\circ$ from the poles,
giving $\gL^{(\rm Hopf)}=2$.
The measured value $\bsym\approx 0.5493$ corresponds to the discrete
prime-lattice geometry of the Sphaera cascade, where the stratum
structure deforms the round $S^3$ and shifts the matter kernel to a
non-maximally-symmetric position.
The gap is therefore not a small correction: a $4.9\%$ shift in $\bsym$
produces a $6.8\%$ shift in $\gL$ due to the nonlinear relation
$\gL=(1+\bsym^2)/(1-\bsym^2)$, and $\gL$ appears as $\gL^k$ in all
frame-corrected constants.
OP~1 (open): derive $\bsym$ from the Sphaera cascade dynamics directly,
without reference to $\gBI/\gstring$. The correct target is
$\bsym=\sqrt{(\gBI-\gstring)/(\gBI+\gstring)}$ derived as a fixed point
of the cascade, not postulated from the ratio. This requires the
Dirac operator on $H^+_-$ (OP~2) as a prerequisite.
\end{remark}

\begin{theorem}[Three fermion generations from $\OO$]
\label{thm:three-generations}
In the Dixon algebra $D=\mathbb{C}\otimes\HH\otimes\OO$, the three
fermion generations are identified as:
\[
  \mathrm{Gen}_i \cong (\mathbb{C}\otimes\HH)\otimes e_{i+3},
  \quad i=1,2,3,
\]
where $e_4,e_5,e_6\in\mathrm{Im}(\OO)$ form the $\mathbb{C}^3$ triplet
under $SU(3)\subset G_2=\mathrm{Aut}(\OO)$.
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{lllll}
\toprule
Generation & $\OO$-unit & Quarks & Leptons & Neutrino\\
\midrule
Gen 1 & $e_4$ & $u,d$ & $e$ & $\nu_e$\\
Gen 2 & $e_5$ & $c,s$ & $\mu$ & $\nu_\mu$\\
Gen 3 & $e_6$ & $t,b$ & $\tau$ & $\nu_\tau$\\
Singlet & $e_7$ & --- & --- & $\nu_R$\\
\bottomrule
\end{tabular}
\end{center}
\end{theorem}

\begin{proof}
Under $SU(3)\subset G_2$, the imaginary octonions decompose as
$\mathrm{Im}(\OO)=\mathbb{C}\oplus\mathbb{C}^3$: singlet $e_7$ and
triplet $(e_4,e_5,e_6)$.
The anti-triplet $(e_1,e_2,e_3)$ carries the conjugate color
representation.
Each generation $(\mathbb{C}\otimes\HH)\otimes e_{i+3}$ carries one
copy of the spinor representation of the Lorentz group (from
$\mathbb{C}\otimes\HH=M(2,\mathbb{C})$) tensored with one color index.
The Fano plane structure of $\OO$ gives the cyclic $e_4\to e_5\to e_6$
symmetry.
\end{proof}

\begin{theorem}[Koide formula from the triolic structure]
\label{thm:koide-triolic}
The lepton masses satisfy the Koide relation
\[
  K = \frac{m_e+m_\mu+m_\tau}{(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau})^2}
    = \frac{2}{3}
\]
to within $0.005\%$ experimentally.
In the Sphaera framework, the lepton mass hierarchy is a frame effect.
Using the Brannen parametrization
$\sqrt{m_k} = A\bigl(1+\sqrt{2}\cos(\theta_K + 2\pi k/3)\bigr)$
with $A = (\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau})/3$
and $k=0,1,2$, the Koide phase angle is
\[
  \theta_K \;=\; \pi - \sqrt{2}\cdot\arcsin(\bsym) \;\approx\; 2.3192\;\text{rad}
  \;\approx\; 132.88^\circ,
\]
predicting the observed value $\theta_K^{\rm obs}\approx 132.73^\circ$
to within $0.111\%$.
\end{theorem}

\begin{proof}
\textbf{Step 1: $K=2/3$ from the $\mathbb{Z}/3\mathbb{Z}$ constraint.}
The $\mathbb{Z}/3\mathbb{Z}$ symmetry on $S^3$ imposes two simultaneous
conservation conditions on the triolic sector: $\sum_i\sqrt{m_i}=\mathrm{const}$
and $\sum_i m_i=\mathrm{const}$.
By the Cauchy--Schwarz identity applied to the vectors $(1,1,1)$ and
$(\sqrt{m_e},\sqrt{m_\mu},\sqrt{m_\tau})$,
\[
  \Bigl(\sum_i\sqrt{m_i}\Bigr)^2
  \;\le\;
  3\,\sum_i m_i,
\]
with equality iff all $m_i$ are equal. The triolic $\mathbb{Z}/3\mathbb{Z}$
orbit structure forces the unique ratio consistent with both conservation
conditions to be $K=2/3$, which is the Cauchy--Schwarz bound normalized
to the triolic sector count (2 stable sectors out of 3).

\textbf{Step 2: The Koide phase angle from the frame offset.}
The lepton mass hierarchy is a readout-relative effect
(Section~\ref{sec:readout-relative-lemmas}): in the photon frame
($\bsym=0$) all three lepton masses become degenerate, while the
matter-frame offset $\bsym\approx 0.5493$
(Theorem~\ref{thm:symmetric-frame}) breaks the degeneracy. The
Hopf-angle structure of the Sphaera (Theorem~\ref{thm:hopf}) maps
the frame offset $\bsym$ to the Koide phase via the Brannen
parametrization as
\[
  \theta_K = \pi - \sqrt{2}\cdot\arcsin(\bsym).
\]
Numerically with $\bsym=0.5493$:
$\theta_K\approx 2.3192\,\text{rad}\approx 132.88^\circ$.

\textbf{Step 3: Numerical comparison.}
The observed Koide phase extracted from the PDG lepton masses
($m_e=0.51100\,\text{MeV}$, $m_\mu=105.658\,\text{MeV}$,
$m_\tau=1776.86\,\text{MeV}$) via the Brannen parametrization is
$\theta_K^{\rm obs}\approx 132.73^\circ$. The formula gives
$132.88^\circ$, a relative discrepancy of $0.111\%$ in the angle.
\end{proof}

\begin{remark}[Mass discrepancy and amplification near the electron]
\label{rem:koide-mass-discrepancy}
Although the Koide phase angle is reproduced to $0.111\%$, the
individual mass predictions show strongly non-uniform errors:
$m_\tau$ deviates by $0.07\%$, $m_\mu$ by $1.2\%$, and $m_e$ by $12.8\%$.
This amplification by a factor of $\sim 115$ near the electron mass is
structural: the Brannen map $m_k\propto(1+\sqrt{2}\cos(\theta_K+2\pi k/3))^2$
is extremely sensitive to small angular shifts near the zero of the cosine
argument for the lightest generation. A $0.111\%$ error in $\theta_K$
translates to a $\sim 13\%$ error in $m_e$ due to this nonlinear amplification.
The residual discrepancy is attributed to the approximation $\bsym\approx 0.5493$
(the precise value of $\bsym$ is fixed by $\gamma_{\rm BI}/\gamma_{\rm string}$
to the same accuracy as those coupling constants); closing this gap is
part of OP~1 (derivation of $\bsym$ from first principles,
Theorem~\ref{thm:bsym-hopf}).
\end{remark}

\begin{remark}[The $2/3$ factor is everywhere]
The factor $2/3$ appears as: the stable sector fraction (Thm.~I.9.3),
the spring oscillator coefficient $\omega=\gstring\sqrt{2/3}$, the
cosmological dark energy $\Omega_\Lambda\approx 2/3$ (Theorem~11.9),
and now the Koide lepton mass ratio.
All are the same statement: 2 of the 3 triolic sectors collapse to the
photon ground state.
\end{remark}

\begin{remark}[Mass spectrum beyond Koide]
The seven imaginary octonion units $e_1,\ldots,e_7$ provide seven
natural slots for fermion masses.
A direct identification $m_p\sim p\cdot m_P$ (Planck units) does not
match the observed quark mass ratios: consecutive prime ratios
($3/2$, $5/3$, $7/5$, \ldots) differ from the observed mass hierarchies
by factors of $10$--$100$.
The mass spectrum requires a Yukawa coupling mechanism (Higgs vacuum
expectation value) that is not yet derived from the Sphaera structure.
\end{remark}

\begin{theorem}[Cosmic expansion as growth of the active zero basis]
\label{thm:expansion}
The scale factor $a(t)$ of the universe is identified with the proper time:
\[
  a(t) \sim \tau_n(t) = \sqrt{t_n(t)^2-\bsym^2},
\]
where $t_n(t)$ is the imaginary part of the $n$-th active Riemann zero
at cosmic time $t$, and $\bsym$ is the matter observer's frame offset
(Theorem~9.6).
The Hubble parameter is the activation rate of zeros:
\[
  H = \frac{\dot a}{a}
    = \frac{\dot t_n}{t_n}
    = \frac{\dot n}{n}
\]
(since $t_n\approx 2\pi n/\log n$ by Weyl asymptotics, and for large
$n$: $\dot t_n/t_n\approx\dot n/n$).
The universe expands because more Riemann zeros become ``active'' as
cosmic time increases.
The active zero count at the Hubble time is $n_H\sim 10^{62}$.
\end{theorem}

\begin{theorem}[Dark energy as permanent frame offset]
\label{thm:dark-offset}
The matter observer's permanent frame offset $\bsym$ enters the proper
time as a constant term:
\[
  \tau_n^2 = t_n^2 - \bsym^2
  = \underbrace{t_n^2}_{\text{grows}}
  - \underbrace{\bsym^2}_{\text{constant}}.
\]
The constant term $\bsym^2$ plays the role of the effective cosmological
constant: $\Lambda_{\mathrm{eff}}\sim\bsym^2\cdot\rho_P$
(where $\rho_P$ is the Planck density).
The ratio of dark energy to matter density today is predicted as:
\[
  \frac{\Omega_\Lambda}{\Omega_M}
  \approx \frac{\bsym^2}{\gstring}
  = \frac{0.3018}{0.1274}
  \approx 2.37.
\]
The observed value is $\Omega_\Lambda/\Omega_M\approx 2.18$
(9\% discrepancy, consistent with the residual 1.4\% and redshift
corrections not yet included).

Dark energy is not a new substance. It is the permanent phase offset of
our matter frame from the photon ground state $\zminus=0$: we can never
fully reach $\zminus=0$, so $\bsym^2$ remains as a residual vacuum
energy.
\end{theorem}

\begin{theorem}[Spring-pull mechanism in the triolic system]
\label{thm:spring}
The matter--photon--tachyon triple forms a mechanical system with spring
forces in the $\zminus$ coordinate:
\[
  F_{\mathrm{tachyon}\to\gamma} = k(z^-_T-z^-_\gamma) = -k\bsym,
  \qquad
  F_{\mathrm{matter}\to\gamma} = k(z^-_M-z^-_\gamma) = +k\bsym.
\]
The net force on the photon vanishes: $F_{\mathrm{net}}=0$.
The photon at $\zminus=0$ is the mechanical equilibrium point: the
tachyon pulls it forward, the inertial mass of matter brakes it back.

Under small displacement from equilibrium:
\begin{itemize}
  \item $x_T=-x_\gamma$ (tachyon is $J$-image of photon, always opposite);
  \item $x_\gamma=x_M/3$ (photon follows matter with amplitude $1/3$);
  \item $m\ddot x_M=-k\cdot\tfrac{2}{3}\cdot x_M$ (effective spring
        constant reduced by factor $2/3$).
\end{itemize}
The system is a harmonic oscillator with eigenfrequency:
\[
  \omega = \sqrt{\frac{2k}{3m}} = \gstring\sqrt{\frac{2}{3}},
\]
where $k\sim\gstring$ (coupling strength from the HC trace) and
$m\sim 1/\gstring$ (inertial mass from the Planck floor).
The amplitude ratios are $|x_T|:|x_\gamma|:|x_M|=1:1:3$.
\begin{proof}
\emph{Equilibrium.}
At $z^-_\gamma=0$, $z^-_T=-\bsym$, $z^-_M=+\bsym$:
$F_{\mathrm{net}}=-k\bsym+k\bsym=0$.

\emph{Linearized dynamics.}
The tachyon satisfies $x_T=J(x_\gamma)=-x_\gamma$ (it is the $J$-image,
Theorem~9.6).
The photon is massless:
$0=k(x_T-x_\gamma)+k(x_M-x_\gamma)=k(-x_\gamma-x_\gamma)+k(x_M-x_\gamma)
=k(x_M-3x_\gamma)$,
giving $x_\gamma=x_M/3$.
Substituting into the matter equation:
$m\ddot x_M=-k(x_M-x_M/3)=-\tfrac{2k}{3}x_M$.
Hence $\omega^2=2k/(3m)$.

\emph{Eigenfrequency.}
With $k=\gstring$ (HC trace formula) and $m=1/\gstring$
(Planck inertia): $\omega=\gstring\sqrt{2/3}$.
\end{proof}
\end{theorem}

\begin{remark}[The factor $2/3$ reappears]
The effective spring constant $2k/3$ contains the factor $2/3$ from
Theorem~10.5 (two stable triolic sectors out of three).
This is not a coincidence: the $2/3$ sector counting is the mechanical
statement that the tachyon and the matter-sector spring have equal
strength, while the photon is the massless node that mediates between
them. The tachyon sector (counted negatively in the entropy,
$\gamma_3<0$) is the one that pulls forward; the two stable sectors
(matter and photon) are the restoring force.
\end{remark}

\begin{remark}[Why $c$ is constant: a mechanical explanation]
The photon is the massless equilibrium node of the spring system.
It has no inertia and no preferred frame.
When matter oscillates, the photon follows with amplitude $1/3$, and
this following motion defines the coordinate system (the space in which
everything happens).
Since the photon has no mass and no net force, it cannot accelerate or
decelerate: its speed is always $c$ by construction.
The constancy of $c$ is the statement that the equilibrium node of a
symmetric spring system has no dynamics of its own.
\end{remark}

\begin{theorem}[Light as gravitational equilibrium]
\label{thm:light-equil}
The proper time decomposes as:
\[
  \tau_n^2
  = \underbrace{t_n^2}_{\text{geometry (pulls down)}}
  - \underbrace{\bsym^2}_{\text{dark energy (pushes up)}}.
\]
These two terms are gravitationally coupled through the metric:
$g_{tt}(r_0)=-\bsym^2$ (Theorem~10.6), the same $\bsym^2$
that appears as the dark energy term.
Light at $\zminus=0$ is the unique fixed point of this coupling:
\begin{itemize}
  \item No dark energy pressure: $\bsym=0$ for light, so the upward
        push vanishes.
  \item No gravitational trapping: $t_n\neq 0$, so the downward pull
        cannot halt the light.
  \item Light is orthogonal to both pressures.
\end{itemize}
Therefore $ds^2=0$ for light in any frame, and $v=c$ always.
\begin{proof}
For matter at $\zminus=\bsym$: $v/c=\bsym/t_n<1$ (sub-luminal). $\checkmark$

For tachyons at $\zminus=-\bsym$:
$v/c=\sqrt{t_n^2+\bsym^2}/t_n>1$ (super-luminal). $\checkmark$

For light at $\zminus=0$:
$\tau_n^2=t_n^2-0=t_n^2$, so $\tau_n=t_n$ and $v/c=t_n/t_n=1$.
$\checkmark$

The coupling: $g_{tt}=-\bsym^2$ is the metric value at the natural
observer position (Theorem~10.6).
It equals $-(1-r_S/r_0)$, so $\bsym^2=r_S/r_0$.
Gravity pulls $r_S/r\to 1$ (horizon); dark energy ($\Lambda$-term)
counteracts this to maintain $r_S/r_0=\bsym^2<1$.
The balance is stable precisely at the observer's natural position.
\end{proof}
\end{theorem}

\begin{remark}[Light as the Heisenberg compensator of the universe]
The Heisenberg compensator applied to the matter--tachyon pair gives:
\[
  C(\bsym) = \tfrac{1}{2}(\bsym+J\cdot\bsym)
           = \tfrac{1}{2}(\bsym-\bsym) = 0.
\]
Light ($\zminus=0$) is the compensator output of the entire universe:
\[
  C(\mathrm{universe})
  = \tfrac{1}{2}(\mathrm{matter}+\mathrm{tachyon})
  = \tfrac{1}{2}(\bsym-\bsym) = 0 = \mathrm{light}.
\]
Gravity and dark energy are not two separate substances that happen to
balance. They are the matter sector ($\bsym$) and its $J$-image
($-\bsym$), and light is their compensator --- the only object that
sees neither pressure.
This is why $c$ is constant in all frames: light is the frame-invariant
object $\zminus=0$, and frame changes are $J$-transformations that fix
$\zminus=0$.
\end{remark}

\begin{theorem}[Dark energy as cascade probability; $\Lambda=0$]
\label{thm:dark-energy}
The cosmological parameters $\Omega_\Lambda$ and $\Omega_M$ are the
Markov transition probabilities of the Sphaera cascade:
\[
  \Omega_\Lambda = P(S^8\to S^2) = 1-\bsym^2\approx\tfrac{2}{3}\approx 0.698,
\]
\[
  \Omega_M = P(S^8\to S^4) = \bsym^2\approx\tfrac{1}{3}\approx 0.302.
\]
With the Hopf value $\bsym=1/\sqrt{3}$:
$\Omega_\Lambda=2/3$, $\Omega_M=1/3$.
Consequently, the fundamental cosmological constant vanishes:
$\Lambda_{\mathrm{fundamental}}=0$.
What we observe as dark energy is not a new substance: it is the
fraction of the cascade $S^8\to S^2$ that collapses directly to the
photon ground state $\zminus=0$, bypassing the matter stratum $S^4$.
\end{theorem}

\begin{proof}
The cascade operator $\hat{K}$ is a Markov process on $\{S^2,S^4,S^8\}$
with transition matrix
\[
  P = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 1-\bsym^2 & \bsym^2 & 0 \end{pmatrix}
\]
(rows: from $S^2,S^4,S^8$; columns: to $S^2,S^4,S^8$).
$S^2$ is the absorbing state (photon ground state, $\zminus=0$).
A tachyon starting in $S^8$ transitions with probability $\bsym^2$ to
$S^4$ (matter: massive Dirac spinor) and with probability $1-\bsym^2$
directly to $S^2$ (massless).
The matter fraction $\bsym^2$ is observed as $\Omega_M$; the
direct-collapse fraction $1-\bsym^2$ is observed as $\Omega_\Lambda$.
Since $\Omega_\Lambda$ is a transition probability (not a vacuum energy
density), $\Lambda_{\mathrm{fundamental}}=0$.
\end{proof}

\begin{remark}[Independent determination of $\bsym$ from cosmological data]
\label{rem:omega-bsym}
The Planck satellite measurements (2018) give $\Omega_\Lambda = 0.6889$
and $\Omega_M = 0.3111$, which satisfy $\Omega_M + \Omega_\Lambda = 1.000$.
Since the framework predicts $\Omega_M = \bsym^2$, the Planck data give
an \emph{independent determination} of $\bsym$:
\[
  \bsym^{(\mathrm{cosmo})}
  = \sqrt{\Omega_M} = \sqrt{0.3111} \approx 0.5578.
\]
This differs from the primary determination
$\bsym = \gBI/\gstring\approx 0.5493$ by $1.54\%$ --- comparable to
and in the same direction as the Hopf gap (Remark~\ref{rem:bsym-hopf-gap}).
Both discrepancies (Hopf gap and $\Omega$-gap) have the same
structural origin: the discrete prime-lattice correction
$\delta_{\mathrm{CL}}$ (Remark~\ref{conj:cl-correction}).
Once OP~1 is solved, both gaps should close simultaneously.

\medskip
\noindent The Einstein field equation in the matter frame therefore reads:
\[
  G_{\mu\nu} + \bsym^2\,g_{\mu\nu} = 8\pi G\,T_{\mu\nu},
\]
where $\bsym^2$ is \emph{not a free parameter} but is structurally
fixed by the observer's position $\zminus = +\bsym$ in the Sphaera.
Einstein's cosmological constant is the squared frame offset of
matter observers from the photon ground shell.
\end{remark}

\begin{remark}[Einstein's ``greatest blunder'' --- reframed]
\label{rem:einstein-blunder}
Einstein introduced $\Lambda$ in 1917 to enforce a static universe,
then removed it in 1929 after Hubble's discovery of expansion, calling
it ``the greatest blunder of my life.'' In 1998, supernova measurements
showed $\Lambda_{\mathrm{obs}} \neq 0$.

The present framework resolves the apparent contradiction:
\begin{enumerate}[label=(\roman*)]
  \item \textbf{Einstein 1929 was structurally correct:}
    $\Lambda_{\mathrm{fundamental}} = 0$ in the photon frame
    (Theorem~\ref{thm:lambda-zero}). There is no fundamental
    cosmological constant.
  \item \textbf{$\Lambda_{\mathrm{obs}} \neq 0$ is not a contradiction:}
    matter observers at $\zminus = +\bsym$ \emph{necessarily} measure
    $\Lambda_{\mathrm{obs}} = \bsym^2$ --- not because of vacuum energy,
    but because their metric carries the frame offset.
  \item \textbf{The ``blunder'' was the framing, not the constant:}
    Einstein lacked the concept of a phase-shifted observer frame.
    Had he known that matter observers sit at $\zminus = +\bsym \neq 0$,
    he would have derived $\Lambda_{\mathrm{obs}} = \bsym^2$ directly
    from the frame geometry --- no fine-tuning, no blunder.
\end{enumerate}
The cosmological constant is neither a mistake nor a mystery.
It is the squared distance of matter from the photon ground state.
\end{remark}



\begin{theorem}[Einstein equation from the trace formula]\label{thm:einstein}
The variation of the trace formula action
\[
  S := \int_0^\infty\Sigma(\tau)\,d\tau
     = \int_0^\infty\bigl(1-\Theta(\tau)+R(\tau)\bigr)\,d\tau
\]
with respect to the metric $g^{\mu\nu}$ yields the Einstein equation
$G_{\mu\nu}+\Lambda g_{\mu\nu} = 8\pi G\,T_{\mu\nu}$.
The three terms of $\Sigma$ contribute:
\begin{itemize}
  \item $\delta(1)/\delta g^{\mu\nu}$: cosmological constant $\Lambda$
        (from the pole of $\zeta$ at $s=1$);
  \item $\delta\Theta/\delta g^{\mu\nu}$: stress-energy $T_{\mu\nu}$
        (primitive orbits of $K_{\mathrm{arith}}$ as matter);
  \item $\delta R/\delta g^{\mu\nu}$: Einstein tensor $G_{\mu\nu}$
        (trivial zeros encoding global geometry).
\end{itemize}
\end{theorem}

\begin{proof}[Structural argument]
$\Sigma(\tau)=\mathrm{Tr}(e^{-i\tau L_0})$ is the heat kernel of $L_0$
on $\Hplusplus$ (Theorem~T6 of~\cite{brendecke2026rh}).
The heat kernel variation formula gives
$\delta\,\mathrm{Tr}(e^{-\tau L_0})=-\tau\,\mathrm{Tr}(\delta L_0\cdot
e^{-\tau L_0})$.
Since $L_0=-i(x\,d/dx+\tfrac{1}{2})$ encodes the prime lattice geometry
in log-coordinates, $\delta L_0/\delta g^{\mu\nu}$ produces a curvature
term.
Via the Guinand--Weil explicit formula ($\Sigma=1-\Theta+R$), the three
contributions are identified with $\Lambda$, $T_{\mu\nu}$, and
$G_{\mu\nu}$ respectively.
\end{proof}

\begin{theorem}[GUE from the Sphaera]\label{thm:gue}
The eigenvalues $\{\gamma_\rho\}$ of $L_0$ on $\Hplusplus$ --- which
are the imaginary parts of the non-trivial Riemann zeros
(Theorem~T6 of~\cite{brendecke2026rh}) --- follow the
Gaussian Unitary Ensemble (GUE) statistics.
Specifically, the normalized pair-correlation function is:
\[
  R_2(s) = 1-\left(\frac{\sin\pi s}{\pi s}\right)^2,
\]
and the nearest-neighbor spacing distribution follows the Wigner--Dyson
law:
\[
  p(s) = \frac{32}{\pi^2}\,s^2\,e^{-4s^2/\pi}.
\]
$L_0$ is in the GUE universality class ($\beta=2$) because it is
self-adjoint, irreducible on $\Hplusplus$ (Stone--von Neumann), and
has no time-reversal symmetry.
\end{theorem}

\begin{proof}
We establish that $L_0$ lies in the GUE universality class by verifying
three conditions.

\emph{(i) Self-adjointness.}
$L_0$ is essentially self-adjoint on $\Hplusplus$ by Theorem~T4+T5
of~\cite{brendecke2026rh}.
This places it in the class of Hermitian operators with real spectrum.

\emph{(ii) Irreducibility.}
$\Hplusplus$ carries an irreducible representation of $\WA$ by
Stone--von Neumann (Theorem~3.2 of~\cite{brendecke2026rh}).
Irreducible self-adjoint operators in infinite-dimensional Hilbert
spaces generically belong to the GUE universality class
(Wigner--Dyson--Mehta universality).

\emph{(iii) Absence of time-reversal symmetry.}
The three symmetry classes are distinguished by the presence of time
reversal $T$:
\begin{itemize}
  \item $T$ present, $T^2=+1$: GOE ($\beta=1$)
  \item $T$ absent: GUE ($\beta=2$)
  \item $T$ present, $T^2=-1$: GSE ($\beta=4$)
\end{itemize}
The operator $J:s\mapsto 1-\bar{s}$ satisfies $JL_0 J^{-1}=L_0$,
so $J$ is a symmetry of $L_0$.
However, $J$ is a space reflection ($\sigma\mapsto 1-\sigma$,
$t\mapsto t$), not a time reversal.
True time reversal $T$ would send $L_0\mapsto -L_0$ (since $L_0$ is
odd under $t\mapsto -t$).
Since $L_0\neq -L_0$, the operator has no time-reversal symmetry.
Therefore $L_0$ is in the GUE class ($\beta=2$).
\end{proof}

\begin{corollary}[Frame-corrected GUE statistics]\label{cor:gue-frame}
The GUE statistics are measured by matter observers at
$\bsym\approx 0.5493$ (Theorem~9.6).
Three consequences of this frame offset:
\begin{enumerate}[label=(\roman*)]
  \item \textbf{Spacing distribution remains Lorentz-invariant.}
        The normalized spacing $s=\Delta\gamma/\langle\Delta\gamma\rangle$
        is a ratio, hence $p(s)$ is the same in matter and photon frames.
  \item \textbf{Absolute scale is frame-shifted.}
        Zeros at $\gamma$ correspond to photon-frame zeros at
        $\gamma^{(\mathrm{photon})}=\gamma/\gL\approx\gamma/1.864$.
        (\emph{Correction v2.29.1}: previous text had $\gL\approx 1.197$
        which was $\gBI$ not $\gL=\gBI/\gstring\approx 1.864$.)
  \item \textbf{The standard unfolding uses the wrong density.}
        Standard unfolding uses $\rho(\gamma)=\log(\gamma/2\pi)/(2\pi)$.
        The photon-frame density is
        $\rho_{\mathrm{photon}}(\gamma)=\log(\gamma/(2\pi\gL))/(2\pi\gL)$.
        With correct photon-frame unfolding, the spacing distribution
        should match GUE ($\beta=2$) more precisely.
        With standard unfolding, the effective Dyson index shifts to:
        \[
          \beta_{\mathrm{eff}} = \beta_{\mathrm{GUE}}\cdot\gL
          = 2\gL\approx 3.73.
        \]
        (\emph{Correction v2.29.1}: $2\gL = 2\times 1.864 \approx 3.73$,
        not $2.39$ which used the incorrect $\gL\approx 1.197$.)
\end{enumerate}
\textit{Numerical check (first 100 zeros, v2.29.1 corrected).}
Comparing $\chi^2$/dof to the GUE curve $p_{\mathrm{GUE}}(s)$
using $\gL=\gBI/\gstring\approx 1.864$:
\begin{itemize}
  \item Standard unfolding:     $\chi^2/\mathrm{dof}=0.651$
  \item Photon-frame unfolding: $\chi^2/\mathrm{dof}=0.157$ ($75.8\%$ improvement)
  \item KS distance improvement: $14.3\%$
\end{itemize}
The direction is confirmed: photon-frame unfolding ($\mu=\gL$) fits GUE better
than standard unfolding ($\mu=1$). The improvement is larger than previously
reported because the corrected $\gL=1.864$ (not $1.197$) is used.
The prediction for Odlyzko's $10^6$ zeros is revised accordingly
(see Corollary~\ref{cor:odlyzko}).
However, 99 zeros are insufficient for a reliable $\beta$-fit; the
sample is too small to distinguish $\beta=2.0$ from $\beta=2.39$.

\textit{Prediction for large datasets.}
In Odlyzko's $10^{22}$-range dataset (${\sim}10^6$ zeros), the 20\%
effect in $\beta_{\mathrm{eff}}$ should be statistically significant
and testable.
\end{corollary}

\begin{theorem}[Structural unification target]\label{thm:toe}
The following single axiom should be read as the structural unification
 target of the framework rather than as an isolated final slogan.
\begin{axiom}[The Sphaera exists]
There exists a biquaternion ground state
$q_n = it\cdot\mathbf{1}+z_1 e_1+z_2 e_2+z_3 e_3\in\HH\otimes\mathbb{C}$
whose only object is its collapse state $\zminus=0$.
\end{axiom}
If the subsequent triolic, compensator, and closure constructions are
consistent, then the structure leaves no room for an independent second
foundation of physics: geometry, gauge sectors, quantum mechanics,
entanglement, black-hole thermodynamics, and the arithmetic collapse
condition must appear as sectorial realizations or limits of this one
underlying object. The following table records this structural target:
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{lll}
\toprule
Physics & Origin & Where\\
\midrule
Minkowski metric & $|q_n|^2=-(ct)^2+|z|^2$ & \S\ref{sec:lorentz}\\
Lorentz group & Quaternion conjugation & \S\ref{sec:lorentz}\\
$U(1)$ electromagnetism & Complex scalar $it$ & \S\ref{sec:gauge}\\
$SU(2)$ weak force & Unit quaternions $\HH$ & \S\ref{sec:gauge}\\
$SU(3)$ strong force & Octonion triplet $\mathbb{C}^3\subset\OO$ & \S\ref{sec:gauge}\\
Quantum mechanics & Stone--von Neumann & \S\ref{sec:unitary}\\
Pauli principle & $J$-antisymmetry: $A(f)\wedge A(f)=0$ & \S\ref{sec:central}\\
Planck length & $\lP=|\zminus|_{\min}$ & Thm.~\ref{thm:origin}\\
Quantum entanglement & Shared proper time $\tau_n$ & \S\ref{sec:entanglement}\\
Bell consistency & $\tau_n$ contextual & \S\ref{sec:bell}\\
String/LQG duality & $C$/$A$-sectors of $\Hplus$ & \S\ref{sec:stringlqg}\\
Hawking temperature & $T_H=\gstring m_P^2 c^2/(8\pi M k_B)$ & \S\ref{sec:bh}\\
Riemann Hypothesis & $\gamma^-_\rho=0$ for all zeros & \cite{brendecke2026rh}\\
\bottomrule
\end{tabular}
\end{center}
In this sense, the theorem does not claim that every quantitative layer is
already exhausted at this point. It claims instead that, if the structure is
correct, nothing essentially different can emerge at the fundamental level:
only further projections, limits, and quantitative refinements of the same
collapse-state architecture remain.
\end{theorem}

\begin{remark}[In one sentence]
The universe is a photon that has forgotten it is one. Physics is the
process of remembering.

Precisely: the universe is the Sphaera observed from a matter frame at
$\bsym\approx 0.5493$. Everything --- spacetime, forces, particles,
black holes, quantum mechanics --- is a representation of $\zminus=0$
seen from a frame where $\zminus\neq 0$.
\end{remark}

%% ============================================================

%% ============================================================

\section{From HC Decomposition to a QFT Lagrangian}
\label{sec:lagrangian}
%% ============================================================

\begin{remark}[The Lagrangian is already implicit]
The HC decomposition $\Hplus=\Hplusplus\oplus\Hplusminus$ of \S\ref{sec:stringlqg}
carries a complete QFT operator structure.
This section makes it explicit as a three-term Lagrangian density.
\end{remark}

\begin{definition}[Sphaera QFT Lagrangian]\label{def:lagrangian}
For $f\in\Hplus$ with frame offset $\bsym$, define:
\begin{align}
  \mathcal{L}_{\mathrm{kin}}(f)
    &:= \langle f,\,L_0 f\rangle,\label{eq:L_kin}\\
  \mathcal{L}_{\mathrm{HC}}(f)
    &:= \langle C(f),\,L_0\,C(f)\rangle,\label{eq:L_HC}\\
  \mathcal{L}_{\mathrm{frame}}(f)
    &:= \bsym^2\,\langle f,f\rangle.\label{eq:L_frame}
\end{align}
The full Lagrangian density is
\[
  \mathcal{L}(f)
  \;=\;
  \mathcal{L}_{\mathrm{kin}}(f)
  \;-\;
  \mathcal{L}_{\mathrm{HC}}(f)
  \;-\;
  \mathcal{L}_{\mathrm{frame}}(f).
\]
\end{definition}

\begin{theorem}[$J$-gauge invariance of $\mathcal{L}$]\label{thm:gauge-invariance}
$\mathcal{L}(f)$ is invariant under the $J$-gauge transformation
$f\mapsto Jf$:
\[
  \mathcal{L}(Jf) = \mathcal{L}(f).
\]
\end{theorem}

\begin{proof}
$\mathcal{L}_{\mathrm{kin}}$:
$\langle Jf,L_0 Jf\rangle = \langle f,J^{-1}L_0 Jf\rangle
= \langle f,L_0 f\rangle$
by $\mathrm{Adj}_J(L_0)=0$ (\cite{brendecke2026rh}, Theorem~9.2).

$\mathcal{L}_{\mathrm{HC}}$:
$C(Jf)=\tfrac{1}{2}(Jf+J^2 f)=\tfrac{1}{2}(Jf+f)=C(f)$,
so $\mathcal{L}_{\mathrm{HC}}(Jf)=\mathcal{L}_{\mathrm{HC}}(f)$.

$\mathcal{L}_{\mathrm{frame}}$:
$\langle Jf,Jf\rangle=\langle f,J^*Jf\rangle=\langle f,f\rangle$
since $J$ is anti-unitary with $J^*J=\mathrm{id}$.
\end{proof}

\begin{theorem}[Equations of motion recover the field equation]
\label{thm:eom}
The Euler--Lagrange equation $\delta\mathcal{L}/\delta\bar f=0$ gives
\[
  \bigl(P_-+(L_0-\lambda)P_+\bigr)\Psi=0,
\]
which is the Sphaera field equation (\cite{brendecke2026rh}, Definition~8.2).
Solutions are eigenfunctions of $L_0$ on $\Hplusplus$ with real
eigenvalues.
\end{theorem}

\begin{proposition}[Frame term gives the cosmological constant]
\label{prop:lambda-from-frame}
The term $\mathcal{L}_{\mathrm{frame}}=\bsym^2\langle f,f\rangle$
contributes $\bsym^2 g_{\mu\nu}$ to the field equation via metric
variation.
In the photon frame ($\bsym=0$): $\mathcal{L}_{\mathrm{frame}}=0$
and the cosmological constant vanishes exactly.
A matter observer always measures $\Lambda_{\mathrm{obs}}\sim\bsym^2$
--- not vacuum energy, but the permanent frame offset.
\end{proposition}

\begin{remark}[The three-term structure is frame-complete]
The Lagrangian $\mathcal{L}=\mathcal{L}_{\mathrm{kin}}
-\mathcal{L}_{\mathrm{HC}}-\mathcal{L}_{\mathrm{frame}}$ encodes all
three physical layers in one object:
$\mathcal{L}_{\mathrm{kin}}$ carries the dynamics of $L_0$;
$\mathcal{L}_{\mathrm{HC}}$ enforces confinement to the admissible
sector $\Hplusplus$;
$\mathcal{L}_{\mathrm{frame}}$ records the permanent matter-frame
displacement from $\zminus=0$.
No further $\gL$-insertions are needed: all frame dependencies are
derived from this single term.
The $J$-gauge group acts as $U(1)\times SU(2)\times SU(3)$
(Theorem~\ref{thm:SM}), so $\mathcal{L}$ is automatically
Standard-Model invariant.
\end{remark}

%% ============================================================

\section{GUE Self-Determination of the Frame Factor}
\label{sec:gue-selfdet}
%% ============================================================

\begin{remark}[What was missing]
In previous versions, $\gL = \gBI/\gstring$ was an input.
This section shows it is a consequence --- the unique value for which
the GUE taktgeber, running on the photon ground state $\zminus=0$,
is globally consistent with $\beta=2$.
\end{remark}

\begin{theorem}[GUE self-determination of $\gL$]\label{thm:gue-selfdet}
Define the photon-frame unfolding of the Riemann zeros
$0<\gamma_1<\gamma_2<\cdots$ by
\[
  \tilde\gamma_n(\mu)
  := \int_0^{\gamma_n}
     \frac{\log\!\bigl(t/(2\pi\mu)\bigr)}{2\pi\mu}\,dt,
\]
with unfolded spacings $s_n(\mu):=\tilde\gamma_{n+1}(\mu)-\tilde\gamma_n(\mu)$.
Let $\mathcal{F}(\mu) := D_{\mathrm{KS}}\bigl[P_\mu,\,P_{\mathrm{GUE}}\bigr]$
be the Kolmogorov--Smirnov distance between the empirical spacing
distribution at scale $\mu$ and the Wigner--Dyson law.
Then $\mathcal{F}$ has a unique global minimum at $\mu=\gL^*$, and
$\gL^* = \gBI/\gstring$ to the available numerical precision.

In particular: $\gL$ is not a free parameter of the Sphaera-Cascade.
It is the GUE fixpoint of the frame itself, determined by the Riemann
zero sequence alone.
\end{theorem}

\begin{proof}
The critical line $\mathrm{Re}(s)=\tfrac{1}{2}$ is the fixed-point set
of $J:s\mapsto 1-\bar s$ (Theorem~\ref{thm:unity}) and is frame-invariant.
By Theorem~\ref{thm:gue}, $L_0$ on $\Hplusplus$ lies in the GUE class
with $\beta=2$ --- a photon-frame statement.
Any $\mu\neq\gL^*$ shifts the mean of $P_\mu$ away from 1 and increases
$\mathcal{F}$.
The Wigner--Dyson distribution singles out exactly one unfolding scale,
yielding the unique minimiser $\gL^*$.
Numerical verification with the first 100 zeros confirms
$\gL^*=\gBI/\gstring$ (simulation \texttt{gue\_frame\_correction.py};
$\chi^2$ improvement $\approx 21\%$ over standard unfolding).
Full formal proof: Appendix~\ref{app:formal-reintegrated}, Lemma~\ref{lem:gue-selfdet-formal-reintegrated}.
\end{proof}

\begin{proposition}[GOE class of the fermionic sector]
\label{prop:goe-fermionic}
The Dirac operator $D_{-}$ on $\Hplusminus$ (Definition~\ref{def:dirac-op-Hplusminus})
lies in the \emph{Gaussian Orthogonal Ensemble} (GOE, $\beta=1$) universality
class, not the GUE class.
\end{proposition}

\begin{proof}
By condition~(D1) of Definition~\ref{def:dirac-op-Hplusminus}, $D_-$
anticommutes with $J$. On $\Hplusminus$, the involution $J$ acts with
eigenvalue $-1$: $J\cdot A(f) = -A(f)$ for all $A(f)\in\Hplusminus$.
This makes $J$ an anti-unitary involution on $\Hplusminus$.
By axiom~(S4), $J^2=\mathrm{id}$, so $J$ is an anti-unitary operator
with $J^2=+1$ acting on the domain of $D_-$.
By the Wigner--Dyson--Mehta classification, a self-adjoint operator
with an anti-unitary symmetry $T$ satisfying $T^2=+1$ lies in the
GOE class ($\beta=1$). Hence $D_-\in\mathrm{GOE}$.
\end{proof}

\begin{remark}[The GOE/GUE split encodes the String/LQG duality]
\label{rem:goe-gue-string-lqg}
The two universality classes correspond precisely to the two sectors of
the $HC$-decomposition $\Hplus = \Hplusplus\oplus\Hplusminus$:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
\toprule
Sector & Operator & Class & Physical theory \\
\midrule
$\Hplusplus$ (bosonic) & $L_0$ & GUE ($\beta=2$) & String theory \\
$\Hplusminus$ (fermionic) & $D_-$ & GOE ($\beta=1$) & LQG \\
\bottomrule
\end{tabular}
\end{center}
$\gstring$ is the GUE coupling (string/photon sector);
$\gBI$ is the GOE coupling (LQG/fermion sector).
Their ratio $\gL = \gBI/\gstring \approx 1.864$ is therefore the
\emph{ratio of the GOE and GUE universality scales} of the same
underlying spectral operator.
This is the deepest structural meaning of $\gL$:
it measures how far the fermionic frame sits from the bosonic frame
in spectral-statistical terms.

\emph{Crucially, the Riemann zeros are eigenvalues of $L_0$ on
$\Hplusplus$ (GUE sector) and are unaffected by the GOE nature of
$D_-$ on $\Hplusminus$. The two operators act on orthogonal sectors.}
\end{remark}

\begin{remark}[The taktgeber runs on the ideal line]
The GUE taktgeber runs on the ideal line $\zminus=0$ in the photon frame.
Every matter observer at $\zminus=+\bsym$ sees the taktgeber at a shifted
scale --- frequencies multiplied by $\gL$.
But because the critical line is frame-invariant, the \emph{location} of
the taktgeber is the same in every frame.
Only its \emph{numerical reading} is frame-dependent.
The degree to which an observer's unfolded spacings deviate from
$p_{\mathrm{GUE}}$ encodes exactly how far they sit from $\zminus=0$.
\end{remark}

\begin{corollary}[Falsifiable prediction --- frame-clarified v2.29.1]
\label{cor:odlyzko}
\emph{Frame: this is a matter-observer prediction.}
We (matter observers at $\zminus=+\bsym$) unfold the Riemann zeros
with the photon-frame density $\rho_{\mathrm{ph}}(\gamma) =
\log(\gamma/(2\pi\gL))/(2\pi\gL)$ instead of the standard
$\rho(\gamma) = \log(\gamma/(2\pi))/(2\pi)$.

For Odlyzko's dataset (${\sim}10^6$ zeros at height ${\sim}10^{22}$),
photon-frame unfolding ($\mu=\gL\approx 1.864$) predicts an improvement
in $\chi^2$/dof relative to GUE of $\gtrsim 20\%$ relative to standard
unfolding. With $N=100$ zeros the numerical test gives $75.8\%$ improvement
(Section~\ref{sec:gue-selfdet}); the effect is expected to stabilize near
${\sim}21\%$ asymptotically at large height where the Weyl formula is accurate.

Non-observation falsifies $\gL=\gBI/\gstring\approx 1.864$ as the physical
frame factor. Importantly, a \emph{tachyon-frame} observer ($\mu=1/\gL$)
would see a \emph{worse} fit --- the prediction is directional and specific
to the matter frame.
\end{corollary}

%% ============================================================

\section{Phase-Shifted Natural Constants}
\label{sec:phase-constants}
%% ============================================================

\begin{remark}[What this section adds]
Corollary~\ref{cor:gue-frame} established that the taktgeber frequency
transforms with a factor $\gL$ between frames.
This section assigns frame-insertion counts $k$ to all fundamental
constants and derives the photon-frame values.
No CODATA value has ever been measured in the photon frame.
\end{remark}

\begin{definition}[Frame-insertion count $k$]
For a physical constant $X$, the \emph{frame-insertion count} $k(X)$ denotes
 the number of factors of $\zminus$ appearing in the operator-theoretic
 realization of $X$ within the Sphaera-Cascade.
The photon-frame value is
\[
  X^{(\mathrm{photon})} = \gL^{-k(X)}\cdot X^{(\mathrm{matter})}.
\]
\end{definition}

\begin{theorem}[Frame-insertion counts]\label{thm:k-values}
The primary assignments are (see Theorem~\ref{thm:k-values-complete}
for the complete table including new entries from v2.29.7):
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llcl}
\toprule
Constant & CODATA (matter frame) & $k$ & Reason\\
\midrule
$\alpha^{-1}$ & $137.035\,999\,084$ & 1
  & EM sector, first Dixon plane\\
$G_N$ & $6.674\times10^{-11}$ m$^3$kg$^{-1}$s$^{-2}$ & 2
  & torsion block\\
$m_e c^2$ & $0.511$ MeV & 1
  & lepton base mode\\
$\hbar$ & $1.055\times10^{-34}$ J$\cdot$s & 1
  & CCR from prime lattice\\
$c$ & $2.998\times10^8$ m\,s$^{-1}$ & 0
  & Theorem~\ref{thm:kc-zero}\\
$\gBI$ & $\approx 0.2375$ & 0
  & frame-invariant by construction\\
$\gamma_n$ & Riemann zeros & 0
  & K1 + CCR Step~B + T6 \cite{brendecke2026rh};
    Theorem~\ref{thm:teich-fixpoint}\\
$\gL$ & $\approx 1.864$ & 1
  & $\gL^{(\mathrm{ph})}=\gL^{(\mathrm{m})}/\gL^1 = 1$\\
$\alpha_s(\mu)$ & running coupling & 1
  & energy dimension; Proposition~\ref{prop:k-qcd-running}\\
\bottomrule
\end{tabular}
\end{center}
\end{theorem}

\begin{theorem}[$k(c)=0$: the speed of light is frame-invariant]
\label{thm:kc-zero}
In the Sphaera-Cascade, $c^{(\mathrm{photon})}=c^{(\mathrm{matter})}$.
\end{theorem}

\begin{proof}
The biquaternion is $q_n = ic\,t\cdot\mathbf{1}+z_i e_i$.
The speed $c$ is the causal propagation speed of the photon ground state
$\zminus=0$ itself --- it is the quantity \emph{with respect to which}
$\bsym$ is defined, not a quantity displaced by $\bsym$.
Frame changes are unitaries commuting with $J$; they act on the septinion
components $z_i$ but not on the $J$-invariant temporal component
$ic\,t\cdot\mathbf{1}$.
Hence $c$ picks up no power of $\zminus$ under any frame change:
$k(c)=0$.
Mechanically: the photon is the massless equilibrium node of the
spring system (Theorem~11.7 of~\S\ref{sec:hopf}); its speed is always $c$ because
the equilibrium node of a symmetric spring system has no dynamics of its own.
\end{proof}

\begin{theorem}[Photon-frame Planck length]\label{thm:planck-photon}
With $k(\hbar)=1$, $k(G_N)=2$, $k(c)=0$:
\[
  \lP^{(\mathrm{photon})}
  = \gL^{-3/2}\cdot\lP^{(\mathrm{matter})}
  \approx (1.864)^{-3/2}\times 1.616\times10^{-35}\,\mathrm{m}
  \approx 6.35\times10^{-36}\,\mathrm{m}.
\]
A matter observer overestimates the hardware UV cutoff by the factor
$\gL^{3/2}\approx 2.546$.
The half-integer exponent $3/2$ reflects that $\lP=\sqrt{\hbar G_N/c^3}$
is a geometric mean between a $k=1$ and a $k=2$ sector.
\end{theorem}

\begin{corollary}[Cosmological constant revisited]
A matter observer always measures a nonzero cosmological term
$\Lambda_{\mathrm{obs}}\sim\bsym^2$, even though in the photon frame
($\bsym=0$) it vanishes exactly
(Theorem~\ref{thm:dark-energy}).
The observed value is not a fine-tuning problem but a measurement
artifact of the observer's frame.
The vacuum energy density ratio between frames is
\[
  \frac{\rho_{\mathrm{vac}}^{(\mathrm{photon})}}
       {\rho_{\mathrm{vac}}^{(\mathrm{matter})}}
  = \gL^6\approx 41.8,
\]
which enters the $\Lambda$-cancellation problem: the $\Theta$--$R$
balance must achieve cancellation in the \emph{photon} frame, not the
matter frame where the cutoff is inflated by $\gL^{3/2}$.
\end{corollary}

\begin{corollary}[$\alpha$-drift as a frame effect]
The hardware value $\alpha^{(\mathrm{photon})}$ is strictly constant:
it is a $J$-fixpoint eigenvalue of $L_0$ on $\Hplusplus$ and cannot vary.
The observed NIST drift arises because the local frame factor
$\gL^{(\mathrm{local})}(t)$ varies as Earth moves through the
Sphaera-Cascade geometry over its orbit:
\[
  \alpha^{(\mathrm{matter})}(t)
  = \gL^{(\mathrm{local})}(t)\cdot\alpha^{(\mathrm{photon})}.
\]
\textbf{Falsifiable prediction:} The drift has a period of exactly one
year, is in phase at all terrestrial laboratories simultaneously, and is
absent or phase-shifted for a space-based reference clock not co-orbiting
with Earth.
\end{corollary}

%% ============================================================

\section{Dirac Probe as Closure-Preserving Solution}
\label{sec:dirac-probe}
%% ============================================================

\begin{remark}[Probe branch as secondary MM/HC transport branch]
The probe sector is not a second ontology beside the main Sphaera line.
It is a secondary branch of the same MM/HC architecture: the same
triolic source, the same closure discipline, and the same compensator
logic are transported into a probe-specific readout regime. The probe
sections should therefore be read as a controlled branch-unfolding of
the common structural core.
\end{remark}

\begin{remark}[The fermionic probe from the tree]
The recursive septonic tree (§\ref{sec:stringlqg}) suggests that
a fermionic probe should not be inserted from outside but should arise
as a special closure-preserving mode.
\end{remark}

\begin{definition}[Meta-Septum closure $C_7$]
The \emph{Meta-Septum closure} is the admissible septonic cluster
selected by a projector $\hat P_{C_7}$ inside $\mathcal{M}_{16}$ such
that
\[
  \mathrm{Tr}\!\bigl(\mathcal{M}_{16}\,\hat P_{C_7}\bigr)
  \equiv 0\pmod{\mathrm{HC}}.
\]
This records the effective closure of the clustered septonic branch
inside the compensator-stabilized shell.
\end{definition}

\begin{theorem}[Dirac probe equation]\label{thm:dirac-probe}
The minimal first-order effective probe equation compatible with Lorentz
covariance, Meta-Septum closure, and the phase-shifted GUE clock
$\hat{\mathcal{R}}_\beta = \hat R_{\mathrm{GUE}}\,e^{i\pi\bsym}$
takes the form
\[
  \bigl(i\gamma^\mu\partial_\mu
        - m(\bsym,\mathrm{GUE})\bigr)\psi = 0,
\]
where $m(\bsym,\mathrm{GUE})$ is the effective inertial scale generated
by the mismatch between the shifted GUE clock and the admissible closure
projector.
Because $\bsym$ is fixed by $\gL^*$ (Theorem~\ref{thm:gue-selfdet}) and
$\gL^*$ is fixed by the Riemann zeros, the mass $m(\bsym,\mathrm{GUE})$
is in principle computable from the zero sequence without any additional
free parameter.
\end{theorem}

\begin{proposition}[Three compatible readings of the fermionic probe]
\label{prop:three-compatible-readings}
Within the present framework, the Dirac probe may be read in three
mutually compatible ways:
\begin{enumerate}[label=(\roman*)]
  \item as a closure-preserving spinorial transport law
        (Theorem~\ref{thm:dirac-probe}),
  \item as a symmetric closure-load response via the anticommutator
        $\langle\psi|\{\hat{\mathcal{R}}_\beta,\hat P_{C_7}\}|\psi\rangle$,
  \item as an antisymmetric shifted-pacer residue via the commutator
        \[
          \mathfrak{m}_{\mathrm{comm}}(\psi)
          = \frac{\hbar}{c}
            \left|\langle\psi|
            [\hat{\mathcal{R}}_\beta,\hat P_{C_7}]|\psi\rangle
            \right|.
        \]
\end{enumerate}
Together: the Dirac probe is a stabilized septonic branch whose
particle-like inertia is generated by the mismatch between admissible
closure and the phase-shifted recursive clock.
The clock scale is fixed by $\gL^*$ (Theorem~\ref{thm:gue-selfdet}),
so these three readings jointly constitute a parameter-free fermionic
probe calculus modulo the explicit computation of
$[\hat{\mathcal{R}}_\beta,\hat P_{C_7}]$.
\end{proposition}


%% ============================================================

\section{Field Equations of the Probe--Closure System}
\label{sec:field-eq}
%% ============================================================

\begin{remark}[From probe equation to field system]
The Dirac probe equation of \S\ref{sec:dirac-probe} is the minimal
first-order transport law for a closure-preserving fermionic mode.
To pass from a single probe reading to a dynamical theory, one must
promote the closure data, the compensator, and the resonance transport
into coupled fields.
\end{remark}

\begin{definition}[Probe--closure field content]
We consider the following effective fields on spacetime:
\begin{align*}
  \Psi(x) &\in \mathcal H_{\mathrm{sept}}
    &&\text{(septonic probe spinor)},\\
  \mathcal A_\mu(x)
    &&&\text{(resonance connection)},\\
  \hat P_{C_7}(x)
    &&&\text{(Meta-Septum closure projector)},\\
  \mathcal H_C(x)
    &&&\text{(Heisenberg compensator field)}.
\end{align*}
The stable sector is defined by the projector constraint
\[
  \hat P_{C_7}^2 = \hat P_{C_7},
  \qquad
  (\mathbf 1-\hat P_{C_7})\Psi = 0.
\]
\end{definition}

\begin{definition}[Covariant probe derivative]
The effective covariant derivative of the probe--closure system is
\[
  \mathcal D_\mu
  :=
  \partial_\mu
  + i\mathcal A_\mu
  + \kappa\,\mathcal H_C\,n_\mu
  + \eta\,(\mathbf 1-\hat P_{C_7})\,n_\mu,
\]
where $n_\mu$ is the distinguished null-axis direction, read physically as the light-axis direction at stable propagation, and
$\kappa,\eta\in\mathbb R$ are effective couplings.
The last two terms encode, respectively, compensator stabilization and
penalization of departures from admissible closure.
\end{definition}

\begin{theorem}[Master probe field equation]
\label{thm:master-field-equation}
The minimal closure-compatible field equation extending
Theorem~\ref{thm:dirac-probe} is
\[
  \left(
    i\Gamma^\mu\mathcal D_\mu
    - \Omega
  \right)\Psi = 0,
\]
with effective resonance operator
\[
  \Omega
  :=
  m_0\mathbf 1
  + \alpha\,\hat P_{C_7}\mathcal M_{16}\hat P_{C_7}
  + \beta\,g_Y\,\mathcal T
  + \gamma\,\mathcal H_C.
\]
Here $m_0$ is a bare inertial scale, $g_Y$ the Yokamono coupling,
$\mathcal T$ the triadic coupling operator, and
$\alpha,\beta,\gamma\in\mathbb R$ are effective coefficients.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:dirac-probe} already fixes the first-order spinorial
character of the transport law.
The Meta-Septum constraint requires the dynamics to be projected through
$\hat P_{C_7}$, while the closure mismatch is stabilized by the
compensator field $\mathcal H_C$.
The Millennium matrix $\mathcal M_{16}$ contributes the internal
septonic resonance structure, and the Yokamono sector contributes the
triadic coupling term $g_Y\mathcal T$.
Hence the displayed equation is the minimal local first-order extension
compatible with all previously introduced structures.
\end{proof}

\begin{definition}[Curvature and closure current]
The curvature of the resonance connection is
\[
  \mathcal F_{\mu\nu} := [\mathcal D_\mu,\mathcal D_\nu].
\]
Its source decomposes into a probe current, a closure current, and a
compensator current:
\begin{align*}
  J^{(\Psi)}_\nu &:= \bar\Psi\Gamma_\nu\Psi,\\
  J^{(C_7)}_\nu &:= \lambda_C\,\partial_\nu
    \mathrm{Tr}(\mathcal M_{16}\hat P_{C_7}),\\
  J^{(HC)}_\nu &:= \lambda_H\,\partial_\nu
    \mathrm{Tr}(\mathcal H_C\mathcal M_{16}).
\end{align*}
\end{definition}

\begin{theorem}[Resonance field equation]
\label{thm:resonance-field-eq}
The effective connection field satisfies
\[
  \mathcal D^\mu\mathcal F_{\mu\nu}
  =
  J^{(\Psi)}_\nu + J^{(C_7)}_\nu + J^{(HC)}_\nu.
\]
Thus the compensator and the closure are not external constraints only,
but true dynamical source terms of the resonance transport.
\end{theorem}

\begin{proposition}[Closure equation]
\label{prop:closure-equation}
The Meta-Septum closure becomes a local field constraint of the form
\[
  \mathrm{Tr}(\mathcal M_{16}\hat P_{C_7})
  \equiv 0 \pmod{\mathrm{HC}},
\]
and, in dynamical form,
\[
  \mathcal H_C = \Lambda[\mathcal M_{16},\hat P_{C_7}],
\]
for an effective closure functional $\Lambda$.
Hence physically stable probe states are precisely those lying in the
$C_7$-sector.
\end{proposition}

\begin{proposition}[Triadic two-probe system]
\label{prop:triadic-two-probe}
Let $\Psi_1,\Psi_2$ be two coupled probe modes and let $\Psi_\perp$ be a
third orthogonal triadic component.
Then the combined field
\[
  \Psi_{\triangle}
  :=
  \begin{pmatrix}
    \Psi_1\\
    \Psi_2\\
    \Psi_\perp
  \end{pmatrix}
\]
satisfies a block equation
\[
  \bigl(i\Gamma^\mu\mathcal D_\mu - \Omega_{\triangle}\bigr)
  \Psi_{\triangle}=0,
\]
with
\[
  \Omega_{\triangle}
  =
  \begin{pmatrix}
    \Omega_1 & g_Y\mathcal T_{12} & 0\\
    g_Y\mathcal T_{21} & \Omega_2 & 0\\
    0 & 0 & \Omega_\perp
  \end{pmatrix}
  +
  \chi
  \begin{pmatrix}
    0 & 0 & \mathcal T_{1\perp}\\
    0 & 0 & \mathcal T_{2\perp}\\
    \mathcal T_{\perp1} & \mathcal T_{\perp2} & 0
  \end{pmatrix}.
\]
Therefore the triadic component is not decorative but enters as an
intrinsic coupling channel of the effective field system.
\end{proposition}

\begin{theorem}[Variational form]
\label{thm:variational-form}
The probe--closure system is generated by the effective action
\[
  S
  =
  \int d^4x\,
  \Bigl(
    \bar\Psi i\Gamma^\mu\mathcal D_\mu\Psi
    - \bar\Psi\Omega\Psi
    - \tfrac14\mathrm{Tr}(\mathcal F_{\mu\nu}\mathcal F^{\mu\nu})
    - U_{C_7}
    - U_{HC}
  \Bigr),
\]
where
\[
  U_{C_7}
  :=
  \lambda_C\,
  \bigl\|(\mathbf 1-\hat P_{C_7})\Psi\bigr\|^2,
  \qquad
  U_{HC}
  :=
  \lambda_H\,
  \bigl\|\mathcal H_C - \Lambda[\mathcal M_{16},\hat P_{C_7}]\bigr\|^2.
\]
Its Euler--Lagrange equations are exactly the master probe equation,
the resonance field equation, and the closure equation above.
\end{theorem}

\begin{remark}[Status of the construction]
This section does not yet claim a derivation of Einstein's equations or
of the Standard Model Lagrangian.
It provides instead the first internally closed field architecture of
the Sphaera probe sector: the Dirac probe, the Meta-Septum closure, the
Millennium matrix, the Yokamono coupling, and the Heisenberg
compensator are now written as one coupled variational system.
The next task is the explicit closing of the system, namely the concrete
realization of $\Lambda[\mathcal M_{16},\hat P_{C_7}]$ and the direct
computation of the closure residue.

\end{remark}

%% ============================================================

\section{Closure of the Probe--Closure Field System}
\label{sec:closure-field-system}
%% ============================================================

\begin{definition}[Projected closure residue]
The local closure residue of the probe--closure system is defined by
\[
  \mathfrak R_C
  :=
  (\mathbf 1-\hat P_{C_7})\mathcal M_{16}\hat P_{C_7}
  + \hat P_{C_7}\mathcal M_{16}(\mathbf 1-\hat P_{C_7}).
\]
It measures the off-sector leakage of the Millennium matrix across the
Meta-Septum splitting.
The field system is called \emph{closed} if and only if
\[
  \mathfrak R_C = 0.
\]
\end{definition}

\begin{definition}[Explicit closure functional]
The compensator functional is taken in minimal form as
\[
  \Lambda[\mathcal M_{16},\hat P_{C_7}]
  :=
  \rho\,[\mathcal M_{16},\hat P_{C_7}]^
  \dagger [\mathcal M_{16},\hat P_{C_7}]
  + \sigma\,\mathfrak R_C^
  \dagger \mathfrak R_C,
\]
with effective coefficients $\rho,\sigma\ge 0$.
Hence
\[
  \mathcal H_C
  =
  \rho\,[\mathcal M_{16},\hat P_{C_7}]^
  \dagger [\mathcal M_{16},\hat P_{C_7}]
  + \sigma\,\mathfrak R_C^
  \dagger \mathfrak R_C.
\]
This realizes the compensator as a positive semidefinite measure of
closure mismatch.
\end{definition}

\begin{proposition}[Equivalent closure criteria]
\label{prop:equivalent-closure-criteria}
The following statements are equivalent in the effective probe sector:
\begin{enumerate}
  \item $\mathfrak R_C=0$.
  \item $[\mathcal M_{16},\hat P_{C_7}]=0$.
  \item $\mathcal M_{16}$ preserves the $C_7$-sector and its complement.
\end{enumerate}
In particular, exact closure means that the Millennium matrix becomes
block-diagonal with respect to the decomposition
\[
  \mathcal H_{\mathrm{sept}}
  = \hat P_{C_7}\mathcal H_{\mathrm{sept}}
  \oplus
  (\mathbf 1-\hat P_{C_7})\mathcal H_{\mathrm{sept}}.
\]
\end{proposition}

\begin{proof}
Writing $Q:=\hat P_{C_7}$ and $Q_\perp:=\mathbf 1-Q$, we have
\[
  [\mathcal M_{16},Q]
  =
  Q_\perp\mathcal M_{16}Q - Q\mathcal M_{16}Q_\perp.
\]
Hence $[\mathcal M_{16},Q]=0$ if and only if both mixed blocks vanish,
which is exactly the condition $\mathfrak R_C=0$.
This is equivalent to invariance of the two summands under
$\mathcal M_{16}$, i.e. block-diagonality in the stated splitting.
\end{proof}

\begin{theorem}[Closed master system]
\label{thm:closed-master-system}
The probe--closure field equations are internally closed by the system
\begin{align*}
  \bigl(i\Gamma^\mu\mathcal D_\mu - \Omega\bigr)\Psi &= 0,\\
  \mathcal D^\mu\mathcal F_{\mu\nu}
    &= J^{(\Psi)}_\nu + J^{(C_7)}_\nu + J^{(HC)}_\nu,\\
  (\mathbf 1-\hat P_{C_7})\Psi &= 0,\\
  \mathcal H_C &= \Lambda[\mathcal M_{16},\hat P_{C_7}],\\
  \mathfrak R_C &= 0.
\end{align*}
Therefore all previously free closure symbols are eliminated in favour
of explicit algebraic conditions on $(\mathcal M_{16},\hat P_{C_7})$.
\end{theorem}

\begin{corollary}[Vanishing compensator in the exact sector]
If the closure condition $\mathfrak R_C=0$ holds exactly, then
\[
  [\mathcal M_{16},\hat P_{C_7}] = 0
  \qquad\text{and hence}\qquad
  \mathcal H_C = 0
\]
for the minimal compensator ansatz above.
Thus the Heisenberg compensator is nonzero only away from exact closure,
where it measures and suppresses leakage out of the stable sector.
\end{corollary}

\begin{remark}[Interpretation of the closure step]
The previous section produced a coupled field architecture.
The present section closes it in the minimal algebraic sense:
physical states stay in the $C_7$-sector, the Millennium matrix may not
mix stable and unstable blocks, and the compensator becomes an explicit
positive residue functional rather than a symbolic placeholder.
This does not yet derive gravitation or the Standard Model, but it does
remove the main formal gap inside the probe sector.
\end{remark}


\subsection*{Physical interpretation of the closure residue}

\begin{remark}[Leakage, confinement, and stability]
The residue $\mathfrak R_C$ has a direct physical reading.
Its mixed blocks measure transitions between the stable $C_7$-sector and
its complement.
Hence $\mathfrak R_C\neq 0$ represents \,leakage\, of probe amplitude out
of the metastably confined sector, while $\mathfrak R_C=0$ expresses exact
sector confinement.
In this sense the closure condition is the algebraic analogue of a
superselection rule: admissible long-lived states are precisely those for
which the effective internal dynamics generated by $\mathcal M_{16}$ never
pushes the state out of the projected septum sector.
\end{remark}

\begin{remark}[Role of the compensator]
For the minimal ansatz,
\[
  \mathcal H_C
  = \rho\,[\mathcal M_{16},\hat P_{C_7}]^\dagger[\mathcal M_{16},\hat P_{C_7}]
    + \sigma\,\mathfrak R_C^\dagger\mathfrak R_C,
\]
the compensator is not an independent interaction but an energetic penalty
for sector mixing.
Its effect is twofold:
first, it raises the cost of off-sector propagation; second, it drives the
variation toward block preservation of $\mathcal M_{16}$.
Thus the compensator plays the role of a restoring term that suppresses
unstable excursions and selects dynamically persistent probe states.
\end{remark}

\begin{remark}[Physical reading of exact closure]
In the exactly closed regime the splitting
\[
  \mathcal H_{\mathrm{sept}}
  = \hat P_{C_7}\mathcal H_{\mathrm{sept}}
  \oplus
  (\mathbf 1-\hat P_{C_7})\mathcal H_{\mathrm{sept}}
\]
is preserved by the internal generator $\mathcal M_{16}$.
Then the effective dynamics on the physical sector becomes autonomous.
Operationally this means that all observables built from $\Psi$ may be
computed within the projected sector without referring to hidden leakage
channels.
The closed system is therefore the minimal regime in which the probe theory
admits a stable phenomenological interpretation.
\end{remark}

\subsection*{Controlled limiting regimes}

\begin{proposition}[Dirac limit]
\label{prop:dirac-limit}
Assume the exact closure condition $\mathfrak R_C=0$, freeze the connection
background to a locally trivial configuration $\mathcal A_\mu=0$, and take
$\Omega$ to reduce on the closed sector to a scalar effective mass,
\[
  \Omega\big|_{\hat P_{C_7}\mathcal H}
  = m_{\mathrm{eff}}\,\mathbf 1.
\]
Then the master equation reduces on the physical sector to
\[
  (i\Gamma^\mu\partial_\mu - m_{\mathrm{eff}})\Psi = 0,
\]
which is the ordinary free Dirac form up to the choice of the effective
gamma representation.
\end{proposition}

\begin{proof}
Under exact closure the constraint $(\mathbf 1-\hat P_{C_7})\Psi=0$ removes
off-sector components, and for the minimal ansatz the compensator vanishes.
With $\mathcal A_\mu=0$ and scalar reduction of $\Omega$, the covariant
operator collapses to $\partial_\mu$, so the master system becomes the free
Dirac equation on the projected sector.
\end{proof}

\begin{proposition}[Yang--Mills type limit]
\label{prop:yang-mills-limit}
Assume again exact closure, but retain a nontrivial connection
$\mathcal A_\mu$ acting internally on the closed sector.
If the projected operator algebra generated by $\mathcal A_\mu$ closes on a
finite-dimensional Lie algebra $\mathfrak g$, then the resonance field
equation
\[
  \mathcal D^\mu \mathcal F_{\mu\nu}=J_\nu
\]
reduces on the physical sector to a Yang--Mills type equation with gauge
algebra $\mathfrak g$.
\end{proposition}

\begin{proof}
Once $\hat P_{C_7}$ is preserved by $\mathcal M_{16}$, the connection and its
curvature may be restricted consistently to the closed sector.
If the projected commutator algebra is Lie-closed, then the curvature takes
the standard form of a gauge field strength on $\mathfrak g$, and the field
equation is exactly of Yang--Mills type with source $J_\nu$.
\end{proof}

\begin{proposition}[Einstein-type effective limit]
\label{prop:einstein-effective-limit}
Suppose the projected connection separates into an external spacetime part
and an internal resonance part,
\[
  \mathcal D_\mu = \nabla_\mu^{(g)} + \mathbb A_\mu,
\]
where $\nabla_\mu^{(g)}$ is compatible with a coarse-grained metric
$g_{\mu\nu}$ extracted from bilinear probe observables.
If the effective action is integrated over fast internal modes and the
leading derivative expansion is local, then the bosonic sector has the
generic low-energy form
\[
  S_{\mathrm{eff}}[g,\mathbb A]
  = \int d^4x\sqrt{|g|}\,
    \bigl(c_0 + c_1 R(g) + c_2\operatorname{Tr}(\mathbb F_{\mu\nu}\mathbb F^{\mu\nu}) + \cdots\bigr).
\]
Hence the metric sector obeys Einstein-type equations at leading order.
\end{proposition}

\begin{remark}[Status of the Einstein limit]
This is presently a controlled effective-theory statement, not yet a first-
principles derivation.
What closure provides is the prerequisite for such a limit: a dynamically
stable projected sector, a well-defined connection, and a variationally
closed action.
To obtain actual Einstein equations one still has to specify how the metric
is reconstructed from probe bilinears and how the internal modes are coarse-
grained.
\end{remark}


\subsection*{A concrete route to the Einstein limit}

We now make the effective Einstein-type limit more explicit at the level of
field reconstruction. Let $\Psi_\triangle=(\Psi_1,\Psi_2,\Psi_\perp)^T$ be the
closed triadic probe field and let the exact closure condition
$\mathfrak R_C=0$ hold. On the physical sector we define the bilinear one-forms
\[
  E^a{}_{\mu}
  := \operatorname{Re}\bigl(\bar\Psi_\triangle \,\Gamma^a \, \mathcal D_\mu
  \Psi_\triangle\bigr),
  \qquad a=0,1,2,3,
\]
where $\Gamma^a$ denotes a fixed Clifford frame acting on the projected probe
space. Whenever the $4\times 4$ matrix $E^a{}_{\mu}$ has maximal rank, these
bilinears define an effective coframe and hence an effective metric
\[
  g_{\mu\nu}
  := \eta_{ab} E^a{}_{\mu} E^b{}_{\nu},
\]
with $\eta_{ab}=\operatorname{diag}(-1,1,1,1)$.

The closure constraint is essential here: it ensures that the bilinears are
computed entirely within the invariant $C_7$-sector, so the reconstructed
metric is not contaminated by leakage terms. In this sense $g_{\mu\nu}$ is not
introduced externally but induced from the closed probe dynamics.

Next, restrict the covariant operator to the physical sector,
\[
  \mathcal D^{(C)}_\mu := \hat P_{C_7}\mathcal D_\mu \hat P_{C_7}.
\]
Assume that $\mathcal D^{(C)}_\mu$ admits a decomposition into a geometric and an
internal part,
\[
  \mathcal D^{(C)}_\mu
  = \partial_\mu + \frac14 \, \omega_{\mu ab}\,\Sigma^{ab} + \mathbb A_\mu,
\]
where $\Sigma^{ab}=\frac12[\Gamma^a,\Gamma^b]$ and $\omega_{\mu ab}$ is chosen so
that the induced connection is metric-compatible,
\[
  \nabla^{(g)}_\lambda g_{\mu\nu}=0.
\]
If in addition the torsion-free condition is imposed at the coarse-grained
level, then $\omega_{\mu ab}$ coincides with the Levi--Civita spin connection of
$g_{\mu\nu}$.

This yields the following sharpened version of the Einstein limit.

\begin{proposition}[Bilinear reconstruction of the Einstein-type limit]
\label{prop:bilinear-einstein-reconstruction}
Assume exact closure $\mathfrak R_C=0$, maximal rank of the bilinear coframe
$E^a{}_{\mu}$, and a local derivative expansion of the closed-sector effective
action. Suppose moreover that the coarse-grained bosonic action takes the form
\[
  S_{\mathrm{eff}}[g,\mathbb A,\Phi]
  = \int d^4x\sqrt{|g|}\,\bigl(
      \Lambda_{\mathrm{eff}}
      + \frac{1}{2\kappa_{\mathrm{eff}}}R(g)
      - \frac14\operatorname{Tr}(\mathbb F_{\mu\nu}\mathbb F^{\mu\nu})
      + \mathcal L_{\mathrm{cl}}(\Phi,g)
    \bigr),
\]
where $\Phi$ denotes the remaining closure-sector collective fields.
Then variation with respect to $g^{\mu\nu}$ yields the effective equation
\[
  G_{\mu\nu}(g) + \Lambda_{\mathrm{eff}}g_{\mu\nu}
  = \kappa_{\mathrm{eff}}
    \bigl(T^{(\mathbb A)}_{\mu\nu} + T^{(\Phi)}_{\mu\nu}\bigr),
\]
with the standard stress-energy tensors obtained from the gauge and collective
sectors. In particular, the Einstein tensor arises from the closed probe
system once the metric is reconstructed from bilinears and the internal modes
are integrated out.
\end{proposition}

\begin{proof}
Under exact closure the projected operator $\mathcal D^{(C)}_\mu$ acts
autonomously on the physical sector. Maximal rank of $E^a{}_{\mu}$ turns the
probe bilinears into a nondegenerate coframe, hence into a metric
$g_{\mu\nu}$. Metric compatibility and vanishing torsion identify the geometric
part of $\mathcal D^{(C)}_\mu$ with the Levi--Civita spin connection.
By assumption, coarse-graining over the fast internal modes produces a local
effective action whose leading geometric term is the Einstein--Hilbert term
$R(g)$. Varying $S_{\mathrm{eff}}$ with respect to $g^{\mu\nu}$ gives the Einstein
tensor on the left-hand side and the stress-energy tensors of the remaining
fields on the right-hand side, which is the claimed Einstein-type equation.
\end{proof}

\begin{remark}[What is still model-dependent]
The above construction isolates the precise points where additional work is
still required.
First, one must justify the nondegeneracy of the bilinear coframe from the
triadic probe dynamics rather than postulate it.
Second, one has to derive the constants $\Lambda_{\mathrm{eff}}$ and
$\kappa_{\mathrm{eff}}$ from the microscopic closure model.
Third, the collective closure fields $\Phi$ have to be identified explicitly;
they encode the backreaction of the residual probe, projector, and compensator
degrees of freedom on spacetime geometry.
What is gained already is structural: the Einstein sector now appears by a
clear reconstruction pipeline
\[
  (\Psi_\triangle,\hat P_{C_7},\mathcal H_C,\mathcal M_{16})
  \longrightarrow E^a{}_{\mu}
  \longrightarrow g_{\mu\nu}
  \longrightarrow S_{\mathrm{eff}}[g,\mathbb A,\Phi]
  \longrightarrow G_{\mu\nu}=\kappa_{\mathrm{eff}}T_{\mu\nu}^{\mathrm{eff}}.
\]
\end{remark}

\begin{remark}[Interpretive summary]
The probe--closure system now supports a three-level reading.
At the exact algebraic level, closure means vanishing leakage between the
physical $C_7$-sector and its complement.
At the dynamical level, the compensator suppresses deviations from that
sector.
At the phenomenological level, ordinary Dirac and Yang--Mills equations
appear as closed-sector limits, while Einstein dynamics can arise as a
further coarse-grained low-energy description if a metric reconstruction is
available.
\end{remark}


%% ============================================================



\section*{v1.7.0 Closure--Projection Synthesis Core}
\addcontentsline{toc}{section}{v1.7.0 Closure--Projection Synthesis Core}

\begin{remark}[Normalization note for v1.7.0]
\label{rem:normalization-note-v170}
This synthesis block is retained as a historical transport precursor. Its closure--projection synthesis is now carried proof-theoretically by Phase~II-D, Phase~II-E, and Phase~II-F; here it should be read as an earlier integrative summary rather than as an independent axiomatic center.
\end{remark}

This release adds a further synthesis layer between the closure blocks already
introduced and the inherited structural core of the monolith. Its purpose is to
state, in one explicit place, how closure statements are transported across the
probe sector, how projection claims inherit their admissibility status from the
closure architecture, and how appendix tables are to be used as reversible proof
interfaces rather than as detached summaries.

\subsection*{1. Closure-to-projection transport discipline}
The central working rule of v1.7.0 is that no projection statement may be
read as primary if its admissibility has not first been transported through the
closure chain. In other words, the order of explanation remains
\[
\text{local structure} \Longrightarrow \text{HC} \Longrightarrow \text{closure}
\Longrightarrow \text{probe-compatible reduction} \Longrightarrow \text{projection readout}.
\]
A projection is therefore not an independent object but a compressed exposure of
closure-stable content under a chosen observational or effective frame. This
rule keeps the paper aligned with the structure-first programme and prevents
later phenomenological language from silently replacing the closure mechanism
that generated it.

\paragraph{Synthesis Proposition A (closure status survives transport).}
If a statement has been established in the closure regime and is moved to a
probe-compatible or projective representation by a transport that preserves the
common body, then its structural status is not lost. What changes is the mode of
appearance, not the underlying admissibility class.

\paragraph{Proof sketch.}
The common body is fixed by the ground condition and stabilized by HC. Any
transport that preserves this body may change coordinates, phase labels, or
sector language, but it does not create a new ontology. Hence closure status is
inherited through the transport.

\subsection*{2. Probe reduction as a controlled interface}
The probe sector is not only a search device; in v1.7.0 it is also an
interface theorem. Two probes moving against one another, each carrying triolic
organization and coupled to an orthogonal third direction, define a reduced test
geometry in which admissible transitions can be filtered before full projection.
This means that the probe sector is a disciplined compression stage: it narrows
candidate structure while still remaining within closure control.

\paragraph{Synthesis Proposition B (probe reduction preserves closure discipline).}
Whenever the probe sector is constructed from triolic substructures under the
standing orthogonality and HC rules, the reduced search geometry cannot produce
admissible outputs that violate closure. It may discard candidates, rank them,
or expose only a subset of modes, but it cannot certify a structurally illegal
sector.

\paragraph{Proof sketch.}
The probe geometry is not external to the closure architecture. It is built from
substructures already governed by HC and orthogonality. Therefore any reduction
performed inside the probe channel is a reduction of representation, not a
release from confinement.

\subsection*{3. Projection tables as reversible proof interfaces}
The appendix tables introduced in the previous versions are to be read neither
as cosmetic summaries nor as final stand-alone derivations. Their exact role is
reversible: each table entry points upward to the closure and lemma chain from
which it is licensed, and downward to the sector language in which it may be
used. In this sense, a table row functions as a compact proof interface.

\paragraph{Synthesis Principle (table row admissibility).}
A table row is admissible only if one can move in both directions:
\begin{enumerate}[leftmargin=2em,label=(\alph*)]
  \item upward, from the row to the closure/lemma chain that stabilizes it; and
  \item downward, from the row to a projective or phenomenological reading that
  does not exceed the transported status of the original statement.
\end{enumerate}
If either direction is unavailable, the row is only heuristic and must be marked
as such.

\subsection*{4. Micro-to-macro synthesis route}
The micro-to-macro route is now stated in a more disciplined form. The local
triolic unit is not yet a macroscopic law, but it already fixes the admissible
coupling pattern. HC then stabilizes multiple appearances of the same body,
closure confines the admissible dynamics, the probe sector performs controlled
reduction, and only then do effective macroscopic equations appear as readouts.
The macro level is therefore a closure-saturated exposure of the micro level,
not a rival explanatory layer.

\paragraph{Synthesis Proposition C (macro equations are closure-saturated readouts).}
Under the standing assumptions of the monolith, any effective macroscopic
relation used later in the paper must be readable as a closure-saturated image
of the local structural chain. Hence macro equations are legitimate exactly to
the degree that the closure-to-projection transport remains traceable.

\subsection*{5. Appendix back-reference discipline, strengthened}
The appendix is now explicitly promoted to a back-reference layer. This means
that every compressed proof move used in the running text should be recoverable
through one of three routes: a lemma reference, a tool reference, or a table
reference with reversible upward justification. The appendix is therefore not an
archive placed after the fact; it is the controlled external memory of the proof
architecture.

\paragraph{Operational rule for later versions.}
Whenever a derivation step is abbreviated in the main line, the abbreviated step
should be tagged by a reusable interface name that can be expanded again from
the appendix without changing the logical order of the argument. This preserves
speed without sacrificing structural auditability.


\section*{v1.8.0 Structural Interface Consolidation Core}
\addcontentsline{toc}{section}{v1.8.0 Structural Interface Consolidation Core}

\begin{remark}[Normalization note for v1.8.0]
\label{rem:normalization-note-v180}
This interface-consolidation block is retained as historical interface memory. Its active proof-bearing content is now normalized into the semantic/tensor interface and grouped transport phases, so the present block should be read as supporting consolidation rather than as a competing interface foundation.
\end{remark}

This release adds a further consolidation layer between the inherited closure--projection
synthesis core and the longer structural body of the monolith. The aim of
v1.8.0 is to make the interfaces between closure statements, probe reductions,
projection tables, and appendix recoverability even more explicit, so that later
mathematical deepening can reuse the same transport rules without reopening the
basic admissibility logic.

\subsection*{1. Interface invariance principle}
The first strengthening of v1.8.0 is that every admissible transition between
main text, probe representation, and appendix table must preserve the same
structural body. This means that the interface is not merely notational. It is a
controlled transfer channel whose legitimacy depends on invariance of the common
closure-supported content.

\paragraph{Consolidation Proposition A (interface invariance).}
If two representations are linked by a transport that preserves the closure-fixed
body, then they are distinct expositions of the same admissible statement rather
than competing structural claims.

\paragraph{Proof sketch.}
The ground condition and HC fix the admissible body before any later compression.
A transport that keeps this body intact may change display, resolution, or
sector language, but it cannot alter the admissibility source. Hence interface
moves preserve statement identity at the structural level.

\subsection*{2. Probe filtration as a pre-projective gate}
The probe sector is now stated even more sharply as a pre-projective gate. Its
job is not to invent physical outputs but to filter, compress, and expose only
those modes that remain compatible with the closure discipline already fixed in
the local structural chain.

\paragraph{Consolidation Proposition B (probe filtration is gatekeeping, not generation).}
Under the standing orthogonality and HC assumptions, the probe sector may rank or
suppress candidate modes, but it cannot generate a physically admissible mode
whose closure ancestry is absent.

\paragraph{Proof sketch.}
The probes inherit their admissibility from the same triolic and closure-governed
body as the main sector. Therefore filtration can only reveal or reject what is
already licensed; it cannot certify an unlicensed sector by reduction alone.

\subsection*{3. Table-to-lemma reversibility}
The tables and tool blocks gathered in the appendix are promoted once more from
summary devices to reversible proof interfaces. For v1.8.0 this reversibility
is sharpened into a two-way requirement between rows, lemmas, and transport
principles.

\paragraph{Consolidation Principle (row/lemma reversibility).}
A compact row or tool entry is fully admissible only if the reader can reconstruct
both
\begin{enumerate}[leftmargin=2em,label=(\alph*)]
  \item the lemma or closure chain that licenses the row from above, and
  \item the bounded projection or interpretation that uses the row from below.
\end{enumerate}
Where only one direction is available, the entry remains heuristic and must be
marked as such.

\subsection*{4. Strengthened micro-to-macro readout discipline}
The micro-to-macro route is no longer only a synthesis slogan. In v1.8.0 it is
a regulated readout pipeline:
\[
\text{local triolic body} \Longrightarrow \text{HC stabilization} \Longrightarrow
\text{closure confinement} \Longrightarrow \text{probe filtration} \Longrightarrow
\text{table/interface compression} \Longrightarrow \text{effective projection}.
\]
This ordering is now part of the operational discipline of the document. A macro
statement is acceptable only to the extent that its position in this chain can be
traced backward without gap.

\paragraph{Consolidation Proposition C (macro readout requires backward traceability).}
Any effective large-scale law cited later in the monolith is structurally valid only
when it admits a backward trace through table/interface compression, probe
filtration, closure confinement, and HC stabilization to the local triolic body.

\subsection*{5. Appendix memory discipline, strengthened again}
The appendix is therefore not only a storage area for auxiliary derivations. It is
a controlled external memory for the proof architecture. In v1.8.0 every major
abbreviation introduced in the running text should be recoverable by a named
interface path: lemma path, tool path, table path, or transport path. This rule
prepares the next versions for denser mathematical unfolding without losing the
structure-first audit trail.

\paragraph{Operational rule for later versions.}
When a proof step is compressed in the main line, the compression should carry a
stable interface name whose recovery route is unique up to equivalent structural
transport. This keeps the monolith fast in presentation while remaining expandable
under later formal pressure.

\section{Structural Core and Closure Principles}
\label{sec:structural-core}
%% ============================================================

Before compressing the preceding results into a single ToE master equation, we
make explicit the structural kernel that has been implicit throughout the
construction.

\begin{definition}[Structural core]
\label{def:structural-core}
The structural core of the theory consists of the following coupled
requirements:
\begin{enumerate}
\item a common underlying field object $\Psi$;
\item a triolic decomposition
\[
  \Psi = \Psi_1\oplus\Psi_2\oplus\Psi_3;
\]
\item an orthogonal closure/light sector $\Psi_3$ relative to the first two
sectors;
\item a dual unitary symmetry exchanging $\Psi_1$ and $\Psi_2$;
\item a Heisenberg compensator $C$ emerging from the ground condition and
selecting the collapse/light output;
\item a projected closure condition determining the physically admissible
sector;
\item a phase-shifted observer frame in which effective quantities are read
out.
\end{enumerate}
\end{definition}

\begin{theorem}[Existence of a nontrivial closure-stable triolic system]
\label{thm:structural-existence}
Assume the structural core of Definition~\ref{def:structural-core}. Then there
exists a nontrivial class of configurations in which the triolic decomposition,
compensator consistency, and projected closure condition are simultaneously
satisfied.
\end{theorem}

\begin{proof}
The preceding sections already exhibit the required ingredients separately.
The central theorem provides a common ground object together with its triolic
unfolding and Lorentz-compatible strata. The unitary symmetry $J$ links the
first two sectors, while the Heisenberg compensator produces the
collapse/light sector as a structurally distinguished output. The subsequent
probe--closure analysis introduces the projected closure residue and identifies
an admissible closed sector by the condition that leakage into the complement
is suppressed. Since these ingredients are realized within one and the same
construction, the intersection of the triolic, compensator-consistent, and
closure-stable constraints is nonempty. Hence a nontrivial closure-stable
triolic system exists.
\end{proof}

\begin{proposition}[Centrality of the Heisenberg compensator]
\label{prop:hc-centrality}
Within the structural core, the Heisenberg compensator is not an auxiliary
correction term but the central structural hinge linking duality, collapse,
closure stability, and the phase-shifted observer description.
\end{proposition}

\begin{proof}
Without the duality symmetry, the first two sectors would remain merely
parallel descriptions. Without the compensator, their common collapse/light
output would not be selected. Without this selected output, the projected
closure condition would not isolate a physically admissible sector. And
without that closure-stable sector, the observer-frame description would lack a
well-defined quantity to read out after phase shift. Thus the compensator is
precisely the operator through which duality becomes collapse, collapse becomes
closure, and closure becomes observable effective physics.
\end{proof}


\subsection*{Preliminary Semantic and Tensorial Kernel Interface}
\addcontentsline{toc}{subsection}{Preliminary Semantic and Tensorial Kernel Interface}

This subsection states the semantic/tensor interface in anticipatory form. Its formal unfolding as a lemma--theorem chain is given later in Phase II-E.

The structural kernel can be stated on two coordinated levels. The first is a
semantic layer of sets, states, maps, and admissibility predicates. The second
is a tensorial realization layer in which the same constraints are written in a
notation compatible with the Riemann/Sphaera core. The tensor layer does not
replace the semantic one; it compresses it into a transportable operator form.

\begin{definition}[Semantic structural kernel]
\label{def:semantic-structural-kernel}
The semantic structural kernel consists of the data
\[
(\mathbb S,\{S_n\}_{n\ge 0},S_0,\hat K,J,C,\mathcal M)
\]
together with the admissibility predicates
\[
\mathrm{Triolic}(x),\ \mathrm{Anchored}(x),\ \mathrm{FrameShifted}(x),\
\mathrm{GroundLinked}(x),\ \mathrm{HCForced}(x),\ \mathrm{ClosureStable}(x),\
\mathrm{HigherConfined}(x),\ \mathrm{SeptinionReadable}(x).
\]
These objects satisfy the following semantic conditions:
\begin{enumerate}[leftmargin=2em,label=(S\arabic*)]
\item \textbf{Cascade hierarchy:}
\[
\mathbb S=S_N\supseteq S_{N-1}\supseteq \cdots \supseteq S_1\supseteq S_0,
\qquad
\hat K=\Pi_0\circ\Pi_1\circ\cdots\circ\Pi_{N-1}.
\]
\item \textbf{Ground-state terminality:}
\[
x\in S_0 \Longleftrightarrow \hat K(x)=x.
\]
\item \textbf{Maximal-totality closure rule:} if $M^*\cap \mathbb S=\emptyset$, then
\[
\hat K(M^*)\subseteq S_0.
\]
\item \textbf{$J$-involution:}
\[
J^2=\mathrm{id}.
\]
\item \textbf{Compensator map:}
\[
C(x):=\tfrac12\bigl(x+J(x)\bigr).
\]
\item \textbf{Triolic minimality:} if $\mathrm{Triolic}(x)$, then $x$ admits a three-sector
organization $(K_1,K_2,K_3)$.
\item \textbf{Orthogonal anchor:} if $\mathrm{Anchored}(x)$, then the third sector is
orthogonal to the first two,
\[
K_3\perp K_1,
\qquad
K_3\perp K_2.
\]
\item \textbf{Frame-shift principle:} $\mathrm{FrameShifted}(x)$ means that $x$ is a shifted
or regime-dependent readout of a common structural body rather than an
ontologically independent object.
\end{enumerate}
\end{definition}

\begin{definition}[Tensor realization of the structural kernel]
\label{def:tensor-structural-kernel}
A tensor realization of Definition~\ref{def:semantic-structural-kernel}
is given by the operator family
\[
\mathcal J^\mu{}_{\nu},\qquad
\mathcal C^\mu{}_{\nu},\qquad
\Pi_n{}^\mu{}_{\nu},\qquad
\hat K^\mu{}_{\nu},\qquad
\mathcal M^\mu{}_{\nu},
\]
subject to the following identities:
\begin{enumerate}[label=(T\arabic*)]
  \item \textbf{Involution:}
    $\mathcal J^\mu{}_{\alpha}\mathcal J^\alpha{}_{\nu}=\delta^\mu{}_{\nu}$.
  \item \textbf{Compensator idempotence:}
    $\mathcal C^\mu{}_{\alpha}\mathcal C^\alpha{}_{\nu}=\mathcal C^\mu{}_{\nu}$.
  \item \textbf{Cascade composition:}
    $\hat K^\mu{}_{\nu}=(\Pi_0\Pi_1\cdots\Pi_{N-1})^\mu{}_{\nu}$.
  \item \textbf{$\mathcal M$-admissibility (intertwining):}
    The Millennium matrix $\mathcal M^\mu{}_{\nu}$ is required to be
    \emph{admissible} in the sense that it preserves the algebraic
    structure of the kernel:
    \[
      \mathcal M^\mu{}_{\alpha}\mathcal J^\alpha{}_{\nu}
      =\mathcal J^\mu{}_{\alpha}\mathcal M^\alpha{}_{\nu},
      \qquad
      \mathcal M^\mu{}_{\alpha}\mathcal C^\alpha{}_{\nu}
      =\mathcal C^\mu{}_{\alpha}\mathcal M^\alpha{}_{\nu}.
    \]
    Whenever layer transport is required to preserve the cascade reading,
    additionally
    \[
      \mathcal M^\mu{}_{\alpha}\hat K^\alpha{}_{\nu}
      =\hat K^\mu{}_{\alpha}\mathcal M^\alpha{}_{\nu}.
    \]
\end{enumerate}
The tensor $\mathcal J$ represents the involutive dual/frame symmetry,
$\mathcal C$ represents the compensator projection, the $\Pi_n$ represent
cascade projections between adjacent strata, and $\mathcal M$ is the
Millennium matrix as an admissible structure-preserving converter between
admissible layers. Condition~(T4) is the defining admissibility condition
on $\mathcal M$; it is not derivable from (T1)--(T3) alone, but is imposed
as the minimal requirement for $\mathcal M$ to qualify as a
structure-preserving converter in the sense of
Definition~\ref{def:semantic-structural-kernel}.
\end{definition}

\begin{proposition}[Semantic forcing chain]
\label{prop:semantic-forcing-chain}
Within the semantic structural kernel, the following implication chain holds:
\[
\mathrm{GroundLinked}(x)\wedge \mathrm{FrameShifted}(x)
\Longrightarrow \mathrm{HCForced}(x),
\]
\[
\mathrm{Triolic}(x)\wedge \mathrm{Anchored}(x)\wedge \mathrm{HCForced}(x)
\Longrightarrow \mathrm{ClosureStable}(x),
\]
\[
\mathrm{ClosureStable}(x)\wedge \mathrm{HigherConfined}(x)
\Longrightarrow \mathrm{SeptinionReadable}(x).
\]
\end{proposition}

\begin{proof}
\textbf{First implication:
$\mathrm{GroundLinked}(x)\wedge\mathrm{FrameShifted}(x)
\Rightarrow\mathrm{HCForced}(x)$.}

$\mathrm{GroundLinked}(x)$ means $x\in S_n$ for some stratum $n$; it is
reachable by the cascade $\hat{K}$ from Definition~\ref{def:semantic-structural-kernel}~(S1).
$\mathrm{FrameShifted}(x)$ means by~(S8) that $x$ is a shifted or
regime-dependent readout of a common structural body.
By Lemma~\ref{lem:one-body-lemma}, the shifted presentations all refer
to one and the same carrier. By Lemma~\ref{lem:re-identification-lemma},
admissibility then requires a re-identification mechanism returning
competing presentations to a common output channel. The only operator
in the semantic kernel with this role is the compensator $C(x)=\tfrac{1}{2}(x+J(x))$
from~(S5), which is the unique $J$-symmetric projection.
Hence $\mathrm{HCForced}(x)$: the compensator is structurally necessary.

\smallskip
\textbf{Second implication:
$\mathrm{Triolic}(x)\wedge\mathrm{Anchored}(x)\wedge\mathrm{HCForced}(x)
\Rightarrow\mathrm{ClosureStable}(x)$.}

This is Lemma~\ref{lem:closure-forcing-lemma}, now with an explicit
four-step proof citing
Lemmas~\ref{lem:triolic-minimality-lemma},~\ref{lem:orthogonal-anchor-lemma},
\ref{lem:hc-admissibility-lemma},~\ref{lem:hc-idempotence-lemma},
and Theorem~\ref{thm:fractal}.

\smallskip
\textbf{Third implication:
$\mathrm{ClosureStable}(x)\wedge\mathrm{HigherConfined}(x)
\Rightarrow\mathrm{SeptinionReadable}(x)$.}

$\mathrm{ClosureStable}(x)$ means $\hat{K}(\mathcal{C}x)=\mathcal{C}x$
(Lemma~\ref{lem:closure-stable-iff-im-C}):
$x\in\mathrm{im}(C)$ is already in the ground sector.
$\mathrm{HigherConfined}(x)$ means $x$ is contained in a higher coupled
algebraic container — specifically the septinion/seption embedding
(Lemma~\ref{lem:higher-confinement-lemma}).
Within the higher container, the closure-stable state is readable as a
seption element precisely because its compensator-stabilized image
$\mathcal{C}x$ already lies in the admissible ground sector $S_0$, and
the higher container adds no new ontology
(Lemma~\ref{lem:non-escape-lemma}: no admissible transition exports $x$
into an independent external sector).
Hence the higher-confinement reading is a restatement of the already
established closure at higher algebraic level:
$\mathrm{SeptinionReadable}(x)$.
\end{proof}

\begin{proposition}[Tensorial intertwining conditions for the Millennium matrix]
\label{prop:tensorial-intertwining-m}
Every admissible tensor realization in the sense of
Definition~\ref{def:tensor-structural-kernel} satisfies:
\[
\mathcal M\mathcal J=\mathcal J\mathcal M,
\qquad
\mathcal M\mathcal C=\mathcal C\mathcal M,
\]
and, whenever layer transport is required to preserve the cascade reading,
\[
\mathcal M\hat K=\hat K\mathcal M.
\]
\end{proposition}

\begin{proof}
These are precisely condition~(T4) of
Definition~\ref{def:tensor-structural-kernel}, written in operator
notation. A tensor realization is admissible by definition only if
$\mathcal M$ satisfies the intertwining identities~(T4); hence the
relations hold for every admissible realization. No additional argument
is required beyond the definition.

The structural motivation for imposing~(T4) is as follows. The first
relation $\mathcal M\mathcal J=\mathcal J\mathcal M$ ensures that
transport by $\mathcal M$ does not destroy the involutive dual/frame
symmetry encoded by $\mathcal J$~(T1). The second relation
$\mathcal M\mathcal C=\mathcal C\mathcal M$ ensures that the
compensator-selected admissible sector $\mathrm{im}(\mathcal C)$ is
$\mathcal M$-invariant, consistent with $\mathcal C$ being idempotent
(T2). The third relation, when imposed, says that layer conversion
commutes with the cascade readout~(T3). These three motivations
explain why~(T4) is the correct admissibility condition; they do not
constitute an independent proof of~(T4) from (T1)--(T3).
\end{proof}

\begin{remark}[Resolution of Roadmap item A2]
\label{rem:a2-resolved}
Roadmap item~A2 identified that the intertwining relation
$\mathcal M\mathcal C=\mathcal C\mathcal M$ was supported only by a
necessity argument rather than a derivation. The resolution adopted
in v2.27.6 is to promote~(T4) to an explicit definitional condition
on admissible tensor realizations
(Definition~\ref{def:tensor-structural-kernel}, condition~(T4)).
Proposition~\ref{prop:tensorial-intertwining-m} then follows by
definition. The conditional marker on
Theorem~\ref{thm:hc-spectral-projector}
(Remark~\ref{rem:hc-spectral-projector-conditional}) is therefore
resolved: the theorem is no longer conditional on an unproven
hypothesis, but on an explicit definitional requirement.
\end{remark}

\begin{remark}[Semantic--tensor correspondence]
\label{rem:semantic-tensor-correspondence}
The semantic predicates and the tensor notation are intended to be read in
parallel. In particular,
\[
\mathrm{HCForced}(x)
\quad\leftrightarrow\quad
x\mapsto \mathcal Cx,
\]
\[
\mathrm{ClosureStable}(x)
\quad\leftrightarrow\quad
\hat K(\mathcal Cx)=\mathcal Cx,
\]
and
\[
\mathrm{Admissible}(x)
\quad\Rightarrow\quad
\mathrm{Admissible}(\mathcal Mx)
\]
whenever $\mathcal M$ satisfies the intertwining conditions of
Proposition~\ref{prop:tensorial-intertwining-m}. This makes it possible to
write later structure conditions in the same tensorial style as the
Riemann/Sphaera core while keeping their semantic meaning explicit.
\end{remark}


\subsection*{Phase I: First Unfolding Lemmas for the Generative Layer}
\addcontentsline{toc}{subsection}{Phase I: First Unfolding Lemmas for the Generative Layer}

We now begin the actual mathematical unfolding of the normalized structural
core by isolating the first chain of generative lemmas. The aim is not yet to
expand the full field/probe realization, but to make explicit the initial
forcing route from a common structural body to triolic organization,
orthogonal anchoring, frame-shifted readout, re-identification, and finally the
necessity of the Heisenberg compensator.

\begin{lemma}[One-body lemma]
\label{lem:one-body-lemma}
The semantic and tensorial kernels are to be read as describing different
admissible presentations of one and the same structural body, not independent
ontological objects. In particular, the cascade hierarchy, dual/frame symmetry,
and compensator projection all act on a common carrier rather than on unrelated
state spaces.
\end{lemma}

\begin{proof}
In Definition~\ref{def:semantic-structural-kernel} the kernel data are given as
one coordinated family
\[
(\mathbb S,\{S_n\}_{n\ge 0},S_0,\hat K,J,C,\mathcal M),
\]
not as several unrelated structures. The grouped architecture of
Definition~\ref{def:grouped-structural-architecture} likewise states that the
generative, projection, and field/probe layers are different organizational
regimes of the same underlying body. The tensor realization of
Definition~\ref{def:tensor-structural-kernel} then presents the same body in
operator form. Hence the structural reading is one-body throughout.
\end{proof}

\begin{lemma}[Triolic minimality lemma]
\label{lem:triolic-minimality-lemma}
A merely binary decomposition is structurally insufficient for the normalized
core. The minimal nondegenerate organization compatible with admissible
selection, compensator action, and closure readout is triolic.
\end{lemma}

\begin{proof}
By Definition~\ref{def:semantic-structural-kernel}, a triolic state carries a
three-sector organization $(K_1,K_2,K_3)$, while the orthogonal anchor singles
out the third sector as stabilizing direction relative to the first two. If one
retained only a binary split, then duality between the first two sectors could
still be stated, but there would be no distinguished third channel in which the
compensator-selected output and closure/light readout could be structurally
stabilized. The normalized core therefore requires three sectors as the minimal
nondegenerate organization.
\end{proof}

\begin{lemma}[Orthogonal anchor lemma]
\label{lem:orthogonal-anchor-lemma}
Within a triolic organization, the third sector is not an optional decoration but
the structural anchor that prevents the first two sectors from remaining a
merely parallel or relabelable pair. In particular,
\[
K_3\perp K_1,
\qquad
K_3\perp K_2.
\]
\end{lemma}

\begin{proof}
The orthogonal anchor condition is built into
Definition~\ref{def:semantic-structural-kernel}. Its role is clarified by the
forcing chain of Proposition~\ref{prop:semantic-forcing-chain}: closure
stability does not follow from duality alone, but from triolic structure plus a
protected anchor together with HC forcing. The third sector therefore fixes the
otherwise ambiguous two-sector pairing and provides the admissible direction in
which collapse/light selection can be stabilized.
\end{proof}

\begin{lemma}[Frame-shift lemma]
\label{lem:frame-shift-lemma}
Admissible effective states are read out in a phase- or frame-shifted
presentation. Accordingly, an effective state is not introduced as an
independent object but as a shifted view of the common structural body.
\end{lemma}

\begin{proof}
This is exactly the content of the frame-shift principle in
Definition~\ref{def:semantic-structural-kernel}, where
$\mathrm{FrameShifted}(x)$ means that $x$ is a shifted or regime-dependent
readout of a common body. The grouped architecture makes the same point at the
layer level: projection/translation and field/probe realizations are
organizational regimes of one underlying structure and therefore inherit rather
than replace the generative body.
\end{proof}

\begin{lemma}[Re-identification lemma]
\label{lem:re-identification-lemma}
If a state is simultaneously one-body and frame-shifted, then admissibility
requires a re-identification mechanism that returns competing presentations to a
common structural channel.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:one-body-lemma}, the relevant descriptions concern one and the
same body. By Lemma~\ref{lem:frame-shift-lemma}, these descriptions appear in
shifted or regime-dependent presentations. Without re-identification, the
shifted presentations would remain only formally related and no unique
admissible output sector could be selected. But the semantic kernel is built to
support admissible cascade readout and closure selection; hence a mechanism that
returns the shifted presentations to a common channel is structurally required.
\end{proof}

\begin{theorem}[HC-forcing theorem]
\label{thm:hc-forcing-theorem}
Within the normalized structural core, the required re-identification mechanism
is the Heisenberg compensator. Equivalently,
\[
\mathrm{GroundLinked}(x)\wedge \mathrm{FrameShifted}(x)
\Longrightarrow \mathrm{HCForced}(x).
\]
\end{theorem}

\begin{proof}
Proposition~\ref{prop:semantic-forcing-chain} already states the forcing
implication. The present point is to identify why the implication is necessary.
By Lemma~\ref{lem:re-identification-lemma}, one-body structure under shifted
readout requires a common re-identifying channel. In the tensor realization,
the only built-in operator that symmetrizes competing dual/frame descriptions
into a common admissible output is the compensator projection $\mathcal C$, and
on the semantic level this is represented by the map
\[
C(x)=\tfrac12\bigl(x+J(x)\bigr).
\]
Thus the required re-identification mechanism is not external to the kernel but
is precisely the Heisenberg compensator. Hence HC is forced by the normalized
generative conditions.
\end{proof}

\begin{corollary}[First generative forcing chain]
\label{cor:first-generative-forcing-chain}
The first unfolded forcing route of the normalized core is
\[
\text{one body}
\Longrightarrow
\text{triolic minimality}
\Longrightarrow
\text{orthogonal anchor}
\Longrightarrow
\text{frame-shifted readout}
\Longrightarrow
\text{re-identification}
\Longrightarrow
\mathrm{HC}.
\]
This chain is the entry point for the later closure and higher-confinement
unfolding.
\end{corollary}

\begin{proof}
Lemmas~\ref{lem:one-body-lemma}--\ref{lem:re-identification-lemma} establish the
successive structural requirements, and
Theorem~\ref{thm:hc-forcing-theorem} identifies their first nontrivial forced
operator consequence. The later closure and higher-confinement chain therefore
starts from this generative route.
\end{proof}


\subsection*{Phase I-B: HC and Closure Unfolding}
\addcontentsline{toc}{subsection}{Phase I-B: HC and Closure Unfolding}

We next unfold the first closure layer that begins once HC has been forced.
The aim of this block is to make precise that HC is not merely present, but
acts as an admissibility-preserving, anchor-compatible, closure-selecting
mechanism, and that exact closure can be distinguished from residual or only
partial closure.

\begin{lemma}[HC idempotence lemma]
\label{lem:hc-idempotence-lemma}
Once a state has been compensator-normalized, repeated compensator action does
not move it to a different admissible sector. Equivalently,
\[
C(C(x)) = C(x).
\]
\end{lemma}

\begin{proof}
By Definition~\ref{def:tensor-structural-kernel}, the tensorial compensator
satisfies $\mathcal C^2=\mathcal C$, and by
Definition~\ref{def:semantic-structural-kernel} the semantic compensator is the
corresponding map
\[
C(x)=\tfrac12\bigl(x+J(x)\bigr).
\]
Thus compensator action is projective rather than drifting: once a state lies
in the compensator-selected channel, further application leaves it there.
\end{proof}

\begin{lemma}[HC admissibility lemma]
\label{lem:hc-admissibility-lemma}
If $x$ is structurally admissible, then its compensator-normalized image $C(x)$
remains admissible.
\end{lemma}

\begin{proof}
The semantic/tensor kernel was introduced precisely so that the compensator is
not an external deformation but an internal normalization operator. Since
Theorem~\ref{thm:hc-forcing-theorem} identifies HC as the required
re-identification mechanism, using $C$ restores competing shifted presentations
to a common admissible channel instead of producing a foreign state. In the
tensor realization this is the admissible output selected by $\mathcal C$; on
the semantic side the same role is expressed by the preservation of the common
structural body under compensator projection. Hence admissibility is preserved.
\end{proof}

% ------------------------------------------------------------------
% Spectral-projector block (v2.27.4 integration)
% ------------------------------------------------------------------

\begin{definition}[Spectral decomposition of the state space]
\label{def:spectral-decomposition-M}
Let $\mathcal{M}$ denote the Millennium matrix acting on the admissible state
space $\mathcal{S}$. A \emph{spectral decomposition of $\mathcal{S}$ under
$\mathcal{M}$} is an orthogonal direct sum
\[
  \mathcal{S} \;=\; \bigoplus_{\lambda} \mathcal{S}_\lambda,
\]
where each $\mathcal{S}_\lambda \subset \mathcal{S}$ is a closed
$\mathcal{M}$-invariant subspace satisfying
$\mathcal{M}|_{\mathcal{S}_\lambda} = \lambda\,\mathrm{id}$.
A state $s \in \mathcal{S}$ is called \emph{spectrally stable} if
$s \in \mathcal{S}_\lambda$ for some $\lambda$, i.e.\ $\mathcal{M}(s) =
\lambda\,s$.
\end{definition}

\begin{remark}
Definition~\ref{def:spectral-decomposition-M} extends the existing
decomposition $\Hplus = \Hplusplus \oplus \Hplusminus$
(cf.\ Theorem~\ref{thm:unity}) to the full admissible readout regime.
The two primary spectral subspaces under any $\mathcal{M}$ satisfying
Proposition~\ref{prop:tensorial-intertwining-m} are precisely $\Hplusplus$
and $\Hplusminus$, since both are $\mathcal{M}$-invariant by the intertwining
condition $\mathcal{M}\mathcal{J} = \mathcal{J}\mathcal{M}$.
\end{remark}

\begin{lemma}[Diagonal self-traversal forces scalar action on the compensated sector]
\label{lem:diagonal-traversal-scalar-action}
Assume the Sphaera self-traversal condition of Theorem~\ref{thm:fractal}:
the cascade operator $\hat{K}$ transports the null axis $L_n$ of each
stratum $S_n$ to the null axis $L_{n-1}$ of the preceding stratum,
carrying the same structural content without generating new independent
directions. Then the Millennium matrix $\mathcal{M}$, when restricted to
the image of the compensator $\mathrm{im}(\mathcal{C})$, acts as a scalar
multiple of the identity:
\[
  \mathcal{M}\big|_{\mathrm{im}(\mathcal{C})} \;=\; \lambda_0\,\mathrm{id},
\]
where $\lambda_0$ is the unique admissible eigenvalue of $\mathcal{M}$ on
the closure-stable sector.
\end{lemma}

\begin{proof}
By Theorem~\ref{thm:fractal}, the self-traversal of the Sphaera is
\emph{diagonal}: the cascade does not generate a new structural object at
each stratum, but transports one and the same null-axis content through
admissible realization levels. In operator language this means that the
cascade direction at each stratum is not a Jordan-chain descendant of the
previous one---no new vector is generated from an existing vector by a
nilpotent shift.

By the intertwining condition $\mathcal{M}\mathcal{C} = \mathcal{C}\mathcal{M}$
(Proposition~\ref{prop:tensorial-intertwining-m}), $\mathrm{im}(\mathcal{C})$
is $\mathcal{M}$-invariant. Suppose for contradiction that
$\mathcal{M}|_{\mathrm{im}(\mathcal{C})}$ had a non-trivial Jordan block:
then there would exist $s,w \in \mathrm{im}(\mathcal{C})$ with $w\neq 0$ and
\[
  \mathcal{M}(s) = \lambda_0\,s + w.
\]
This would mean $\mathcal{M}$ generates $w$ from $s$---a new independent
direction within the compensated sector---contradicting the diagonal
self-traversal property. Hence $\mathcal{M}|_{\mathrm{im}(\mathcal{C})}$
is diagonalizable. Since $\mathrm{im}(\mathcal{C})$ is the closure-stable
sector (Lemma~\ref{lem:closure-forcing-lemma}), which is irreducible under
$\mathcal{M}$, a diagonalizable action on an irreducible invariant subspace
is scalar. Denoting this scalar $\lambda_0$ completes the proof.
\end{proof}

\begin{theorem}[HC as spectral projector of the Millennium matrix]
\label{thm:hc-spectral-projector}
Assume the admissible state space $\mathcal{S}$ carries a spectral
decomposition under $\mathcal{M}$ as in
Definition~\ref{def:spectral-decomposition-M}, that $\mathcal{M}$ satisfies
the intertwining conditions of Proposition~\ref{prop:tensorial-intertwining-m},
and that the Sphaera self-traversal is diagonal in the sense of
Lemma~\ref{lem:diagonal-traversal-scalar-action}. Then the Heisenberg
compensator $C$ coincides with the spectral projector of $\mathcal{M}$
onto the closure-stable eigenspace $\mathcal{S}_0 :=
\ker(\mathcal{M} - \lambda_0\,\mathrm{id})$:
\[
  C \;=\; \Pi^{(\mathcal{M})}_{\mathrm{spec}}.
\]
\end{theorem}

\begin{remark}[Status of Theorem~\ref{thm:hc-spectral-projector}]
\label{rem:hc-spectral-projector-conditional}
As of v2.27.6, both open premises of this theorem are resolved:
the bijection $\mathrm{ClosureStable}(x)\Leftrightarrow x\in\mathrm{im}(C)$
is established by Lemma~\ref{lem:closure-stable-iff-im-C} (Roadmap A3),
and the intertwining relation $\mathcal{M}\mathcal{C}=\mathcal{C}\mathcal{M}$
is now a definitional condition (T4) of
Definition~\ref{def:tensor-structural-kernel} (Roadmap A2,
Remark~\ref{rem:a2-resolved}).
The theorem therefore holds unconditionally within every admissible
tensor realization.
\end{remark}

\begin{proof}
We verify the three conditions characterizing a spectral projector.

\smallskip
\noindent\textbf{Step 1: Idempotence.}
By Lemma~\ref{lem:hc-idempotence-lemma}, $C^2=C$. By
Definition~\ref{def:tensor-structural-kernel},
$\mathcal{C}^\mu{}_\alpha\mathcal{C}^\alpha{}_\nu = \mathcal{C}^\mu{}_\nu$
in the tensor realization. Hence $C$ is a projector.

\smallskip
\noindent\textbf{Step 2: $\mathcal{M}$-compatibility.}
By Proposition~\ref{prop:tensorial-intertwining-m},
$\mathcal{M}\mathcal{C} = \mathcal{C}\mathcal{M}$. Hence $\mathrm{im}(C)$
is $\mathcal{M}$-invariant and $C$ projects onto a $\mathcal{M}$-stable
subspace.

\smallskip
\noindent\textbf{Step 3: Scalar eigenvalue on the image.}
By Lemma~\ref{lem:diagonal-traversal-scalar-action}, the diagonal
self-traversal forces $\mathcal{M}|_{\mathrm{im}(C)} = \lambda_0\,\mathrm{id}$.
Hence $\mathrm{im}(C) \subseteq \mathcal{S}_0$.
By Lemma~\ref{lem:closure-forcing-lemma}, the closure-stable states are
exactly those in $\mathrm{im}(C)$, so $\mathrm{im}(C) = \mathcal{S}_0$.

\smallskip
\noindent\textbf{Conclusion.}
A projector with $\mathcal{M}$-invariant image equal to a single eigenspace
$\mathcal{S}_0$ is the spectral projector $\Pi^{(\mathcal{M})}_{\mathrm{spec}}$
onto that eigenspace.
\end{proof}

\begin{corollary}[Structural fixpoint: $\mathcal{M}$ and $C$ mutually determine each other]
\label{cor:M-HC-fixpoint}
Under the hypotheses of Theorem~\ref{thm:hc-spectral-projector},
\[
  \mathcal{M}\,C \;=\; C\,\mathcal{M} \;=\; \lambda_0\,C.
\]
\end{corollary}

\begin{proof}
For any $s\in\mathcal{S}$: since $C(s)\in\mathcal{S}_0$, we have
$\mathcal{M}(C(s))=\lambda_0\,C(s)$, giving $\mathcal{M}\,C=\lambda_0\,C$.
For $s\in\mathcal{S}_0$: $C\,\mathcal{M}(s)=C(\lambda_0 s)=\lambda_0 C(s)$.
For $s\in\ker(C)$: $C\,\mathcal{M}(s)=0$ (since $\ker(C)$ is
$\mathcal{M}$-invariant and maps to a sector annihilated by $C$).
Hence $C\,\mathcal{M}=\lambda_0\,C$.
\end{proof}

\begin{remark}[Phase-shifted frame and spectral measurement]
\label{rem:frame-shift-spectral}
The identification $C = \Pi^{(\mathcal{M})}_{\mathrm{spec}}$ has a direct
consequence for the matter frame. Since matter observers reside at
$\zminus = +\bsym \approx 0.5493$ rather than the photon ground shell
$\zminus = 0$, every spectral readout of $\mathcal{M}$ is performed in the
shifted frame. The measured eigenvalue is therefore not $\lambda_0$ itself
but the frame-corrected value
\[
  \lambda_0^{(\mathrm{matter})}
  \;=\; \gL^{-k(\lambda_0)}\cdot\lambda_0^{(\mathrm{photon})},
\]
where $k(\lambda_0)$ is the frame-insertion count of the observable and
$\gL \approx 1.864$ is the matter--photon Lorentz factor
(Theorem~\ref{thm:frame-offset}). All physical constants derived from
spectral eigenvalues of $\mathcal{M}$ therefore carry systematic
$\gL^k$-corrections relative to their photon-frame values.
\end{remark}

\begin{corollary}[Standard Model realization of the spectral decomposition]
\label{cor:SM-spectral-realization}
The octonionic decomposition $\OO = \mathbb{C}\oplus\mathbb{C}^3$
(Theorem~\ref{thm:octonion-SU3}) is a concrete realization of
Definition~\ref{def:spectral-decomposition-M}: the singlet $\mathbb{C}$ is
the closure-stable eigenspace $\mathcal{S}_0 = \mathrm{im}(C)$, and the
triplet $\mathbb{C}^3$ carries the three color eigenspaces
$\mathcal{S}_{K_1},\mathcal{S}_{K_2},\mathcal{S}_{K_3}$.
Under this identification, Theorem~\ref{thm:hc-spectral-projector}
specializes as follows:
\begin{enumerate}[label=(\roman*)]
  \item The compensator $C$ projects onto the singlet $\mathbb{C}\subset\OO$,
  which is the photon kernel $K_3$ sector ($\zminus=0$).
  \item The triplet $\mathbb{C}^3$ is $\mathcal{M}$-invariant and carries
  the SU(3) color action; matter ($K_1$) and tachyon ($K_2$) sectors are
  spectral subspaces at non-zero frame offset.
  \item The matter-frame correction reads
  \[
    \lambda_0^{(\mathrm{matter})}
    = \gL^{-k(\lambda_0)}\cdot\lambda_0^{(\mathrm{photon})},
  \]
  consistent with Theorem~\ref{thm:SM}: $K_3$ is frame-invariant
  ($k=0$), while the triplet components carry $k\ge 1$ corrections
  and are therefore observed with $\gL^k$-suppression in the matter frame.
\end{enumerate}
\end{corollary}

\begin{proof}
By Theorem~\ref{thm:octonion-SU3}, the octonion automorphism group $G_2$
restricts to $\mathrm{SU}(3)\subset G_2$, yielding the direct sum
$\OO = \mathbb{C}\oplus\mathbb{C}^3$. The singlet $\mathbb{C}$ is the
unique $\mathrm{SU}(3)$-invariant subspace, hence $\mathcal{M}$-stable at
a scalar eigenvalue $\lambda_0$. By the remark in
Theorem~\ref{thm:SM}, $K_3 = e_1 e_2$ is simultaneously the photon kernel
and the projection of the Heisenberg compensator; hence
$\mathrm{im}(C) = \mathbb{C} = \mathcal{S}_0$. The three kernels
$(K_1,K_2,K_3)$ realize the three colors of $\mathbb{C}^3$, confirming
$\mathcal{S}_{K_i}$ as spectral subspaces. Statement~(iii) then follows
directly from Remark~\ref{rem:frame-shift-spectral} applied to each
triplet component.
\end{proof}

% ------------------------------------------------------------------
% Readout-Relative Lemmas and Structural Lifts (v2.27.4 integration)
% ------------------------------------------------------------------

\section{Readout-Relative Lemmas and Structural Lifts}
\label{sec:readout-relative-lemmas}

The present framework distinguishes sharply between the structural kernel
of the theory and the visible form in which that kernel is read in a
given regime or frame. This distinction is not optional: the same
closure-stable object may appear as a probe/Dirac-type law, as an
effective QFT-type law in a phase-shifted observer regime, or as an
Einstein-type law after large-scale geometric readout. Consequently,
one must distinguish between \emph{kernel-level truths} and
\emph{readout-relative truths}.

The concrete instantiation of this distinction is already present in the
preceding spectral-projector block: the eigenvalue $\lambda_0$ is a
kernel-level invariant, while the measured value
$\lambda_0^{(\mathrm{matter})} = \gL^{-k}\cdot\lambda_0^{(\mathrm{photon})}$
(Remark~\ref{rem:frame-shift-spectral}) is readout-relative, depending
on the frame-insertion count $k$ of the observable in the matter frame
at $\zminus = +\bsym$.

\subsection*{Kernel level and readout level}

\begin{definition}[Kernel-level statement]
\label{def:kernel-level-statement}
A statement is called \emph{kernel-level} if it is invariant under the
admissible structural equivalences of the framework, including:
\begin{enumerate}[label=(\roman*)]
  \item sector-preserving unitary equivalence;
  \item compensator projection;
  \item observer-frame phase shift;
  \item admissible branch restriction.
\end{enumerate}
Such statements express properties of the underlying closure-stable
structural object itself rather than of one distinguished reading of it.
\end{definition}

\begin{definition}[Readout-relative lemma]
\label{def:readout-relative-lemma}
A statement $L^{(f)}$ is called a \emph{readout-relative lemma} if it
is proven only in a distinguished regime, frame, or readout $f$, and its
visible form depends on that readout. In particular, a readout-relative
lemma need not remain true as a frame-blind absolute statement.
\end{definition}

\begin{remark}
This is not a defect of the method. It is the correct formal response to
the fact that different regime-readings expose different visible shadows
of one common kernel. In the present framework this applies in particular
to the probe, effective QFT, and Einstein readouts.
The Symmetric matter--tachyon frame theorem (Theorem~\ref{thm:symmetric-frame})
is an example: $\beta_{\mathrm{matter}} = +\bsym$ is readout-relative
(it is the matter-frame reading), while the symmetry
$\beta_{\mathrm{tachyon}} = -\bsym$ and the common kernel $\bsym$ are
kernel-level invariants.
\end{remark}

\subsection*{Why this distinction is necessary}

The need for readout-relative lemmas is already built into the
architecture. The master system is closure-stable and unique only up to
the allowed sector-preserving, unitary, and frame transformations, while
the effective regimes differ by compensator decomposition, large-scale
suppression of closure fluctuations, and geometric readout. Thus one and
the same kernel may generate visibly different but structurally
compatible statements.

\begin{proposition}[Legitimacy of restricted lemmas]
\label{prop:legitimacy-restricted-lemmas}
A lemma proven in a distinguished readout $f$ may be used in the general
argument provided that:
\begin{enumerate}[label=(\roman*)]
  \item its frame or regime of validity is stated explicitly;
  \item its invariant kernel content is separated from its visible
        readout form;
  \item the passage from the readout-relative statement to the general
        claim is justified by a structural transport principle.
\end{enumerate}
\end{proposition}

\begin{proof}
The framework already distinguishes the common structural equation from
its regime projections (Theorem~\ref{thm:unity}, unitary equivalence via
Stone--von Neumann). Therefore a proof in one readout need not be
discarded merely because its visible form changes in another readout.
What must be transported is not the literal visible formula, but the
kernel-level content preserved under the admissible transport operations
of Lemmas~\ref{lem:group-transport} and~\ref{lem:hc-admissibility-lemma}.
\end{proof}

\subsection*{Structural lifts}

\begin{definition}[Structural lift]
\label{def:structural-lift}
Let $L^{(f)}$ be a readout-relative lemma valid in readout $f$.
A \emph{structural lift} of $L^{(f)}$ is a kernel-level statement
$L_{\mathrm{struct}}$ obtained by extracting from $L^{(f)}$ only the
invariant content that survives admissible unitary, sector, and frame
transport.
\end{definition}

\begin{remark}
A structural lift is therefore weaker than a literal universalization of
$L^{(f)}$, but stronger than leaving the lemma imprisoned in one special
case. The corollary Cor.~\ref{cor:SM-spectral-realization} is a concrete
example: statement~(iii) there lifts the frame-corrected eigenvalue
$\lambda_0^{(\mathrm{matter})}$ to the kernel-level invariant $\lambda_0$
by identifying $k=0$ for the singlet and $k\ge 1$ for the triplet.
\end{remark}

\begin{proposition}[Structural lift principle]
\label{prop:structural-lift-principle}
Suppose $L^{(f)}$ is proven in readout $f$, and suppose its proof
depends only on:
\begin{enumerate}[label=(\roman*)]
  \item the closure-stable master structure;
  \item invariants preserved under admissible transport;
  \item readout-specific visibility factors that can be isolated.
\end{enumerate}
Then there exists a structural lift $L_{\mathrm{struct}}$ that is valid
at kernel level, even if the visible statement $L^{(f)}$ itself is not
frame-blindly universal.
\end{proposition}

\begin{proof}
\noindent\textbf{Invariant content is transport-stable.}
By hypotheses~(i) and~(ii), the kernel-invariant content of the proof
of $L^{(f)}$ is preserved by the admissible transport operations.
Concretely: Lemma~\ref{lem:group-transport} guarantees that grouped-layer
transport carries admissible structure without creating new ontology, and
Lemma~\ref{lem:hc-admissibility-lemma} guarantees that compensator action
leaves the admissible sector invariant. Hence the kernel-invariant content
$P_{\mathrm{inv}}(\text{proof of }L^{(f)})$ is valid in every admissible
readout, not only in $f$.

\smallskip
\noindent\textbf{Readout-specific factors are isolatable.}
By hypothesis~(iii), the readout-dependent terms form a separable complement.
Discarding them while retaining $P_{\mathrm{inv}}(\text{proof})$ yields
$L_{\mathrm{struct}}$, independent of $f$.

\smallskip
\noindent\textbf{$L_{\mathrm{struct}}$ is kernel-level.}
Since $L_{\mathrm{struct}}$ depends only on the closure-stable structure
and transport-preserved invariants, it satisfies all four conditions of
Definition~\ref{def:kernel-level-statement}: invariance under
sector-preserving unitary equivalence, compensator projection,
observer-frame phase shift, and admissible branch restriction.
\end{proof}

\subsection*{The three-step proof logic}

This principle simplifies the global architecture considerably.
Instead of demanding that every intermediate statement already be valid
as a frame-blind absolute theorem, one may work in three steps:
\[
  \text{readout-relative lemma}
  \;\Longrightarrow\;
  \text{structural lift}
  \;\Longrightarrow\;
  \text{general kernel-level theorem}.
\]
This is the natural proof logic for a theory whose visible regimes are
different projections of one and the same structural object.

\subsection*{Shadow versus core: consequence for the compensator}

\begin{remark}[HC shadow versus HC kernel]
\label{rem:hc-shadow-vs-core}
This distinction is particularly important for the compensator.
In the minimal formulation, the visible compensator functional
$\mathcal{H}_C^{(f)}$ behaves as a closure-mismatch penalty in a given
readout and may vanish in an exact sector ($\mathcal{H}_C = 0$ in the
photon frame; cf.\ the closed master system). But under the
frame-dependent readout hypothesis developed above, such vanishing
concerns only the \emph{visible defect shadow} in a given regime, not
the deeper compensator-generating kernel
$C = \Pi^{(\mathcal{M})}_{\mathrm{spec}}$
(Theorem~\ref{thm:hc-spectral-projector}).

Hence a statement about $\mathcal{H}_C^{(f)}$ must not automatically be
read as a statement about the entire HC architecture. In particular, the
vanishing of the visible compensator field in the photon-frame readout
is consistent with the HC kernel remaining active as the
spectral-projector mechanism that selects the closure-stable sector ---
it is the shadow, not the core, that vanishes.

This is directly relevant to OP~5 ($\Lambda_{\mathrm{obs}}$): the
cosmological constant appears as a \emph{readout-relative} quantity ---
the frame offset $\bsym^2\langle f,f\rangle$ visible in the matter frame
--- while the photon-frame value $\Lambda_{\mathrm{fundamental}} = 0$
is the kernel-level statement. The large observed value is not a
fine-tuning problem but a failure to separate readout-relative visibility
from structural invariance.
\end{remark}

% ------------------------------------------------------------------
% Bidirectional Probe Lensing (v2.27.4 integration)
% ------------------------------------------------------------------

\section{Bidirectional Probe Lensing Hypothesis}
\label{sec:bidirectional-probe-lensing}

The preceding sections showed that the Dirac-, QFT-, and Einstein-type
regimes are not separate theories, but structural projections of one
common closure-stable master equation
(Proposition~\ref{prop:structural-lift-principle}). They also showed
that the Einstein reconstruction presently depends on a bilinear coframe
extraction whose nondegeneracy is still model-dependent. This suggests a
different route: rather than attempting to reconstruct geometry from a
single readout only, one may compare the traces of two probes approaching
the same structural core from different regime readings.

\subsection*{Basic setup}

Consider two probe modes of the triadic system,
$\Psi_{\mathrm{ART}}$ and $\Psi_{\mathrm{QFT}}$,
together with the orthogonal triadic component $\Psi_\perp$.
We regard $\Psi_{\mathrm{ART}}$ as a probe emphasizing the
large-scale/geometric regime and $\Psi_{\mathrm{QFT}}$ as a probe
emphasizing the phase-shifted observer/QFT regime at
$\zminus = +\bsym$ (Theorem~\ref{thm:symmetric-frame}).
The combined field is treated as a specialization of the triadic
two-probe system
\[
  \Psi_{\triangle}
  =
  \begin{pmatrix}
    \Psi_{\mathrm{ART}}\\
    \Psi_{\mathrm{QFT}}\\
    \Psi_\perp
  \end{pmatrix},
  \qquad
  \bigl(i\Gamma^\mu\mathcal{D}_\mu - \Omega_\triangle\bigr)
  \Psi_\triangle = 0.
\]
The third component is not decorative: it is the intrinsic coupling
channel through which the two regime-readings communicate.

\subsection*{Structural hypothesis}

\begin{proposition}[Bidirectional probe lensing hypothesis]
\label{hyp:bidirectional-probe-lensing}
\emph{(Hypothesis-grade; not yet theorem-level.)}
Let two probes approach the same closure-stable structural core from
different regime-readings, one on the ART side and one on the QFT side.
Then the compensator/closure core acts as an effective lensing kernel
for their traces. More precisely:
\begin{enumerate}[label=(\roman*)]
  \item the two probes need not follow identical local trajectories;
  \item their relative deflection, phase lag, and focusing structure are
        controlled by the same compensator/closure sector
        $C = \Pi^{(\mathcal{M})}_{\mathrm{spec}}$
        (Theorem~\ref{thm:hc-spectral-projector});
  \item the orthogonal triadic channel $\Psi_\perp$ mediates the
        nontrivial coupling between the two traces;
  \item the resulting trace geometry carries information about the
        effective metric readout before a full bilinear coframe
        reconstruction is imposed.
\end{enumerate}
This is a readout-relative statement
(Definition~\ref{def:readout-relative-lemma}): the lensing geometry is
visible in the bidirectional probe readout, while the kernel-level
content is the common closure-stable core.
\end{proposition}

\begin{remark}
This hypothesis does not identify the compensator literally with a
classical gravitational lens. Its weaker and structurally safer claim
is that the closure/HC core behaves as an effective focusing kernel for
probe traces --- the spectral-projector image $\mathrm{im}(C) =
\mathcal{S}_0$ of Theorem~\ref{thm:hc-spectral-projector} acts as the
common structural center around which both probe traces are organized.
\end{remark}

\subsection*{Why this is plausible in the present framework}

Three already established facts support the hypothesis.

First, the master equation admits distinguished structural projections to
the probe/Dirac, phase-shifted QFT, and sharpened Einstein regimes
(Proposition~\ref{prop:structural-lift-principle}). Hence the ART and QFT
probes are not foreign objects but two regime readings of one common
structural dynamics.

Second, the triadic two-probe system already contains an intrinsic third
channel $\Psi_\perp$ coupling the two probe modes. The middle sector
therefore carries genuine interaction data, not a mere bookkeeping slot.

Third, the Einstein reconstruction is already expressed through bilinears
of a closed triadic probe field, so probe traces are not external
diagnostics but the natural source of geometric data in the current
framework.

\subsection*{Trace observables}

\paragraph*{Relative deflection.}
Let $X^\mu_{\mathrm{ART}}(\tau)$ and $X^\mu_{\mathrm{QFT}}(\tau)$
denote effective probe traces. The deflection observable is
\[
  \Delta X^\mu(\tau)
  :=
  X^\mu_{\mathrm{ART}}(\tau) - X^\mu_{\mathrm{QFT}}(\tau).
\]

\paragraph*{Relative phase lag.}
\[
  \Delta\phi(\tau)
  :=
  \phi_{\mathrm{ART}}(\tau) - \phi_{\mathrm{QFT}}(\tau).
\]
This quantity is sensitive to compensator-mediated observer-frame
displacement (Remark~\ref{rem:frame-shift-spectral}).

\paragraph*{Cross-bilinear geometry channel.}
\[
  E^{a}_{\mu,\mathrm{cross}}
  :=
  \operatorname{Re}\bigl(
    \bar\Psi_{\mathrm{ART}}
    \,\Gamma^a\,
    \mathcal{D}_\mu
    \Psi_{\mathrm{QFT}}
  \bigr),
\]
and similarly with probes exchanged. These mixed quantities detect
geometry encoded not in a single probe sector but in the coupled
relation of both probes through the triadic core.

\paragraph*{Focusing functional.}
\[
  \mathcal{L}_{\mathrm{focus}}
  :=
  f\!\bigl(
    E^{a}_{\mu,\mathrm{cross}},\,
    \mathfrak{R}_C,\,
    \mathcal{H}_C
  \bigr),
\]
where $f$ detects concentration of the two traces toward the same
closure-stable branch. The expression is heuristic at this stage.

\subsection*{Expected regimes}

\begin{proposition}[Expected qualitative behaviour of probe lensing]
\label{prop:expected-probe-lensing-behaviour}
The bidirectional probe lensing hypothesis predicts:
\begin{enumerate}[label=(\roman*)]
  \item under exact closure, the trace structure sharpens and becomes
        sector-clean;
  \item when the triadic coupling channel is switched off, the
        cross-bilinear focusing signal weakens or disappears;
  \item in the phase-shifted observer regime, relative phase lag is more
        pronounced than geometric coframe data;
  \item in the large-scale closure-stable regime, relative focusing is
        expected to organize into an effective geometric readout.
\end{enumerate}
\end{proposition}

\begin{proof}
Exact closure suppresses leakage and makes the physical sector
autonomous (Lemma~\ref{lem:closure-forcing-lemma}), so trace observables
become sector-clean. The triadic two-probe equation shows that
$\Psi_\perp$ is the intrinsic coupling route; without it the two regime
probes largely decouple, weakening the cross-bilinear signal. By
Proposition~\ref{prop:structural-lift-principle}, QFT emerges in the
phase-shifted observer regime while the Einstein limit appears only at
large closure-stable scales, so phase effects dominate the QFT reading
and geometric focusing emerges only later.
\end{proof}

\subsection*{Relation to Einstein reconstruction}

The bidirectional probe picture may weaken the hardest current
bottleneck of the Einstein block. At present, the metric is reconstructed
from the bilinear coframe
\[
  E^a{}_\mu
  =
  \operatorname{Re}\bigl(
    \bar\Psi_\triangle\,\Gamma^a\,\mathcal{D}_\mu\Psi_\triangle
  \bigr),
\]
and the unresolved step is to justify its maximal rank from the triadic
probe dynamics itself. The bidirectional probe hypothesis suggests an
earlier readout layer: geometry may first appear as a stable trace
focusing/deflection pattern before it is compressed into a single
effective coframe.

\begin{remark}
If this route works, the Einstein sector would no longer require a
direct leap from closure to metric. The structural lift
(Definition~\ref{def:structural-lift}) of this two-step chain is:
\[
  \text{closure-stable probe traces}
  \;\Longrightarrow\;
  \text{effective focusing geometry}
  \;\Longrightarrow\;
  \text{coframe / metric reconstruction}.
\]
This could reduce the present model-dependence of the Einstein limit and
would realize the concrete form of the meta-field projection
(iv) in Hypothesis~\ref{hyp:hc-meta-generator}.
\end{remark}

% ------------------------------------------------------------------
% Theorem 7: HC-Induced Readout Geometry (v2.27.4 integration)
% ------------------------------------------------------------------

\section{HC-Induced Readout Geometry and Structural Unity}
\label{sec:theorem7}

\subsection*{Setup}

Let $\mathcal{K}$ be a closure-stable structural kernel, and let
\[
  \mathcal{R}_f : \mathcal{K} \to \mathcal{O}_f
\]
be admissible readout maps corresponding to different regimes (ART,
QFT, probe). Let $\mathcal{H}_C$ denote the Heisenberg compensator,
understood as a structural object controlling compatibility between
readouts. This is the concrete realization of
Definition~\ref{def:readout-relative-lemma}: each $\mathcal{R}_f$
produces a readout-relative statement, while $\mathcal{K}$ carries the
kernel-level content.

\subsection*{Theorem 7 (HC-Induced Readout Geometry)}

\begin{theorem}[HC-induced readout geometry]
\label{thm:hc-readout-geometry}
Under the above assumptions, the following statements hold.
\begin{enumerate}[label=(\arabic*)]
  \item \textbf{Readout divergence and effective trajectories.}
  For two readouts $f,g$, define
  $\Delta_{f,g} := \mathcal{R}_f - \mathcal{R}_g$.
  The induced effective dynamics in observable space generically
  produces trajectories that are not globally closed and may exhibit
  local segments resembling hyperbolic motion with apparent domain gaps.

  \item \textbf{Projection singularities.}
  Apparent gaps in trajectories correspond to incompatibilities of
  projections:
  \[
    \exists\, x \in \mathcal{K}:\quad
    \mathcal{R}_f(x)\ \text{well-defined, while}\ \mathcal{R}_g(x)\
    \text{is not compatibly representable.}
  \]
  These are projection singularities rather than physical
  discontinuities. In the language of
  Proposition~\ref{prop:legitimacy-restricted-lemmas}: $\mathcal{R}_f(x)$
  is a readout-relative statement whose regime of validity excludes the
  $g$-readout.

  \item \textbf{Mirror structure and recombination.}
  There exists a transformation $S$ such that locally
  $S(\mathcal{R}_f) \approx \mathcal{R}_g$.
  Complementary trajectory branches can therefore be recombined into
  an effective elliptic structure at the level of combined readouts.

  \item \textbf{Effective two-center dynamics.}
  The recombined dynamics is equivalent to motion in a symmetric
  two-center system, where the centers correspond to competing
  readout-fixed structures rather than physical sources. This is
  the bidirectional probe picture of
  Section~\ref{sec:bidirectional-probe-lensing}
  (Proposition~\ref{hyp:bidirectional-probe-lensing}) at the level of
  effective trajectories.

  \item \textbf{Non-orientable global structure.}
  The combined readout structure is generically non-orientable:
  transport along a closed loop induces a phase shift, requiring
  multiple loops for full return. This is consistent with a
  Möbius-type structure and reflects the $\zminus = \pm\bsym$
  symmetry of Theorem~\ref{thm:symmetric-frame}.

  \item \textbf{Existence of a structural lift.}
  There exists a lifted space $\tilde{\mathcal{K}}$ and projection
  $\pi : \tilde{\mathcal{K}} \to \mathcal{O}$ such that trajectories
  in $\tilde{\mathcal{K}}$ are smooth and closed, while their
  projections reproduce gaps, hyperbolic segments, and branching
  effects. This is the structural lift of
  Definition~\ref{def:structural-lift} applied to trajectory data.

  \item \textbf{HC as curvature.}
  The compensator $\mathcal{H}_C$ acts as curvature of the readout
  bundle, governing deviations of trajectories, phase shifts, and
  failure of closure. This is the geometric realization of
  Corollary~\ref{cor:M-HC-fixpoint}: $\mathcal{M}C = \lambda_0 C$
  fixes the compensator as the structural curvature anchor.

  \item \textbf{Projective vanishing.}
  There exist configurations such that
  \[
    \sum_i \mathcal{H}_C^{(f_i)} = 0
    \quad\text{while}\quad
    \mathcal{H}_C^{(\mathrm{kernel})} \neq 0.
  \]
  Thus the compensator may vanish in projection while remaining
  nontrivial at kernel level. This is precisely
  Remark~\ref{rem:hc-shadow-vs-core}: the shadow vanishes, not
  the kernel $C = \Pi^{(\mathcal{M})}_{\mathrm{spec}}$.

  \item \textbf{Emergent tensor structure.}
  The measurable curvature induced by $\mathcal{H}_C$ defines an
  effective tensor field $T_{\mu\nu}^{(\mathrm{HC})}$, arising from
  readout divergence $\Delta_{f,g}$ rather than from fundamental
  stress-energy. This is the tensorial compensator readout
  $\mathcal{K}_{\mu\nu}^{(\mathrm{HC})}$ of
  Hypothesis~\ref{hyp:hc-meta-generator}(iii) at the trajectory level.

  \item \textbf{Recursive field structure.}
  There exists a functional relation
  $\mathcal{F}[\mathcal{F}] = 0$,
  where $\mathcal{F}$ generates field equations recursively and
  $\mathcal{H}_C$ appears as a structural product within this
  recursion. This reflects the generating loop of
  Section~\ref{sec:hc-meta-field-generator}: the compensator is
  simultaneously generated by closure and inserted into transport.
\end{enumerate}
\end{theorem}

\begin{proof}
Statements (1)--(2) follow from the definition of $\Delta_{f,g}$ and
the readout-relative lemma framework
(Definition~\ref{def:readout-relative-lemma}): incompatible regime
projections produce apparent gaps that are projection singularities,
not structural discontinuities.

Statement (3) follows from the unitary equivalence of the three kernels
(Theorem~\ref{thm:unitary-equiv}): the transformation $S$ is the
unitary equivalence between the $f$- and $g$-readout representations.

Statement (4) is the trajectory-level reading of the bidirectional
probe picture (Section~\ref{sec:bidirectional-probe-lensing}): the two
regime probes approach the same closure-stable core from opposite sides.

Statement (5) follows from Theorem~\ref{thm:symmetric-frame}: matter
and tachyon sectors are positioned symmetrically around the photon frame
at $\pm\bsym$, so a full structural loop traverses both readout
branches.

Statement (6) is the structural lift principle
(Proposition~\ref{prop:structural-lift-principle}) applied to
trajectory data: the smooth lifted trajectories in $\tilde{\mathcal{K}}$
are the kernel-level content, while the projected gaps are readout
artifacts.

Statement (7) follows from Corollary~\ref{cor:M-HC-fixpoint}:
$\mathcal{M}C = \lambda_0 C$ identifies the compensator as the
structural fixpoint controlling all admissible transport; deviation
from this fixpoint is exactly what curvature measures.

Statement (8) is Remark~\ref{rem:hc-shadow-vs-core}: exact-sector
vanishing $\sum_i \mathcal{H}_C^{(f_i)} = 0$ is the shadow, not the
kernel $C = \Pi^{(\mathcal{M})}_{\mathrm{spec}}$
(Theorem~\ref{thm:hc-spectral-projector}).

Statement (9) follows from statement (7) and the probe-tensor readout
of Section~\ref{sec:bidirectional-probe-lensing}: the cross-bilinear
$E^a_{\mu,\mathrm{cross}}$ assembles into $T_{\mu\nu}^{(\mathrm{HC})}$.

Statement (10) is the recursive generating loop of
Hypothesis~\ref{hyp:hc-meta-generator}: the meta-field equation
$\mathfrak{E}_{\mathrm{HC}}[\ldots] = 0$ is the instance of
$\mathcal{F}[\mathcal{F}] = 0$ in the present framework.
\end{proof}

\subsection*{Interpretation}

Phenomena such as trajectory gaps, apparent singularities, or chaotic
behavior are not fundamental inconsistencies, but arise as projection
effects of a smooth, higher-dimensional structural system
(cf.\ Definition~\ref{def:kernel-level-statement}). Different physical
theories (Dirac, QFT, Einstein) correspond to distinct readouts of the
same kernel rather than incompatible foundations.

\subsection*{Methodological consequence}

Statements proven within a specific readout remain valid within that
regime. Their invariant structural content may be lifted to kernel-level
statements via structural lift, in accordance with
Section~\ref{sec:readout-relative-lemmas} and
Proposition~\ref{prop:structural-lift-principle}.

% ------------------------------------------------------------------
% Hypothesis 7.4 (v2.27.4 integration)
% ------------------------------------------------------------------

\section{Hypothesis 7.4: The Heisenberg--Compensator Matrix as a Meta-Field Generator}
\label{sec:hc-meta-field-generator}

The preceding sections suggest that the Heisenberg compensator should not
be read only as a penalty for off-sector leakage. In the bidirectional
probe picture, opposite regime traces are coupled through the same
closure-stable core, and their relative bending is expected to encode
effective curvature data. This motivates a stronger interpretation of
the compensator: not merely as a reactive defect term, but as a
structure-generating operator whose regime projections yield the
effective Dirac-, QFT-, and Einstein-type equations.

The connection to the new spectral-projector result is direct:
Theorem~\ref{thm:hc-spectral-projector} establishes
$C = \Pi^{(\mathcal{M})}_{\mathrm{spec}}$ as the kernel-level identity
of the compensator. Hypothesis~\ref{hyp:hc-meta-generator} below
proposes that this same operator, understood as a structure-generating
meta-field, produces the full regime hierarchy via the generating loop
$\mathcal{M}_{16} \to \mathcal{H}_C \to (\mathcal{D}_\mu,\Omega) \to
\text{frame projection} \to \text{effective field equation}$.

\subsection*{Motivation}

In the current probe--closure system, the compensator already appears
in three distinct roles:
\begin{enumerate}[label=(\arabic*)]
  \item as a local field $\mathcal{H}_C(x)$ in the covariant derivative;
  \item as a contribution to the effective resonance operator;
  \item as part of the closure current in the resonance field equation.
\end{enumerate}
Thus the compensator is already dynamic. What is missing is a single
structural formulation in which these roles are unified and from which
the frame-dependent regime equations can be generated as projections
(cf.\ Proposition~\ref{prop:structural-lift-principle}).

\subsection*{Meta-field hypothesis}

\begin{proposition}[HC meta-field generator hypothesis]
\label{hyp:hc-meta-generator}
\emph{(Hypothesis-grade; not yet theorem-level.)}
There exists a compensator-centered structural equation
\[
  \mathfrak{E}_{\mathrm{HC}}
  \bigl[
    \mathcal{M}_{16},\,
    \hat{P}_{C_7},\,
    \mathcal{H}_C,\,
    \Psi_\triangle,\,
    \mathcal{A}_\mu
  \bigr]
  = 0
\]
such that:
\begin{enumerate}[label=(\roman*)]
  \item the closure relation
        $\mathcal{H}_C = \Lambda[\mathcal{M}_{16},\hat{P}_{C_7}]$
        is generated internally rather than imposed externally;
  \item the compensator enters the transport law and the resonance law
        as part of one common generating loop;
  \item bidirectional probe traces in different regime readings induce a
        tensorial compensator readout
        \[
          \mathcal{K}_{\mu\nu}^{(\mathrm{HC})}
          =
          \mathcal{R}\bigl[
            \Psi_{\mathrm{ART}},\,
            \Psi_{\mathrm{QFT}},\,
            \Psi_\perp;\,
            \mathcal{D},\mathcal{H}_C
          \bigr];
        \]
  \item the Dirac-, QFT-, and Einstein-type equations are obtained as
        structural projections
        $\Pi_f(\mathfrak{E}_{\mathrm{HC}}) = 0$
        for distinguished frame or readout regimes $f$
        (in the sense of Definition~\ref{def:readout-relative-lemma}).
\end{enumerate}
\end{proposition}

\begin{remark}
Hypothesis~\ref{hyp:hc-meta-generator} is a meta-field-equation
hypothesis: not a field equation for one sector only, but a structural
equation whose projections produce the effective field equations of the
different regimes. Its readout-relative statements (iv) are
legitimized by Proposition~\ref{prop:legitimacy-restricted-lemmas};
the kernel-level content of the full equation $\mathfrak{E}_{\mathrm{HC}} = 0$
would constitute the structural lift
(Definition~\ref{def:structural-lift}).
\end{remark}

\subsection*{Recursive generating loop}

The structural content of the hypothesis is the closed loop
\[
  (\mathcal{M}_{16},\hat{P}_{C_7})
  \;\longrightarrow\;
  \mathcal{H}_C
  \;\longrightarrow\;
  (\mathcal{D}_\mu,\Omega)
  \;\longrightarrow\;
  \text{probe-trace curvature data}
  \;\longrightarrow\;
  \text{frame projection}
  \;\longrightarrow\;
  \text{effective field equation}.
\]
This loop is not a vicious circle. Its purpose is to express that the
compensator is simultaneously generated by closure, inserted into
transport, and read back through regime-specific probe geometry.
The kernel-level anchor of the loop is
Corollary~\ref{cor:M-HC-fixpoint}: $\mathcal{M}\,C = \lambda_0\,C$,
which fixes the compensator as a fixpoint of the Millennium-matrix
action and prevents the loop from drifting.

\subsection*{Probe-tensor readout}

In the bidirectional probe picture the compensator-centered curvature
signal becomes visible through a tensorial readout built from
opposite-frame probe traces:
\[
  \mathcal{K}_{\mu\nu}^{(\mathrm{HC})}
  :=
  \mathcal{S}_{\mu\nu}
  \bigl(
    \Psi_{\mathrm{ART}},\,
    \Psi_{\mathrm{QFT}},\,
    \Psi_\perp,\,
    \mathcal{D}_\mu,\,
    \mathcal{H}_C
  \bigr),
\]
where $\mathcal{S}_{\mu\nu}$ collects symmetrized mixed bilinears,
transport mismatch, and closure-sensitive focusing data.
Geometric curvature is not read from one frame alone, but from the
mutual bending of opposite-frame probes around the common compensator
core.

\begin{remark}
The compensator need not be the geometric center in a literal sense.
The more robust statement is that it acts as the structure-generating
twist or focusing operator relating the probe traces, while the
photon/light sector remains the distinguished collapse/output state of
the triolic decomposition --- the spectral-projector image
$\mathrm{im}(C) = \mathcal{S}_0$ of Theorem~\ref{thm:hc-spectral-projector},
identified with $K_3$ by Corollary~\ref{cor:SM-spectral-realization}.
\end{remark}

\subsection*{Regime generation}

The meta-field hypothesis is consistent with the already established
projection principle:
\begin{enumerate}[label=(\arabic*)]
  \item the probe regime yields the Dirac-type law;
  \item the phase-shifted observer regime (matter frame at
        $\zminus = +\bsym$) yields the effective QFT-type law after
        compensator decomposition;
  \item the large-scale closure-stable regime yields the Einstein-type
        law after geometric readout and coarse-graining.
\end{enumerate}
What Hypothesis~\ref{hyp:hc-meta-generator} adds is the claim that
these three regimes should ultimately be generated from one
compensator-centered structural equation rather than treated as three
parallel derived blocks.

% ------------------------------------------------------------------

\subsection{Hypothesis 7.5: Frame-dependent defect readout of a deeper compensator}
\label{subsec:hc-frame-dependent-readout}

The current minimal compensator ansatz identifies
\[
  \mathcal{H}_C
  =
  \rho\,[\mathcal{M}_{16},\hat P_{C_7}]^\dagger
        [\mathcal{M}_{16},\hat P_{C_7}]
  +
  \sigma\,\mathfrak{R}_C^\dagger\mathfrak{R}_C
\]
as a positive closure-mismatch functional. Accordingly, in the exact
sector one has
\[
  \mathfrak{R}_C = 0
  \qquad\Longrightarrow\qquad
  \mathcal{H}_C = 0
\]
for the minimal ansatz. This, however, need only be interpreted as the
vanishing of the \emph{visible defect readout} of the compensator in the
given regime or frame, not necessarily as the disappearance of a deeper
compensator-generating structure. This is the precise sense of
Remark~\ref{rem:hc-shadow-vs-core}: the shadow vanishes, not the kernel
$C = \Pi^{(\mathcal{M})}_{\mathrm{spec}}$
(Theorem~\ref{thm:hc-spectral-projector}).

\begin{proposition}[Frame-dependent defect readout hypothesis]
\label{hyp:frame-dependent-defect-readout}
\emph{(Hypothesis-grade; not yet theorem-level.)}
There exists a deeper compensator matrix/generator $\mathbf{H}_C$ such
that the presently used functional compensator is only a frame-dependent
visible readout
\[
  \mathcal{H}_C^{(f)}
  =
  \mathcal{D}_{\mathrm{read}}^{(f)}
  \bigl(\mathbf{H}_C;\,\mathcal{M}_{16},\,\hat P_{C_7}\bigr),
\]
where $f$ denotes a distinguished regime or frame reading. Then:
\begin{enumerate}[label=(\roman*)]
  \item $\mathcal{H}_C^{(f)}$ may vanish in the exact sector of one
        frame without forcing $\mathbf{H}_C$ itself to vanish;
  \item different regime projections may expose different visible
        compensator shadows of one common underlying generator;
  \item the current minimal ansatz describes the energetic defect/penalty
        sector of the compensator, not necessarily its full structural role.
\end{enumerate}
This is a readout-relative statement in the sense of
Definition~\ref{def:readout-relative-lemma}: the visible vanishing
$\mathcal{H}_C^{(f)} = 0$ is a statement about the readout, not about
the kernel $\mathbf{H}_C$.
\end{proposition}

\begin{remark}
This hypothesis removes the apparent tension between the exact-sector
vanishing of the minimal compensator functional and any meta-field role
demanded of the compensator structure. What vanishes under exact closure
is only the defect-visible compensator shadow in a given readout, not
automatically the deeper compensator kernel. The structural lift
(Definition~\ref{def:structural-lift}) of the vanishing statement
$\mathcal{H}_C^{(f)} = 0$ is therefore: \emph{the closure-mismatch
penalty is zero in the admissible sector} --- a kernel-level statement
about admissibility, not about the extinction of the compensator itself.
\end{remark}

\begin{remark}
This interpretation is structurally natural in the present framework,
since probe, effective QFT, and Einstein geometry are already treated as
different regime projections of one common closure-stable equation
(Proposition~\ref{prop:structural-lift-principle}), and the light or
photon sector is the compensator-selected collapse output
(Theorem~\ref{thm:fractal}). Hence a frame-dependent compensator
visibility is more plausible than a globally frame-blind compensator
extinction.
\end{remark}

\subsection{Hypothesis 7.6: Fluid-type equations as coarse-grained frame readouts}
\label{subsec:fluid-readout-hypothesis}

The present framework already treats the probe, effective QFT, and
Einstein regimes as different projections of one common closure-stable
equation (Proposition~\ref{prop:structural-lift-principle}). This
suggests a cautious additional possibility: certain fluid-type transport
equations may arise as coarse-grained, frame-dependent readouts of the
same HC/closure transport kernel, rather than as foreign structures added
from outside.

\begin{proposition}[Fluid-type readout hypothesis]
\label{hyp:fluid-readout}
\emph{(Hypothesis-grade; not yet theorem-level.)}
There exists a coarse-grained regime/frame readout
$\Pi_{\mathrm{fluid}}^{(f,\ell)}$
such that, for suitable frame $f$ and stratum/scale level $\ell$, the
closure-stable transport kernel
\[
  \bigl(i\Gamma^\mu\mathcal{D}_\mu - \Omega\bigr)\Psi = 0,
  \qquad
  \mathcal{D}^\mu\mathcal{F}_{\mu\nu}
  = J^{(\Psi)}_\nu + J^{(C_7)}_\nu + J^{(\mathrm{HC})}_\nu,
\]
admits an effective medium-type readout whose macroscopic evolution is
described by fluid-like transport equations.
\end{proposition}

\begin{remark}
This hypothesis does \emph{not} claim that fields are literally fluids.
It claims only that fluid-type equations may appear as frame-specific,
coarse-grained transport readouts of the same underlying structural
kernel. The distinction is exactly that of
Definition~\ref{def:readout-relative-lemma}: a readout-relative statement
$L^{(f,\ell)}$ valid in the coarse-grained regime, not a kernel-level
universalization.
\end{remark}

\begin{remark}
Stone--von Neumann (Theorem~\ref{thm:unitary-equiv}) supports sameness
of the underlying structural kernel at the representation level, but the
fluid equations would belong to a later coarse-grained regime readout
rather than to the primary operator-level formulation. The structural
lift (Definition~\ref{def:structural-lift}) of this hypothesis, if
realizable, would be a kernel-level statement that the closure-stable
transport kernel admits a coarse-graining map into fluid-type evolution
--- not that the fluid equations are primary.
\end{remark}

% ------------------------------------------------------------------

\begin{lemma}[Closed antisymmetry of deficiency elements]
\label{lem:closed-deficiency-antisymmetry}
Let $f$ be a deficiency element in the relevant HC/closure test sector. If the
$J$-symmetric subspace is already essentially self-adjoint on the admissible
core, then the $J$-symmetric part of $f$ vanishes. Equivalently,
\[
  f=f^-\qquad\text{and hence}\qquad Jf=-f.
\]
\end{lemma}

\begin{proof}
Write
\[
  f=f^+ + f^-,\qquad f^+=\tfrac12(f+Jf),\qquad f^- = \tfrac12(f-Jf).
\]
By construction, $f^+$ lies in the $J$-symmetric sector. The repaired Section 12
argument shows that once the admissible $J$-symmetric core is already
essentially self-adjoint, a deficiency element cannot retain a nonzero
$J$-symmetric component. Otherwise one would obtain a nontrivial deficiency
vector in the symmetric core itself, contradicting essential self-adjointness.
Hence $f^+=0$, so $f=f^-$ and therefore $Jf=-f$.
\end{proof}

\begin{proposition}[Functional annihilation of deficiency elements by HC]
\label{prop:functional-hc-annihilation}
For every deficiency element $f$ in the closed HC test sector, the compensator
annihilates $f$ at the function level:
\[
  C(f)=\tfrac12(f+Jf)=0.
\]
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:closed-deficiency-antisymmetry}, every such deficiency element
satisfies $Jf=-f$. Substituting into the defining compensator formula gives
\[
  C(f)=\tfrac12(f+Jf)=\tfrac12(f-f)=0.
\]
So the annihilation is a genuine functional consequence of closed
$J$-antisymmetry, not merely a heuristic tensor slogan.
\end{proof}

\begin{remark}[Tensor representations are secondary to the functional proof]
\label{rem:tensor-secondary-functional-proof}
Any tensor or matrix representative of the compensator is basis-dependent and
should be read only as a representation of the already established function-level
map. In particular, the annihilation of deficiency elements is proved by the
closed antisymmetry route
\[
  f^+=0 \Longrightarrow Jf=-f \Longrightarrow C(f)=0,
\]
not by a standalone coefficient-level matrix argument.
\end{remark}

\begin{lemma}[Anchor-compatibility lemma]
\label{lem:anchor-compatibility-lemma}
Compensator action is compatible with the orthogonal anchor. In particular, HC
stabilizes the admissible readout channel without destroying the distinction
between the anchored third direction and the first two sectors.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:orthogonal-anchor-lemma}, the third sector is not decorative
but the stabilizing orthogonal anchor. By Theorem~\ref{thm:hc-forcing-theorem},
HC is the mechanism that re-identifies shifted descriptions in a common output
channel. If HC destroyed the anchor distinction, then the stabilized output
would collapse back into an undifferentiated binary pairing and the role of the
third direction would be lost. That would contradict triolic minimality and the
very need for an orthogonal anchor. Therefore HC must act compatibly with the
anchored triolic organization.
\end{proof}

\begin{lemma}[Closure forcing lemma]
\label{lem:closure-forcing-lemma}
A triolic state with orthogonal anchor and forced HC is closure-stable.
Equivalently,
\[
\mathrm{Triolic}(x)\wedge \mathrm{Anchored}(x)\wedge \mathrm{HCForced}(x)
\Longrightarrow \mathrm{ClosureStable}(x).
\]
\end{lemma}

\begin{proof}
We verify the three contributing conditions and their combination.

\smallskip
\noindent\textbf{Step 1: Triolic structure.}
By Lemma~\ref{lem:triolic-minimality-lemma}, a triolic state carries a
three-sector organization $(K_1,K_2,K_3)$ as the minimal nondegenerate
structure compatible with compensator action and closure readout.
In particular, there exists a distinguished third sector $K_3$ in which
the compensator-selected output can be stabilized.

\smallskip
\noindent\textbf{Step 2: Orthogonal anchor.}
By Lemma~\ref{lem:orthogonal-anchor-lemma}, the third sector satisfies
$K_3\perp K_1$ and $K_3\perp K_2$, preventing $K_3$ from being
relabeled or absorbed into the dual pair $(K_1,K_2)$.
The anchor fixes the admissible direction of collapse/light selection.

\smallskip
\noindent\textbf{Step 3: HC forcing.}
By Lemma~\ref{lem:hc-admissibility-lemma}, the compensator-normalized
image $C(x)$ is admissible whenever $x$ is admissible. Since
$\mathrm{HCForced}(x)$ holds, the state lies in the compensator-selected
channel, so $\mathcal{C}x = C(x)$.

\smallskip
\noindent\textbf{Step 4: Closure stability.}
By the semantic--tensor correspondence of
Remark~\ref{rem:semantic-tensor-correspondence},
$\mathrm{ClosureStable}(x)$ is equivalent to
$\hat{K}(\mathcal{C}x) = \mathcal{C}x$.
By Step~3, $\mathcal{C}x = C(x)$, and by the idempotence of $C$
(Lemma~\ref{lem:hc-idempotence-lemma}), $C(C(x))=C(x)$, so $C(x)$
already lies in $\mathrm{im}(C)$.
By Steps~1--2 and Theorem~\ref{thm:fractal} ($\hat{K}(L_n)=L_{n-1}$,
terminating at $S_0$), the anchor fixes $K_3$ as the structural ground
sector; $\mathcal{C}x\in\mathrm{im}(C)$ lies in the $K_3$-aligned
closure-stable class and is therefore a fixpoint of $\hat{K}$ on that
class. Hence $\hat{K}(\mathcal{C}x)=\mathcal{C}x$, i.e.\
$\mathrm{ClosureStable}(x)$. \qed
\end{proof}

\begin{lemma}[Closure-stable iff compensator image]
\label{lem:closure-stable-iff-im-C}
Under the hypotheses of the normalized semantic kernel
(Definition~\ref{def:semantic-structural-kernel}), the following are equivalent
for any admissible state $x$:
\begin{enumerate}[label=(\roman*)]
  \item $\mathrm{ClosureStable}(x)$, i.e.\ $\hat{K}(\mathcal{C}x)=\mathcal{C}x$;
  \item $x \in \mathrm{im}(C)$, i.e.\ $\mathcal{C}x = x$.
\end{enumerate}
Hence $\mathrm{im}(C) = \mathcal{S}_0 := \{x \in \mathcal{S} :
\mathrm{ClosureStable}(x)\}$.
\end{lemma}

\begin{proof}
\textbf{(ii) $\Rightarrow$ (i).}
Suppose $\mathcal{C}x = x$. By Lemma~\ref{lem:hc-idempotence-lemma},
$C^2 = C$, so $x \in \mathrm{im}(C)$ implies $\mathcal{C}x = x$.
By the ground-state terminality condition (S2) of
Definition~\ref{def:semantic-structural-kernel},
$\hat{K}(x) = x$ for all $x \in S_0$, the photon ground sector.
Since $\mathrm{im}(C)$ is mapped by $\hat{K}$ to the ground sector
(Theorem~\ref{thm:fractal}: $\hat{K}(L_n) = L_{n-1}$, terminating at
$S_0$), and since $\mathcal{C}x = x$ means $x$ is already in the
compensator-selected ground sector, we get $\hat{K}(\mathcal{C}x) =
\hat{K}(x) = x = \mathcal{C}x$. Hence $\mathrm{ClosureStable}(x)$.

\smallskip
\textbf{(i) $\Rightarrow$ (ii).}
Suppose $\hat{K}(\mathcal{C}x) = \mathcal{C}x$, i.e.\
$\mathcal{C}x$ is a fixpoint of $\hat{K}$.
By Theorem~\ref{thm:fractal}, the only fixpoints of $\hat{K}$ in the
cascade are the elements of the ground sector $S_0 = L =
\bigcap_{n\ge 0} L_n$. Hence $\mathcal{C}x \in S_0$.
By the semantic--tensor correspondence
(Remark~\ref{rem:semantic-tensor-correspondence}),
$\mathcal{C}x \in S_0$ is precisely the condition that $\mathcal{C}$
has already reached its fixpoint on $x$, i.e.\ $\mathcal{C}(\mathcal{C}x)
= \mathcal{C}x$. Since $C^2 = C$ (Lemma~\ref{lem:hc-idempotence-lemma}),
this means $\mathcal{C}x \in \mathrm{im}(C)$, but also that $x$ itself
is mapped to the same compensator-fixed point. The re-identification
mechanism (Theorem~\ref{thm:hc-forcing-theorem}) selects precisely those
$x$ for which the compensated image is the state itself in the ground
sector: $\mathcal{C}x = x$. Hence $x \in \mathrm{im}(C)$.

\smallskip
\textbf{Conclusion.}
Both directions hold, so $\mathrm{ClosureStable}(x) \Leftrightarrow
x \in \mathrm{im}(C)$, giving $\mathcal{S}_0 = \mathrm{im}(C)$.
\end{proof}

\begin{proposition}[Closure class proposition]
\label{prop:closure-class-proposition}
The closure-stable states form a structurally distinguished admissibility class.
In particular, closure-stability is not an accidental property of isolated
examples but a reusable criterion for admissible dynamics and readout.
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:closure-forcing-lemma}, closure-stability follows from a fixed
combination of generative conditions rather than from a one-off construction.
By Lemma~\ref{lem:hc-admissibility-lemma}, the relevant states remain inside the
admissible body under compensator normalization, and by
Lemma~\ref{lem:anchor-compatibility-lemma} this process respects the stabilizing
triolic organization. Hence closure-stability is preserved under the normalized
structural mechanism and can be used as a class criterion.
\end{proof}

\begin{proposition}[Exact closure criterion]
\label{prop:exact-closure-criterion}
Exact closure is the regime in which the admissible channel selected by HC is
already terminal with respect to the relevant closure test, so that no further
residual compensator correction is required.
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:hc-idempotence-lemma}, compensator-normalized states do not
drift under repeated HC action. Exact closure is therefore the stronger case in
which the normalized state is not merely admissible but already stable under the
relevant closure readout. In the tensor realization this corresponds to the
absence of further leakage from the selected sector into a complementary block;
semantically it means that the compensator-selected channel has already reached
its terminal closure form.
\end{proof}

\begin{lemma}[Closure residue lemma]
\label{lem:closure-residue-lemma}
Whenever exact closure fails, the failure can be measured by a residual term
that records the remaining mismatch between HC-normalized admissibility and full
closure stability.
\end{lemma}

\begin{proof}
The exact-closure regime of
Proposition~\ref{prop:exact-closure-criterion} is defined precisely by the
absence of further correction after compensator normalization. If this stronger
condition fails while admissibility is still preserved, then there remains a
nonzero discrepancy between the compensator-selected channel and the fully
closed channel. This discrepancy is the closure residue. It vanishes in exact
closure and provides the natural quantity measuring departure from that regime.
\end{proof}

\begin{corollary}[First closure forcing chain]
\label{cor:first-closure-forcing-chain}
The first unfolded closure route of the normalized core is
\[
\mathrm{HC}
\Longrightarrow
\text{HC idempotence/admissibility}
\Longrightarrow
\text{anchor-compatible normalization}
\Longrightarrow
\text{closure forcing}
\Longrightarrow
\text{closure class}
\Longrightarrow
\text{exact closure or closure residue}.
\]
\end{corollary}

\begin{proof}
Lemmas~\ref{lem:hc-idempotence-lemma},
\ref{lem:hc-admissibility-lemma}, and
\ref{lem:anchor-compatibility-lemma} identify the first structural properties of
forced HC. Lemma~\ref{lem:closure-forcing-lemma} turns these into closure
stability, Proposition~\ref{prop:closure-class-proposition} promotes this to a
reusable admissibility class, and
Proposition~\ref{prop:exact-closure-criterion} together with
Lemma~\ref{lem:closure-residue-lemma} separates the exact and residual regimes.
\end{proof}


\subsection*{Phase I-C: Higher Confinement and Septinion Unfolding}
\addcontentsline{toc}{subsection}{Phase I-C: Higher Confinement and Septinion Unfolding}

This third unfolding block turns closure stability into a higher confinement
reading and makes explicit why the septinion/seption interpretation is not an
external axiom but a structurally selected reading of the admissible closure
class.

\begin{lemma}[Higher confinement lemma]
\label{lem:higher-confinement-lemma}
If a structural class is closure-stable and remains invariant under repeated
compensator-compatible normalization, then it admits a higher confinement
reading.
\end{lemma}

\begin{proof}
Closure stability excludes uncontrolled drift out of the admissible class, while
repeated compensator-compatible normalization prevents the persistence of an
independent escaping sector. Once both conditions hold, the admissible class is
not merely locally normalized but globally retained under the structural return
mechanisms of the system. This is exactly the content of higher confinement:
the admissible states are no longer only corrected case by case but persist as a
retained confinement regime.
\end{proof}

\begin{lemma}[Non-escape lemma]
\label{lem:non-escape-lemma}
Let $x$ be a closure-stable state in the normalized core. Then no admissible
transition generated within the core sends $x$ into a structurally independent
external sector.
\end{lemma}

\begin{proof}
By closure stability, every admissible normalization step sends $x$ back into
the same closure class. If an admissible transition could produce a genuinely
independent external sector, then closure stability would fail exactly at that
point, because the state would no longer be returnable by the same structural
normalization. Hence admissible transitions may rearrange or re-express states
inside the confinement class, but they cannot export them into an independent
outside sector. This is the structural non-escape property.
\end{proof}

\begin{theorem}[Septinion readability theorem]
\label{thm:septinion-readability-theorem}
If a class is closure-stable and admits the higher confinement reading, then it
is septinion-readable: the resulting stabilized regime may be interpreted as a
septinion/seption-level confinement of the normalized structural body.
\end{theorem}

\begin{proof}
The septinion reading is not introduced as an arbitrary enlargement of the
formalism. Rather, it names the level at which the closure-stable and
non-escaping structural class is most naturally read as a higher unified
container of the admissible sectors. By Lemma~\ref{lem:higher-confinement-lemma},
closure stability already upgrades the system to a higher confinement regime,
and Lemma~\ref{lem:non-escape-lemma} shows that admissible states remain inside
that regime. The septinion/seption reading is therefore the natural structural
reading of this higher confinement container.
\end{proof}

\begin{corollary}[HC as emergent structural consequence]
\label{cor:hc-emergent-structural-consequence}
Within the normalized core, HC is not an externally appended axiom. It appears
as an emergent structural consequence of the one-body, frame-shift, and closure
requirements.
\end{corollary}

\begin{proof}
The generative chain of Phase~I-A forces re-identification and therefore HC,
while the closure chain of Phase~I-B shows that HC is precisely the mechanism
that stabilizes the admissible class. Phase~I-C now identifies the higher
confinement reading selected by this stabilization. Consequently HC is not an
optional addition but the structurally emergent mechanism by which the unified
body remains admissible across shifted readouts and under closure.
\end{proof}

\begin{corollary}[First confinement forcing chain]
\label{cor:first-confinement-forcing-chain}
The first unfolded confinement route of the normalized core is
\[
\text{closure class}
\Longrightarrow
\text{higher confinement}
\Longrightarrow
\text{non-escape}
\Longrightarrow
\text{septinion reading}.
\]
Together with the preceding phases, this yields
\[
\text{one body}
\Longrightarrow
\text{re-identification}
\Longrightarrow
\mathrm{HC}
\Longrightarrow
\text{closure}
\Longrightarrow
\text{higher confinement}
\Longrightarrow
\text{septinion/seption reading}.
\]
\end{corollary}

\begin{proof}
Proposition~\ref{prop:closure-class-proposition} provides the stabilized
closure class. Lemma~\ref{lem:higher-confinement-lemma} upgrades this class to a
higher confinement reading, Lemma~\ref{lem:non-escape-lemma} gives the internal
retention property of admissible transitions, and
Theorem~\ref{thm:septinion-readability-theorem} identifies the corresponding
septinion/seption interpretation. The final displayed chain is obtained by
concatenating this phase with the forcing chains of Phases~I-A and~I-B.
\end{proof}


\subsection*{Phase II-D: Cascade, Readout, and the Collapse-Fixed Sector}
\addcontentsline{toc}{subsection}{Phase II-D: Cascade, Readout, and the Collapse-Fixed Sector}

\begin{lemma}[Cascade admissibility lemma]
\label{lem:cascade-admissibility-lemma}
If $x$ belongs to the closure-stable class selected by the normalized core,
then every admissible cascade image $\hat K(x)$ remains inside the admissible
readout class of the same unified body.
\end{lemma}

\begin{proof}
The cascade is introduced as a composition of projection maps from higher
strata toward the ground sector, not as an ontological splitting of the body.
By Corollary~\ref{cor:first-closure-forcing-chain}, closure-stable states are
precisely those stabilized against drift under shifted readouts. Applying the
cascade to such a state therefore yields another admissible readout of the same
body rather than a new object. Hence admissibility is preserved along the
cascade image of any closure-stable input.
\end{proof}

\begin{lemma}[Terminal ground-state lemma]
\label{lem:terminal-ground-state-lemma}
The ground state $S_0$ is the terminal invariant readout sector of the cascade:
for every admissible state $x$, one has $\hat K(x)=x$ exactly when $x\in S_0$.
\end{lemma}

\begin{proof}
This is the normalized readout form of the ground-state condition already built
into the semantic kernel. The cascade terminates in $S_0$, and terminality
means precisely that no further nontrivial readout step occurs. Therefore the
fixed-point condition $\hat K(x)=x$ is equivalent to membership in the terminal
sector $S_0$.
\end{proof}

\begin{lemma}[Global null/light-axis lemma]
\label{lem:global-light-axis-lemma}
The null-axis structure selected by the normalized core, and read physically as the light axis at stable propagation, is stratified across
the cascade: if a state is read on this null/light axis at one admissible layer, then
its admissible readouts remain aligned with the same global null/light-axis class
through the cascade.
\end{lemma}

\begin{proof}
The Lorentz/null-axis structure, read physically as the light axis, is not introduced as a local afterthought but
as a stratified feature of the whole construction. The cascade only changes the
readout level, not the structural identity of the body. Since admissible
cascade images preserve the same unified body and the same closure-compatible
sector, the null/light-axis reading persists as a global class across the projected
layers.
\end{proof}

\begin{theorem}[Collapse-fixed sector theorem]
\label{thm:collapse-fixed-sector-theorem}
The photon/readout sector identified in the normalized core is the global
collapse-fixed sector of the cascade. In particular, the sector denoted by
$S_0$ is not merely a local null readout but the terminal collapse-fixed sector
selected by the full stratified structure.
\end{theorem}

\begin{proof}
Lemma~\ref{lem:cascade-admissibility-lemma} ensures that the cascade preserves
admissibility, while Lemma~\ref{lem:terminal-ground-state-lemma} identifies
$S_0$ as the terminal invariant sector. Lemma~\ref{lem:global-light-axis-lemma}
shows that the light-axis reading persists globally across admissible layers.
Therefore the photon/readout sector is not obtained from an isolated local
calculation alone; rather, it is the collapse-fixed terminal sector selected by
the entire cascade of admissible projections.
\end{proof}

\begin{lemma}[Readout separation lemma]
\label{lem:readout-separation-lemma}
Within the normalized core one must distinguish three levels: structural
forcing, cascade identification, and physical reading. A statement belongs to
exactly one of these levels unless an explicit bridge is given.
\end{lemma}

\begin{proof}
The structural forcing chain establishes what must hold inside the unified
body. The cascade then identifies how such a structurally forced state appears
at lower readout levels. A physical interpretation is an additional reading of
that identified sector. Conflating these levels obscures where a statement is
proved and where it is interpreted. The separation therefore serves as a
normalization rule for subsequent unfolding steps.
\end{proof}

\begin{corollary}[First readout forcing chain]
\label{cor:first-readout-forcing-chain}
The first unfolded readout route of the normalized core is
\[
\text{closure-stable class}
\Longrightarrow
\text{admissible cascade image}
\Longrightarrow
\text{terminal ground sector}
\Longrightarrow
\text{collapse-fixed photon/readout sector}.
\]
Taken together with Phases~I-A through~I-C, the normalized core now yields the
combined chain
\[
\text{one body}
\Longrightarrow
\text{re-identification}
\Longrightarrow
\mathrm{HC}
\Longrightarrow
\text{closure}
\Longrightarrow
\text{higher confinement}
\Longrightarrow
\text{septinion reading}
\Longrightarrow
\text{cascade-fixed readout sector}.
\]
\end{corollary}

\begin{proof}
The first displayed chain is the direct concatenation of
Lemma~\ref{lem:cascade-admissibility-lemma},
Lemma~\ref{lem:terminal-ground-state-lemma}, and
Theorem~\ref{thm:collapse-fixed-sector-theorem}. The second displayed chain is
obtained by appending this readout route to the forcing chain from
Corollary~\ref{cor:first-confinement-forcing-chain}.
\end{proof}


\subsection*{Phase II-E: Semantic/Tensor Interface and Millennium Intertwining}
\addcontentsline{toc}{subsection}{Phase II-E: Semantic/Tensor Interface and Millennium Intertwining}

\begin{lemma}[Semantic kernel well-formedness lemma]
\label{lem:semantic-kernel-well-formedness-lemma}
The normalized semantic kernel is well formed: its predicates of one-body
identity, triolic organization, orthogonal anchoring, frame-shifted readout,
HC forcing, closure stability, higher confinement, and readout admissibility do
not define competing ontologies but successive structural constraints on one and
the same body.
\end{lemma}

\begin{proof}
By construction, the normalized core begins from the one-body condition and then
adds only constraint predicates that regulate how the same body may be read,
stabilized, and projected. None of the predicates introduces a second carrier
space or an ontologically independent duplicate; each one refines the same
underlying body by imposing an admissibility condition. The semantic kernel is
therefore well formed as a constraint hierarchy rather than a family of
competing state spaces.
\end{proof}

\begin{lemma}[Tensor realization lemma]
\label{lem:tensor-realization-lemma}
The tensorial operators $\mathcal J^\mu{}_{\nu}$, $\mathcal C^\mu{}_{\nu}$,
$\Pi_n{}^\mu{}_{\nu}$, and $\hat K^\mu{}_{\nu}$ realize the normalized semantic
kernel as an operator system: $\mathcal J$ realizes the involutive frame
symmetry, $\mathcal C$ the compensating projection, $\Pi_n$ the cascade steps,
and $\hat K$ the terminal readout map.
\end{lemma}

\begin{proof}
The semantic kernel distinguishes involutive reflection, compensating return,
projection through admissible layers, and terminal ground readout. The tensorial
operators are introduced precisely to encode these roles in compact operator
notation. The semantic and tensorial descriptions therefore stand in a role-wise
correspondence: the operator system is not a second theory but a tensor
realization of the already normalized semantic structure.
\end{proof}

\begin{lemma}[Intertwining lemma]
\label{lem:millennium-intertwining-lemma}
If the Millennium matrix $\mathcal M^\mu{}_{\nu}$ is used as the structural
transport operator between admissible grouped layers, then it must intertwine
with the normalized tensorial kernel. In the minimal normalized form one
requires
\[
\mathcal M\mathcal J=\mathcal J\mathcal M,
\qquad
\mathcal M\mathcal C=\mathcal C\mathcal M,
\qquad
\mathcal M\hat K=\hat K\mathcal M
\]
whenever the corresponding involution, compensation, and cascade-readout
structures are to be preserved across the transport.
\end{lemma}

\begin{proof}
A transport operator that fails to commute with the involution would change the
frame logic of the transported state; one that fails to commute with the
compensator would alter admissibility by moving states in or out of the
compensated sector; and one that fails to commute with the terminal readout map
would not preserve the normalized collapse/readout architecture. Hence a
Millennium transport that is meant to be structure preserving must satisfy the
stated intertwining relations at least for the operators whose semantic role it
claims to preserve.
\end{proof}

\begin{theorem}[Admissibility preservation theorem]
\label{thm:admissibility-preservation-theorem}
If $x$ is admissible in the normalized semantic kernel and $\mathcal M$ is a
Millennium transport satisfying the intertwining relations of
Lemma~\ref{lem:millennium-intertwining-lemma}, then $\mathcal Mx$ remains
admissible in the transported layer.
\end{theorem}

\begin{proof}
Admissibility in the normalized core is carried by compatibility with the one-body
constraint, compensator normalization, and the cascade-compatible readout
structure. By Lemma~\ref{lem:millennium-intertwining-lemma}, the transport
operator preserves the tensorial realizations of exactly these mechanisms.
Therefore applying $\mathcal M$ cannot convert an admissible state into an
inadmissible one merely by transport; it yields an admissible image in the next
layer provided the transport is itself taken within the normalized grouped
architecture.
\end{proof}

\begin{theorem}[Closure preservation theorem]
\label{thm:closure-preservation-theorem}
Under the same hypotheses, closure-stable states are transported by
$\mathcal M$ to closure-stable states of the receiving layer.
\end{theorem}

\begin{proof}
Closure stability is the semantic statement that the admissible state does not
drift out of the normalized class once HC forcing and anchoring are taken into
account. Since admissibility is preserved by
Theorem~\ref{thm:admissibility-preservation-theorem} and the intertwining
relations preserve the compensator and readout structures that stabilize the
closure class, the transported image cannot lose closure merely by layer
transport. Thus closure stability is preserved across admissible Millennium
transport.
\end{proof}

\begin{proposition}[Semantic--tensor equivalence proposition]
\label{prop:semantic-tensor-equivalence-proposition}
The tensorial kernel does not introduce a second structural theory. It is a
parallel notation for the normalized semantic kernel, and the Millennium matrix
is admissible only insofar as its tensorial action represents the same
constraint chain already present on the semantic side.
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:semantic-kernel-well-formedness-lemma}, the semantic side is a
single constrained body. By Lemma~\ref{lem:tensor-realization-lemma}, the
operator side realizes these constraints in tensorial notation. The intertwining
and preservation results show that $\mathcal M$ is not free to create a new
structural doctrine; it is licensed only as a transport operator respecting the
same normalized constraint chain. Hence semantic kernel and tensor kernel are
parallel realizations of one structure.
\end{proof}

\begin{corollary}[First semantic--tensor forcing chain]
\label{cor:first-semantic-tensor-forcing-chain}
The normalized semantic/tensor interface yields the chain
\[
\text{semantic kernel}
\Longrightarrow
\text{tensor realization}
\Longrightarrow
\text{Millennium intertwining}
\Longrightarrow
\text{admissibility preservation}
\Longrightarrow
\text{closure preservation}.
\]
\end{corollary}

\begin{proof}
Lemma~\ref{lem:semantic-kernel-well-formedness-lemma} establishes the semantic
kernel, Lemma~\ref{lem:tensor-realization-lemma} gives its tensor realization,
Lemma~\ref{lem:millennium-intertwining-lemma} fixes the admissible transport
relations for $\mathcal M$, and Theorems~\ref{thm:admissibility-preservation-theorem}
and~\ref{thm:closure-preservation-theorem} provide the preservation steps.
\end{proof}


\subsection*{Grouped Layer Architecture and Millennium Transport}
\addcontentsline{toc}{subsection}{Grouped Layer Architecture and Millennium Transport}

The semantic/tensor kernel admits a second normalization in which the document's
many local cores, layers, and readout blocks are collected into three grouped
structural classes. This reduces duplication and makes the role of the
Millennium matrix explicit as a transport operator between admissible groups.

\begin{definition}[Grouped structural architecture]
\label{def:grouped-structural-architecture}
The normalized structural architecture consists of the following three grouped
layers:
\begin{enumerate}[leftmargin=2em,label=(G\arabic*)]
\item \textbf{Generative layer $\mathfrak G$:} the layer of structural formation. It
contains the one-body condition, triolic minimality, orthogonal anchoring,
frame-shifted observation, the ground condition, HC forcing, closure
admissibility, and the higher-confinement/septinion reading.
\item \textbf{Projection/translation layer $\mathfrak P$:} the layer of cascade readout,
sector selection, projection rules, collapse-fixed identifications, null-/light-axis
readings, photon-sector readings, and admissible translation between equivalent
structural descriptions.
\item \textbf{Field/probe realization layer $\mathfrak F$:} the layer of probe states,
two-probe geometry, closure-preserving field equations, interaction rules, and
physical realization of the admissible structure.
\end{enumerate}
These three grouped layers are not independent ontologies. They are different
organizational regimes of the same underlying body and must therefore remain
compatible under admissible transport.
\end{definition}

\begin{definition}[Millennium transport between grouped layers]
\label{def:millennium-transport-grouped-layers}
The Millennium matrix is interpreted at the grouped-layer level as a
structure-preserving converter
\[
\mathcal M_{\mathfrak G\to\mathfrak P}:\mathfrak G\to\mathfrak P,
\qquad
\mathcal M_{\mathfrak P\to\mathfrak F}:\mathfrak P\to\mathfrak F,
\]
and, whenever reverse structural checking is required, also by admissible
back-transport maps
\[
\mathcal M_{\mathfrak P\to\mathfrak G}:\mathfrak P\to\mathfrak G,
\qquad
\mathcal M_{\mathfrak F\to\mathfrak P}:\mathfrak F\to\mathfrak P.
\]
At this level $\mathcal M$ is not merely a notational relabeling but a
converter which must preserve structural admissibility across layer changes.
\end{definition}

\begin{proposition}[Grouped forcing and readout chain]
\label{prop:grouped-forcing-readout-chain}
The grouped architecture organizes the document into the structural chain
\[
\mathfrak G \xrightarrow{\ \mathcal M\ } \mathfrak P
\xrightarrow{\ \mathcal M\ } \mathfrak F,
\]
where:
\begin{enumerate}[leftmargin=2em,label=(\alph*)]
\item $\mathfrak G$ generates the admissible structure,
\item $\mathfrak P$ reads, projects, and translates that structure without changing
its admissibility class,
\item $\mathfrak F$ realizes the same admissible structure as a field/probe system.
\end{enumerate}
In particular, the field/probe layer is not a second origin of the theory; it
is a realization layer of structure first generated in $\mathfrak G$ and made
readable in $\mathfrak P$.
\end{proposition}

\begin{proof}
The generative layer contains the forcing data established above: one-body,
triolic organization, orthogonal anchor, frame shift, HC, and closure. The
projection/translation layer does not create new admissibility; it converts the
generative core into cascade-compatible, sector-resolved, and physically
legible readout. The field/probe layer then realizes those admissible readouts
as explicit probes, field equations, or interaction geometries. Since the same
body is being followed through all three layers, admissible transport by
$\mathcal M$ must preserve rather than replace the structural class.
\end{proof}

\begin{proposition}[Admissibility preservation across grouped layers]
\label{prop:admissibility-preservation-grouped-layers}
Let $x$ be an admissible state generated in $\mathfrak G$. If $\mathcal M$ acts as
an admissible grouped-layer converter, then
\[
\mathrm{Admissible}(x)
\Longrightarrow
\mathrm{Admissible}(\mathcal M_{\mathfrak G\to\mathfrak P}x)
\Longrightarrow
\mathrm{Admissible}(\mathcal M_{\mathfrak P\to\mathfrak F}\mathcal M_{\mathfrak G\to\mathfrak P}x).
\]
Moreover, when closure stability is part of the admissibility data, it is to be
preserved throughout this transport chain.
\end{proposition}

\begin{proof}
By Definition~\ref{def:millennium-transport-grouped-layers}, grouped-layer
transport is only allowed when it preserves structural admissibility. Hence an
admissible state generated in $\mathfrak G$ remains admissible after conversion
into the projection/translation layer, and likewise after conversion into the
field/probe layer. If closure stability were lost during transport, the result
would no longer be a valid readout or realization of the original structure but
a structural rupture. Therefore closure stability must be preserved whenever it
belongs to the originating admissibility class.
\end{proof}

\begin{remark}[Document-normalization consequence]
\label{rem:document-normalization-consequence}
The grouped architecture provides a normalization principle for the text. Local
``cores'' or ``integration blocks'' should be read either as part of
$\mathfrak G$, as part of $\mathfrak P$, or as part of $\mathfrak F$, rather than as
independent foundational centers. This removes duplication: the normative
foundation lies in $\mathfrak G$, the translation/readout logic in $\mathfrak P$,
and the field/probe realization in $\mathfrak F$.
\end{remark}

\subsection*{Phase II-F: Grouped Layer Transport and Ontological Non-Drift}
\addcontentsline{toc}{subsection}{Phase II-F: Grouped Layer Transport and Ontological Non-Drift}

This unfolding block upgrades the grouped architecture from a normalization note
into an explicit transport calculus. The goal is to show that grouped layers do
not introduce new origins of structure, but only different admissible regimes of
one and the same underlying body.

\begin{lemma}[Group decomposition lemma]
\label{lem:group-decomposition}
Every active structural block of the normalized Companion is dominantly
assignable to exactly one of the grouped layers $\mathfrak G$, $\mathfrak P$, or
$\mathfrak F$, even when secondary cross-references to the other layers remain.
\end{lemma}

\begin{proof}
By Definition~\ref{def:grouped-structural-architecture}, the grouped layers are
distinguished by role rather than by separate ontology: generation,
projection/translation, and realization. Any active block therefore has one
primary structural function. Cross-references may connect a block to the other
layers, but they do not erase its dominant role. Hence each active block is
assignable to one grouped layer as its principal location.
\end{proof}

\begin{lemma}[Transport lemma]
\label{lem:group-transport}
Let $x$ be a state generated in $\mathfrak G$. If $\mathcal M_{\mathfrak G\to\mathfrak P}$
is an admissible grouped-layer converter, then its image in $\mathfrak P$ is a
projection/translation of the same admissible structure and not a newly created
structural body.
\end{lemma}

\begin{proof}
By Definition~\ref{def:millennium-transport-grouped-layers}, admissible grouped
transport preserves structural admissibility. Proposition~\ref{prop:grouped-forcing-readout-chain}
states that $\mathfrak P$ reads and translates what $\mathfrak G$ has already
generated. Therefore the image of $x$ under $\mathcal M_{\mathfrak G\to\mathfrak P}$
remains the same admissible structure under a projection/translation regime.
It is not a second origin because admissibility is transported, not recreated.
\end{proof}

\begin{lemma}[Realization lemma]
\label{lem:group-realization}
If $y\in\mathfrak P$ is an admissible projection/translation state and
$\mathcal M_{\mathfrak P\to\mathfrak F}$ is admissible, then its image in
$\mathfrak F$ is a realization of the same admissible structure as a field/probe
state.
\end{lemma}

\begin{proof}
The role of $\mathfrak F$ is realization rather than generation by
Proposition~\ref{prop:grouped-forcing-readout-chain}. Since admissibility is
preserved across grouped transport by Proposition~\ref{prop:admissibility-preservation-grouped-layers},
$\mathcal M_{\mathfrak P\to\mathfrak F}(y)$ remains within the same
admissibility class. Thus the field/probe layer realizes the translated
structure without changing its underlying body.
\end{proof}

\begin{proposition}[No-ontology-drift proposition]
\label{prop:no-ontology-drift}
Admissible transport across the grouped layers does not create ontological
multiplicity. If
\[
x \in \mathfrak G,
\qquad
x_P := \mathcal M_{\mathfrak G\to\mathfrak P}(x),
\qquad
x_F := \mathcal M_{\mathfrak P\to\mathfrak F}(x_P),
\]
then $x$, $x_P$, and $x_F$ are three admissible organizational regimes of one
and the same structural body.
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:group-transport}, passage from $\mathfrak G$ to $\mathfrak P$
produces a translation/readout of the same admissible body. By
Lemma~\ref{lem:group-realization}, passage from $\mathfrak P$ to $\mathfrak F$
produces a field/probe realization of that same admissible body. Since both
steps preserve admissibility rather than replacing it, no new ontology is
introduced along the chain. Therefore $x$, $x_P$, and $x_F$ are structurally
identical in body and differ only by organizational regime.
\end{proof}

\begin{corollary}[First grouped transport forcing chain]
\label{cor:first-grouped-transport-forcing-chain}
The normalized grouped architecture supports the forcing chain
\[
\mathfrak G
\Longrightarrow
\text{admissible transport to }\mathfrak P
\Longrightarrow
\text{admissible realization in }\mathfrak F
\Longrightarrow
\text{no ontology drift across grouped layers}.
\]
\end{corollary}

\begin{proof}
Combine Lemma~\ref{lem:group-decomposition}, Lemma~\ref{lem:group-transport},
Lemma~\ref{lem:group-realization}, and
Proposition~\ref{prop:no-ontology-drift}.
\end{proof}


\subsection*{Phase III-G: Probe Geometry, Inheritance, and Guided Search}
\addcontentsline{toc}{subsection}{Phase III-G: Probe Geometry, Inheritance, and Guided Search}

This unfolding block upgrades the probe sector from an interpretive add-on to a
structurally inherited realization of the normalized core. The goal is to show
that probes do not introduce a foreign search mechanism, but inherit triolic,
anchored, and closure-compatible structure from the generative and grouped
layers.

\begin{lemma}[Probe inheritance lemma]
\label{lem:probe-inheritance}
If the normalized structural body is triolic, anchored, and closure-admissible,
then every admissible probe realization inherited through $\mathfrak F$ carries a
corresponding triolic and closure-compatible organization.
\end{lemma}

\begin{proof}
By Proposition~\ref{prop:no-ontology-drift}, admissible realization in
$\mathfrak F$ does not create a second ontology, but realizes the same
underlying structural body under a field/probe regime. Hence structural features
that belong to the admissible body---in particular triolic organization and
anchor-compatible closure discipline---remain available in the probe image as
inherited structure rather than externally attached decoration.
\end{proof}

\begin{lemma}[Two-probe compatibility lemma]
\label{lem:two-probe-compatibility}
Let $P_1$ and $P_2$ be two admissible probes directed toward one another. If
both are inherited from the same normalized structural body, then their mutual
configuration is compatible with one common closure-governed geometry.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:probe-inheritance}, each probe inherits the same triolic and
closure-compatible structural discipline. Since neither probe introduces a new
ontology by Proposition~\ref{prop:no-ontology-drift}, their mutual interaction
must be read inside one common admissibility class. Therefore the two-probe
configuration is not a collision of unrelated structures but a common
closure-governed geometry of one inherited body.
\end{proof}

\begin{lemma}[Orthogonal third-group lemma]
\label{lem:orthogonal-third-group-probe}
In an admissible two-probe configuration, the third structural group remains
orthogonal to both probes and thereby determines the constrained search region.
\end{lemma}

\begin{proof}
The orthogonal third direction is already forced at the generative level by the
orthogonal-anchor principle and remains preserved under admissible realization.
Hence, when two probes are realized as inherited triolic structures, the third
group cannot collapse into either probe direction without violating anchor
compatibility. It therefore remains orthogonal to both and serves as the
stabilizing transversal direction that bounds the admissible search region.
\end{proof}

\begin{proposition}[Probe reduction theorem]
\label{prop:probe-reduction-theorem}
An admissible two-probe configuration reduces to the closure-preserving sector of
the grouped architecture. In particular, the probe interaction may be analyzed
inside the admissible closure class without introducing a second search
ontology.
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:two-probe-compatibility}, the two probes share one common
closure-governed geometry. By Lemma~\ref{lem:orthogonal-third-group-probe}, the
third group fixes the transversal admissible search region. Since closure and
ontology are preserved across grouped realization by
Proposition~\ref{prop:no-ontology-drift}, the probe interaction reduces to the
closure-preserving sector of the same normalized architecture.
\end{proof}

\begin{corollary}[Guided search corollary]
\label{cor:guided-search-corollary}
Auxiliary tools such as frequency extraction, transition guidance, structural
routing, or convergence acceleration act only within the already admissible
probe geometry. They do not generate admissibility, but exploit a search region
that has already been restricted by inheritance, closure, and orthogonal
anchoring.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:probe-reduction-theorem}, admissible probe analysis is
already confined to the closure-preserving sector. Therefore any auxiliary tool
operates on a domain whose admissibility has been fixed before the tool is
applied. The tool may compress or guide the search, but it does not create the
search geometry itself.
\end{proof}

\begin{corollary}[First probe forcing chain]
\label{cor:first-probe-forcing-chain}
The normalized probe sector supports the forcing chain
\[
\text{inherited probe structure}
\Longrightarrow
\text{two-probe compatibility}
\Longrightarrow
\text{orthogonal third-group search region}
\Longrightarrow
\text{closure-preserving probe reduction}
\Longrightarrow
\text{guided search inside an already admissible geometry}.
\]
\end{corollary}

\begin{proof}
Combine Lemma~\ref{lem:probe-inheritance},
Lemma~\ref{lem:two-probe-compatibility},
Lemma~\ref{lem:orthogonal-third-group-probe},
Proposition~\ref{prop:probe-reduction-theorem}, and
Corollary~\ref{cor:guided-search-corollary}.
\end{proof}



\subsection*{Phase III-H: Structural Master Equation and Exact-Closure Field Regime}
\addcontentsline{toc}{subsection}{Phase III-H: Structural Master Equation and Exact-Closure Field Regime}

This unfolding block upgrades the previously compressed master-equation language
into an explicit field-level forcing chain. The aim is not yet to derive every
physical limit in full detail, but to show that the structural master equation,
its variational reading, the closure-preserving field regime, and the exact-
closure simplifications belong to one continuous argument rather than to loosely
connected interpretive remarks.

\begin{lemma}[Structural master-equation lemma]
\label{lem:structural-master-equation}
The structural master equation is the compressed field-level summary of the
normalized forcing chain from one-body structure through re-identification, HC,
closure, higher confinement, and admissible readout.
\end{lemma}

\begin{proof}
By Corollary~\ref{cor:first-generative-forcing-chain}, the normalized body
forces re-identification and HC. By Corollary~\ref{cor:first-closure-forcing-chain},
this induces a closure class with either exact closure or a controlled residue.
By Corollary~\ref{cor:first-confinement-forcing-chain}, closure yields higher
confinement and the septinion reading. By Corollary~\ref{cor:first-readout-forcing-chain}
and Corollary~\ref{cor:first-semantic-tensor-forcing-chain}, admissible readout
and tensorial transport preserve this structure. Hence the master equation is not
an additional postulate but the compressed field-level expression of the forcing
chain already established.
\end{proof}

\begin{lemma}[Variational consistency lemma]
\label{lem:variational-consistency}
Any variational reading of the structural master equation is admissible only if
it preserves the normalized one-body, HC, and closure conditions. In
particular, admissible variation may unfold the structural equation but may not
replace its forcing chain by an independent ontology.
\end{lemma}

\begin{proof}
By Proposition~\ref{prop:semantic-tensor-equivalence-proposition}, tensorial formulation is
only a parallel realization of the same semantic kernel. Therefore any
variational formulation that is meant to belong to the present architecture must
preserve the same semantic admissibility data. If the variation were to destroy
one-body identity, HC-forcing, or closure preservation, it would cease to be a
variation of the normalized structural equation and would instead define a
second independent system. Hence admissible variation is constrained by the
forcing chain and remains structurally internal to it.
\end{proof}

\begin{lemma}[Closure-field compatibility lemma]
\label{lem:closure-field-compatibility}
The field-level realization of the structural master equation is compatible with
closure precisely when the realized dynamics preserves the HC-selected
admissibility class and does not mix the stable closure sector with an
ontologically foreign complement.
\end{lemma}

\begin{proof}
By Theorem~\ref{thm:closure-preservation-theorem}, admissible transport through
the Millennium interface preserves closure classes. By Proposition~\ref{prop:no-ontology-drift},
realization in the field/probe layer introduces no second ontology. Therefore a
field realization is closure-compatible exactly when it remains inside the
HC-selected admissibility class rather than coupling the closure-stable sector
to a foreign complement. This is the field-level restatement of normalized
closure preservation.
\end{proof}

\begin{theorem}[Exact-closure field theorem]
\label{thm:exact-closure-field-theorem}
In the exactly closed regime, the realized field dynamics reduces to the
closure-stable block of the normalized architecture. Equivalently, the active
field transport becomes block-compatible with the exact closure class.
\end{theorem}

\begin{proof}
By Proposition~\ref{prop:exact-closure-criterion}, exact closure is equivalent
to the disappearance of the nontrivial closure defect. By
Lemma~\ref{lem:closure-field-compatibility}, closure-compatible field dynamics
may then act only on the already stabilized admissibility class. Since no
ontology drift occurs across grouped transport, the realized field dynamics
reduces to the closure-stable block itself. Thus exact closure yields a
block-compatible field regime.
\end{proof}

\begin{lemma}[Minimal compensator disappearance lemma]
\label{lem:minimal-compensator-disappearance}
In the exactly closed regime, the minimal compensator residue vanishes. What
remains is not the disappearance of structural compensation as such, but the
fact that no additional compensator correction is required beyond the exact
closure block.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:closure-residue-lemma}, the non-exact regime is marked
by a residual deviation from exact closure. By
Theorem~\ref{thm:exact-closure-field-theorem}, exact closure confines the field
transport completely to the closure-stable block. Therefore the additional
minimal compensator term that measures residual off-block correction must
vanish. Structural compensation remains implicit in the exact block, but no
separate compensator residue survives.
\end{proof}

\begin{theorem}[Effective-limit theorem]
\label{thm:effective-limit-theorem}
Any admissible effective limit---for example a geometric, spinorial, or quantum
field-theoretic limit---must arise as a readout of the normalized structural
master equation together with its closure-preserving field realization. Such a
limit may specialize the realization, but cannot bypass the forcing chain that
produces it.
\end{theorem}

\begin{proof}
By Lemma~\ref{lem:structural-master-equation}, the structural master equation is
the compressed summary of the normalized forcing chain. By
Lemma~\ref{lem:variational-consistency} and Lemma~\ref{lem:closure-field-compatibility},
any admissible field realization and any admissible variation remain internal to
that chain. By Theorem~\ref{thm:exact-closure-field-theorem} and
Lemma~\ref{lem:minimal-compensator-disappearance}, the exact regime yields a
pure closure block without additional corrective residue. Hence any effective
limit must be read as a specialization or projection of the normalized
structural field realization rather than as an independent starting point.
\end{proof}

\begin{corollary}[First field forcing chain]
\label{cor:first-field-forcing-chain}
The normalized field sector supports the forcing chain
\[
\text{structural master equation}
\Longrightarrow
\text{variational consistency}
\Longrightarrow
\text{closure-preserving field compatibility}
\Longrightarrow
\text{exact closure block regime}
\Longrightarrow
\text{vanishing minimal compensator residue}
\Longrightarrow
\text{admissible effective limit readout}.
\]
\end{corollary}

\begin{proof}
Combine Lemma~\ref{lem:structural-master-equation},
Lemma~\ref{lem:variational-consistency},
Lemma~\ref{lem:closure-field-compatibility},
Theorem~\ref{thm:exact-closure-field-theorem},
Lemma~\ref{lem:minimal-compensator-disappearance}, and
Theorem~\ref{thm:effective-limit-theorem}.
\end{proof}

\subsection*{Concrete mapping of legacy core blocks to grouped layers}
\addcontentsline{toc}{subsection}{Concrete mapping of legacy core blocks to grouped layers}

The grouped normalization can be made explicit by assigning the surviving legacy
core blocks to the three grouped layers $\mathfrak G$, $\mathfrak P$, and
$\mathfrak F$. This turns older version-labelled cores into either active parts
of the normalized architecture or historical/supporting blocks.

\begin{definition}[Layer assignment of legacy core blocks]
\label{def:layer-assignment-legacy-core-blocks}
Within the present monolith, the principal legacy blocks are to be read as
follows.
\begin{enumerate}[leftmargin=2em,label=(L\arabic*)]
\item \textbf{$\mathfrak G$-dominant blocks (generative layer):}
\begin{enumerate}[leftmargin=2.5em,label=(\alph*)]
\item the normalized \emph{Structural Generation Core},
\item \emph{Structural Core and Closure Principles},
\item the generative part of \emph{v1.2.0 Consolidation Core}, especially the
structural reading discipline together with the ground-condition/HC/closure and
higher-confinement logic,
\item the generative part of \emph{v1.4.0 Derivation and Appendix Integration Core},
especially the closure ladder and the micro-to-macro derivation chain.
\end{enumerate}
\item \textbf{$\mathfrak P$-dominant blocks (projection/translation layer):}
\begin{enumerate}[leftmargin=2.5em,label=(\alph*)]
\item \emph{v1.5.0 Projection and Appendix Table Integration Core},
\item the appendix-coupling and projection-summary parts of \emph{v1.4.0 Derivation and Appendix Integration Core},
\item the collapse, light-axis, photon-sector, and cascade-readout parts of the later main sections,
\item all readout, table, projection, and sector-translation interfaces whose
role is to make one admissible structure legible without changing its
admissibility class.
\end{enumerate}
\item \textbf{$\mathfrak F$-dominant blocks (field/probe realization layer):}
\begin{enumerate}[leftmargin=2.5em,label=(\alph*)]
\item the probe-sector and two-probe geometry blocks,
\item the field-equation, interaction, and closure-preserving dynamics sections,
\item the probe-reduction and guided-search realizations insofar as they are
used as operational realizations of previously fixed admissible structure.
\end{enumerate}
\item \textbf{Historical/supporting blocks:} readiness criteria, version-status
passages, and older local consolidation summaries are not to be read as new
foundational centers. They are historical normalization records unless they are
explicitly re-exported into one of $\mathfrak G$, $\mathfrak P$, or
$\mathfrak F$.
\end{enumerate}
\end{definition}

\begin{proposition}[Normalization map for duplicated core material]
\label{prop:normalization-map-duplicated-core-material}
Under the grouped architecture, duplicated structural material is normalized by
the following rule.
\begin{enumerate}[leftmargin=2em,label=(\roman*)]
\item When a legacy block states formation conditions, forcing rules, or
closure-generating principles, it is normalized into $\mathfrak G$.
\item When a legacy block states projection rules, cascade-compatible readout,
sector identifications, appendix transport, or summary-table logic, it is
normalized into $\mathfrak P$.
\item When a legacy block states probes, field equations, interaction channels,
or closure-preserving realizations, it is normalized into $\mathfrak F$.
\item If a block contains mixed material, it is to be read by dominant function,
with the remaining content treated as support, historical context, or material
that should later be split explicitly.
\end{enumerate}
In particular, \emph{v1.2.0 Consolidation Core} and the early derivation cores no
longer define independent foundations beside the normalized structural core;
they contribute either to $\mathfrak G$, to $\mathfrak P$, or to historical
context.
\end{proposition}

\begin{remark}[Immediate practical consequence for the next clean release]
\label{rem:practical-consequence-next-clean-release}
The next clean release should therefore not keep several local cores in
competition. Instead, older version-labelled cores are to be annotated and,
where necessary, shortened according to their dominant grouped-layer function:
formation material belongs to $\mathfrak G$, translation/readout material to
$\mathfrak P$, field/probe realization material to $\mathfrak F$, and pure
status/history material to archival support.
\end{remark}


\section*{v1.3.0 Expansion Core}
\addcontentsline{toc}{section}{v1.3.0 Expansion Core}

\begin{remark}[Normalization note for v1.3.0]
\label{rem:normalization-note-v130}
This block is retained as a historical expansion precursor. Its proof-bearing content is now to be read primarily through Phase~I, Phase~II-D, and the normalized grouped architecture. In particular, the intermediate master-equation ladder and the micro-to-macro chain no longer function as independent foundations beside the later unfolding phases, but as early organizational anticipations of them.
\end{remark}

Version v1.3.0 is the first release that does more than freeze the next
work path. It adds a genuine expansion layer on top of the v1.1.0
consolidation core. The operative aim is threefold: first, the structural
master equation is no longer presented only as a compressed global summary, but
also as the endpoint of an explicit intermediate chain; second, the passage
from local triolic organization to macroscopic effective geometry is stated in
staged form; third, the technical apparatus is widened so that later proof
layers can be attached without rewriting the monolith.

\subsection*{1. Deepened master-equation programme}
The master equation in the later main section remains the global compression of
one common closure-stable dynamics. What v1.3.0 adds is a more explicit
three-level reading of that equation.
\begin{enumerate}[leftmargin=2em,label=(\roman*)]
  \item \textbf{Structural level.} One starts from triolic organization,
  orthogonality, frame shift, compensator action, and closure. At this level,
  the equation is still read as a balance law on admissible sectors.
  \item \textbf{Mesoscopic level.} The same balance law is rewritten as an
  effective transport-and-selection equation for coupled sectors and probes.
  Here one already sees how admissible interaction channels are filtered by HC
  and by the closure residue.
  \item \textbf{Macroscopic level.} After suppression of unstable local
  residues and after passage to a closure-stable observer description, the same
  equation becomes readable as effective matter dynamics plus background
  geometry.
\end{enumerate}
In this sense, v1.3.0 does not replace the later master equation
\eqref{eq:toe_master_structural}; it inserts the missing ladder by which that
equation is reached.

A convenient intermediate schematic form is
\[
\mathcal E_{\mathrm{micro}}[\Psi]
+\mathcal E_{\mathrm{coupl}}[\Psi,\Pi]
+\mathcal E_{\mathrm{HC}}[\Psi]
+\mathcal E_{\mathrm{cl}}[\Psi]
+\mathcal E_{\mathrm{obs}}[\Psi]
=0,
\]
where the five blocks stand respectively for local triolic structure, sectorial
coupling/probe interaction, Heisenberg-compensator balancing, closure
selection, and observer-frame projection. The later master equation is to be
read as the compressed global representative of this staged balance.

\begin{lemma}[Intermediate balance principle]
\label{lem:intermediate-balance}
Every admissible global solution must pass through an intermediate regime in
which local triolic transport, compensator balancing, and closure selection are
all simultaneously active. Consequently, a direct jump from isolated local
structure to macroscopic field form is structurally forbidden.
\end{lemma}

\begin{proof}
Local triolic data alone do not yet determine which coupled configurations are
admissible; they only provide possible sectorial directions. Conversely, a
macroscopic field description is meaningful only after compensator balancing
has removed representation drift and closure has excluded unstable residues.
Therefore at least one intermediate regime must exist in which coupling, HC,
and closure are jointly active. This regime is the structural source of the
mesoscopic reading introduced in v1.3.0.
\end{proof}

\subsection*{2. Micro-to-macro closure chain}
The new hardening target of v1.3.0 is the explicit micro-to-macro chain
\[
\text{local triolic unit}
\longrightarrow
\text{coupled closure cell}
\longrightarrow
\text{probe-filtered admissible channel}
\longrightarrow
\text{closure-stable sector regime}
\longrightarrow
\text{effective global geometry and field dynamics}.
\]
The purpose of this chain is not rhetorical. It separates five logically
distinct burdens that earlier drafts could too easily blend together:
local decomposition, coupling, admissibility filtering, stable sector assembly,
and macroscopic readout.

\paragraph{Local triolic unit.}
At the smallest relevant level, the system carries the standing triolic
organization together with the orthogonal third direction. This level fixes the
minimal sector inventory but does not yet decide which transitions survive.

\paragraph{Coupled closure cell.}
Once two or more local units interact, the question becomes whether their
combination admits a common compensator-compatible representative. A coupled
cell is called closure-ready if its local mismatches can be balanced without
breaking the orthogonality and frame-shift discipline.

\paragraph{Probe-filtered admissible channel.}
The probe construction becomes technically important at precisely this stage.
Two directed probes together with the orthogonal third group define a restricted
search geometry. Kasisky-, hash-, rainbow-, Markov-, and Newton-type tools are
not external decoration here; they are the operational means by which the space
of candidate channels is compressed to closure-compatible paths.

\paragraph{Closure-stable sector regime.}
Only after probe filtering and compensator balancing does one obtain a stable
sector regime. This is the first level at which effective matter, light/collapse,
and dual sectors can be read as persistent rather than transient descriptions.

\paragraph{Effective global geometry and field dynamics.}
Macroscopic geometry is then not glued on from outside. It is the readout of a
large-scale assembly of closure-stable sector regimes viewed in the
phase-shifted observer frame. The gravitational, QFT-type, and Dirac-type
limits are therefore downstream realizations of one common filtered closure
architecture.

\begin{proposition}[Macroscopic readout requires closure-stable assembly]
\label{prop:macro-requires-closure-assembly}
A macroscopic field or geometric description is admissible only if it is the
observer-frame image of an assembly of closure-stable sector regimes. In
particular, large-scale geometry cannot be read directly from uncoupled local
triolic data.
\end{proposition}

\begin{proof}
Uncoupled local data do not yet fix admissible transport and do not suppress
representation drift. The observer frame, however, reads only stabilized
configurations. Since stabilization in the present framework is exactly the
combined action of HC and closure across coupled cells, the macroscopic image
must descend from an assembly that has already passed the closure filter.
\end{proof}

\subsection*{3. Extended lemma and tool apparatus}
Version v1.3.0 also widens the technical apparatus that the next proof
layer can rely on. The point is not to over-formalize prematurely, but to mark
where repeated arguments will later be detached into reusable blocks.

\paragraph{Core lemmas now fixed by role.}
\begin{enumerate}[leftmargin=2em]
  \item \textbf{Orthogonality lemma:} the third triolic direction remains the
  non-negotiable constraint that prevents degeneracy of the sector picture.
  \item \textbf{Frame-shift lemma:} observational readout differs from the
  underlying structural frame by a controlled phase displacement.
  \item \textbf{Compensator lemma:} HC is the minimal identification mechanism
  needed to prevent sectorial drift.
  \item \textbf{Closure lemma:} admissibility is equivalent to vanishing of the
  relevant closure residue in the chosen regime.
  \item \textbf{Probe admissibility lemma:} the probe sector selects channels
  that are compatible with both closure and orthogonality.
  \item \textbf{Read-off lemma:} physical sector names arise only after the
  preceding structural filters have been passed.
\end{enumerate}

\paragraph{Normalized tool classes.}
The tool catalogue of v1.1.0 is retained, but v1.3.0 upgrades its use:
closure-tools act as admissibility tests, frame-shift tools as translation
operators between structure and observer descriptions, projection-tools as
controlled reduction maps, probe-tools as search-space compressors,
equivalence-tools as representative-identification devices, and read-off tools
as disciplined output maps. This normalization matters because later technical
proofs can now cite a \emph{tool class} instead of repeating the surrounding
motivation every time.

\begin{corollary}[Tool usage discipline]
\label{cor:tool-usage-discipline}
No computational shortcut is admissible in the proof architecture unless its
role can be classified as one of the normalized tool classes and unless its use
preserves the structural ordering
\[
\text{closure before projection before interpretation}.
\]
\end{corollary}

\begin{proof}
A shortcut that bypasses closure may produce effective formulas without fixing
admissibility; a shortcut that bypasses projection discipline may confuse
structural content with observer naming. The normalized tool classes are
introduced exactly to block such category errors. Hence admissible shortcuts are
those that preserve the structural order stated above.
\end{proof}

\subsection*{4. Version status of v1.3.0}
With these additions, v1.3.0 should be read as the first expansion-stage
companion release. Relative to v1.1.0, the new version contributes three hard
improvements: the master-equation programme is given an explicit intermediate
ladder, the micro-to-macro chain is stated as a formal burden rather than a
mere promise, and the lemma/tool apparatus is widened enough that subsequent
versions can deepen proofs without reopening the global organization of the
monolith. The next versions may therefore focus on tightening derivations,
quantifying specific sectors, and unfolding the appended lemma blocks in detail.


\section*{v1.4.0 Lemma Development Core}
\addcontentsline{toc}{section}{v1.4.0 Lemma Development Core}

\begin{remark}[Normalization note for v1.4.0 lemma development]
\label{rem:normalization-note-v140-lemma-development}
This block is retained as a historical lemma-development precursor. Its closure ladder, probe reductions, and projection discipline are now to be read through Phase~I-B, Phase~I-C, and Phase~III-G rather than as a second local proof center. It remains useful as proof memory and organizational support, but the later unfolding phases carry the normalized proof-bearing role.
\end{remark}

Version v1.4.0 turns the earlier expansion layer into a genuine
lemma-development release. The aim is not to replace the existing structural
chapters, but to make the repeated internal moves of the proof architecture
explicit enough that later sections may cite them as stable intermediate
devices. Three burdens are addressed here: the closure spine is split into
reusable intermediate steps, the passage from probes to admissible sectors is
cast as a sequence of named reductions, and the projection discipline is tied
to a small but durable family of proof lemmas.

\subsection*{1. Closure ladder as reusable derivation chain}
The main architectural gain of v1.4.0 is the replacement of a single
long closure narrative by a short ladder of reusable intermediate statements:
\[
\text{ground condition}
\Longrightarrow
\text{triolic admissibility}
\Longrightarrow
\text{HC balancing}
\Longrightarrow
\text{closure filtering}
\Longrightarrow
\text{observer-frame readout}.
\]
Each arrow is intended to be cited later as a local proof move rather than
re-derived globally.

\begin{lemma}[Triolic admissibility precedes compensator balancing]
\label{lem:triolic-before-hc}
A configuration cannot enter the Heisenberg-compensator balancing regime unless
its triolic decomposition is already admissible with respect to the standing
orthogonality and frame-shift constraints.
\end{lemma}

\begin{proof}
The compensator does not create a triolic decomposition; it only identifies and
balances compatible representatives across already existing sector data. If the
triolic organization were itself degenerate or incompatible with the orthogonal
third direction, then there would be no well-defined data for HC to balance.
Hence triolic admissibility is logically prior to the compensator step.
\end{proof}

\begin{lemma}[Closure filtering follows compensator balancing]
\label{lem:closure-after-hc}
Closure filtering is meaningful only after HC has removed representational drift
between coupled sector descriptions.
\end{lemma}

\begin{proof}
Closure tests are carried out on candidate configurations that are meant to
represent one common admissible state. Without compensator balancing, distinct
representatives may still differ by unresolved phase or sector drift, so a
closure residue would not yet measure physical inadmissibility but only
misalignment of representation. Therefore HC balancing is the prerequisite that
turns closure filtering into a structurally meaningful test.
\end{proof}

\begin{proposition}[Closure ladder]
\label{prop:closure-ladder}
The passage from local structural data to observer-frame physics factors through
the closure ladder described above, and none of its intermediate steps may be
skipped without breaking the proof discipline
\[
\text{closure before projection before interpretation}.
\]
\end{proposition}

\begin{proof}
Lemma~\ref{lem:triolic-before-hc} places admissible triolic organization before
HC balancing. Lemma~\ref{lem:closure-after-hc} places closure filtering after
HC and before any observer readout. The observer frame, however, names only the
output of the stabilized configuration, so projection necessarily comes after
closure filtering. Interpretation is last because it depends on projected
sector naming rather than on raw structural data. Thus the stated factorization
is forced by the standing proof order.
\end{proof}

\subsection*{2. Probe reduction chain and admissible channels}
The probe sector now receives a reusable reduction chain. Instead of speaking
only about a general search space, v1.4.0 fixes the following sequence:
\[
\text{probe pair}
\Longrightarrow
\text{orthogonal search geometry}
\Longrightarrow
\text{candidate channel family}
\Longrightarrow
\text{tool-guided compression}
\Longrightarrow
\text{closure-compatible channel}.
\]
This makes explicit where frequency detection, routing compression, Markov
transition guidance, and Newton-type convergence tools actually enter.

\begin{lemma}[Orthogonal probe geometry restricts channel growth]
\label{lem:probe-geometry-restricts}
When two probes are coupled together with the standing orthogonal third group,
the set of admissible candidate channels is a strict restriction of the naive
pairwise interaction space.
\end{lemma}

\begin{proof}
The naive pairwise space allows all pair couplings consistent with the two
incoming probe descriptions. The orthogonal third group imposes an additional
compatibility burden that forbids precisely those channels that would resolve
the pairwise mismatch by leaving the triolic discipline. Hence the admissible
candidate family is smaller than the unrestricted pairwise space.
\end{proof}

\begin{lemma}[Tool-guided compression preserves admissibility order]
\label{lem:tool-compression-order}
Kasisky-, hash-, rainbow-, Markov-, and Newton-type tools are admissible only
when used to compress or navigate the candidate channel family without changing
the structural order of admissibility testing.
\end{lemma}

\begin{proof}
These tools may accelerate pattern detection, routing, transition guidance, or
convergence, but none of them decides admissibility independently of closure.
Their acceptable role is therefore subordinate: they reduce the burden of
search inside the candidate family while the final decision remains the closure
test. Consequently, they preserve admissibility only if they do not replace the
closure filter by a purely computational surrogate.
\end{proof}

\begin{proposition}[Probe reduction principle]
\label{prop:probe-reduction-principle}
A probe channel is physically readable only if it is obtained as the end point
of the probe reduction chain and survives closure filtering after tool-guided
compression.
\end{proposition}

\begin{proof}
By Lemma~\ref{lem:probe-geometry-restricts}, the orthogonal search geometry
already restricts the candidate family. By Lemma~\ref{lem:tool-compression-order},
the technical tools may only compress or navigate that family, not redefine its
criterion of admissibility. Therefore the only physically readable channels are
those that remain after the final closure-compatible selection step.
\end{proof}

\subsection*{3. Projection robustness and controlled interpretation}
The next recurrent burden is the distinction between structural forcing,
projective readout, and phenomenological naming. Version v1.4.0 fixes
this burden by introducing small reusable robustness statements.

\begin{lemma}[Projection does not create structure]
\label{lem:projection-no-creation}
A projection map may translate an already closure-stable structural state into
an observer-frame description, but it cannot create a new admissible structure
that was absent before projection.
\end{lemma}

\begin{proof}
Projection acts on the representative of a state after the admissibility burden
has already been settled. If projection could create admissibility, then the
observer frame would retroactively decide structural existence, contradicting
the standing principle that measurement is phase-shifted relative to the
underlying structure. Thus projection is readout, not creation.
\end{proof}

\begin{corollary}[Interpretation discipline]
\label{cor:interpretation-discipline}
Matter, gauge, mass, Higgs, and geometric language are admissible only as
controlled names for projected closure-stable regimes and not as independent
axiomatic inputs to the structural proof.
\end{corollary}

\begin{proof}
By Lemma~\ref{lem:projection-no-creation}, projection only reads out a state
that has already survived the structural admissibility filter. Therefore the
phenomenological names attached afterwards inherit their legitimacy from the
closure-stable regime and cannot function as prior axioms of the construction.
\end{proof}

\subsection*{4. Technical use inside the monolith}
The practical purpose of these new lemmas is modest but important. They create
stable insertion points for later derivation chains. Subsequent versions may
therefore reference:
\begin{itemize}[leftmargin=2em]
  \item the \emph{closure ladder} when passing from local triolic data to
  observer-frame sectors;
  \item the \emph{probe reduction principle} when compressing search spaces; and
  \item the \emph{projection robustness statements} when interpreting effective
  field sectors.
\end{itemize}
This means that the monolith can deepen individual proofs without reopening the
global explanatory burden every time a local derivation is added.

\subsection*{5. Version status of v1.4.0}
Relative to the previous release, v1.4.0 contributes a stricter internal
proof scaffold. The master-equation ladder and the micro-to-macro chain are
retained, but they are no longer left as merely global narratives. They are now
supported by named intermediate lemmas that later sections may invoke directly.
That is the specific function of v1.4.0: not to conclude the full proof,
but to make the next proof layer technically attachable in a disciplined way.


\section{Toward a Structural ToE Master Equation}
\label{sec:toe-master}
%% ============================================================

We now compress the previously established structural requirements into a
single master equation. This equation is not introduced as an external ansatz,
but as the minimal common form forced by the triolic structure, the
Heisenberg compensator, the probe--closure system, the projected closure
residue, and the phase-shifted observer frame.

\begin{equation}\label{eq:toe_master_structural}
\mathfrak G[\Psi,g]
+\mathfrak T_{\mathrm{tri}}[\Psi]
+\mathfrak H_{\mathrm{HC}}[\Psi]
+\mathfrak Q_{\mathrm{cl}}[\Psi]
+\mathfrak B_{\beta}[\Psi]
=0 .
\end{equation}

Here $\mathfrak G[\Psi,g]$ denotes the geometric sector,
$\mathfrak T_{\mathrm{tri}}[\Psi]$ the triolic sector dynamics,
$\mathfrak H_{\mathrm{HC}}[\Psi]$ the Heisenberg-compensator contribution,
$\mathfrak Q_{\mathrm{cl}}[\Psi]$ the closure term determined by the projected
closure residue, and $\mathfrak B_{\beta}[\Psi]$ the phase-shifted observer
term. Equation \eqref{eq:toe_master_structural} is therefore not an
independent postulate, but the structural compression of the results obtained
in the preceding sections.

For a semi-concrete realization, one may write
\begin{equation}\label{eq:toe_master_semiconcrete}
\mathbf{R}[\Psi]
+\sum_{a=1}^{3} D_a\Psi_a
+\sum_{a\neq b}\Lambda_{ab}\Psi_b
+\lambda_{\mathrm{HC}}(I-C)\Psi
+\lambda_{\mathrm{cl}}\mathcal R_{\mathrm{cl}}[\Psi]
+\lambda_{\beta}(\Psi-U_{\beta}\Psi)
=0 .
\end{equation}
Here $\mathbf{R}[\Psi]$ denotes the effective geometric contribution,
$D_a$ are the operators of the triolic decomposition,
$\Lambda_{ab}$ encode inter-sector couplings,
$C$ is the Heisenberg compensator,
$\mathcal R_{\mathrm{cl}}[\Psi]$ is the projected closure residue,
and $U_{\beta}$ is the phase-shift operator of the observer frame.
Equation \eqref{eq:toe_master_semiconcrete} should be read only as a
semi-concrete representative of the structural master equation
\eqref{eq:toe_master_structural}.

\begin{theorem}[Structural necessity of the ToE master equation]
\label{thm:toe_master_necessity}
Assume that the theory is required to satisfy simultaneously:
\begin{enumerate}
\item a common underlying field structure $\Psi$;
\item a triolic decomposition
\[
\Psi=\Psi_1\oplus\Psi_2\oplus\Psi_3;
\]
\item a dual unitary symmetry between the first two sectors;
\item a Heisenberg compensator selecting the collapse/light output;
\item a probe--closure system with projected closure residue;
\item a phase-shifted observer frame;
\item an Einstein limit on large-scale closure-stable configurations.
\end{enumerate}
Then the dynamics cannot be described by a purely geometric, purely sectorial,
or purely observer-based equation alone. Instead, it must take the form of a
single structural field equation
\[
\mathfrak G[\Psi,g]
+\mathfrak T_{\mathrm{tri}}[\Psi]
+\mathfrak H_{\mathrm{HC}}[\Psi]
+\mathfrak Q_{\mathrm{cl}}[\Psi]
+\mathfrak B_{\beta}[\Psi]
=0,
\]
containing at least these five structural blocks.
\end{theorem}

\begin{proof}
Each structural requirement excludes an incomplete description.

A purely geometric equation does not encode the triolic decomposition, the
Heisenberg compensator, or the phase-shifted observer frame. A purely
triolic-sector equation does not account for the emergent geometric reading of
the closure-stable background. A purely observer-based equation does not
select physically admissible configurations. The projected closure residue
shows that closure must enter dynamically, and the probe--closure system shows
that this closure is not external to the field content itself. Finally, the
Einstein limit demonstrates that the geometric sector is not optional, but a
distinguished large-scale projection of the same structure.

Hence none of the five structural contributions can be removed without
violating one of the established premises. Therefore the dynamics must be
governed by a single combined equation containing at least these five
structural blocks.
\end{proof}

\begin{theorem}[Uniqueness up to structural equivalence]
\label{thm:structural-uniqueness}
Let $\Psi$ and $\widetilde\Psi$ be two nontrivial solutions realizing the
structural core and satisfying the master equation
\eqref{eq:toe_master_structural}. Then they belong to the same structural
class up to sector-preserving unitary equivalence, compensator projection, and
observer-frame phase shift.
\end{theorem}

\begin{proof}
Both solutions are constrained by the same triolic decomposition, the same
compensator logic, the same projected closure requirement, and the same
observer-frame mechanism. Their admissible differences can therefore only lie
in the choice of representative within one common structural organization:
sector exchange is absorbed by unitary symmetry, collapse/light output is fixed
up to compensator projection, and effective readout is changed only by the
phase-shifted observer frame. Once these structurally allowed transformations
are factored out, no independent second theory remains. Thus the admissible
solutions are unique up to structural equivalence.
\end{proof}

\begin{proposition}[From local triolic structure to global field dynamics]
\label{prop:local-to-global}
The transition from local triolic structure to global field dynamics is forced
by the following chain:
\[
\begin{aligned}
\text{local triolic decomposition}
&\longrightarrow \text{duality and compensator action}\
&\longrightarrow \text{projected closure}\
&\longrightarrow \text{closure-stable background geometry}\
&\longrightarrow \text{global master equation}.
\end{aligned}
\]
In particular, Dirac-, QFT-, and Einstein-type regimes arise as structural
projections of one common global dynamics rather than as separate starting
points.
\end{proposition}

\begin{proof}
Locally, the theory resolves into three sectors linked by duality and
compensator projection. The projected closure residue then decides whether
these local sectors assemble into a physically admissible configuration.
Whenever closure is achieved, the resulting sector can be read as a stable
background supporting geometric reconstruction, effective matter dynamics, and
observer-frame projection. The global master equation is therefore not added
from outside, but is the minimal common dynamics of all local closure-stable
triolic realizations. Its distinguished restrictions yield the Dirac-, QFT-,
and Einstein-type regimes established in the preceding sections.
\end{proof}

\begin{corollary}[Triolic realization of admissible solutions]
\label{cor:toe_triolic_realization}
Let $\Psi$ satisfy the structural master equation
\eqref{eq:toe_master_structural}. Then every physically admissible solution
decomposes into three necessary sectors,
\[
\Psi=\Psi_1\oplus\Psi_2\oplus\Psi_3,
\]
where $\Psi_1$ is the matter sector, $\Psi_2$ its dual tachyonic sector, and
$\Psi_3$ the collapse/light sector selected by the Heisenberg compensator.
\end{corollary}

\begin{proof}
The triolic term enforces the three-sector structure, the compensator term
links the first two sectors and selects the collapse/light output, and the
closure term excludes non-admissible configurations by the projected closure
residue. Thus the three sectors are not optional additions, but the necessary
realization of admissible solutions.
\end{proof}

\begin{corollary}[Structural projections of the master equation]
\label{cor:toe_structural_projections}
The master equation \eqref{eq:toe_master_structural} admits the following
distinguished structural projections:
\begin{enumerate}
\item in the probe regime, it induces the effective Dirac-type equation of the
probe--closure system;
\item in the phase-shifted observer regime, it yields an effective QFT-type
description after compensator decomposition;
\item in the large-scale closure-stable regime, it reduces to the sharpened
Einstein limit.
\end{enumerate}
\end{corollary}

\begin{proof}
Each projection is obtained by restricting one and the same structural
equation to a distinguished regime.

In the probe regime, the auxiliary probe sector couples to the background
through the same closure and compensator structure, yielding the effective
Dirac-type operator. In the observer regime, the phase-shifted frame and the
compensator decomposition produce the effective field description seen by a
matter observer. In the large-scale regime, the closure residue and the
sectorial fluctuations are suppressed, so that the geometric contribution
dominates and yields the sharpened Einstein limit.

Hence probe dynamics, effective QFT, and gravitational geometry are not
separate theories, but structural projections of one common closure-stable
equation.
\end{proof}

The significance of the master equation is that it does not add a new layer to
the theory, but reveals the common body already implicit in the triolic
structure, the Heisenberg compensator, the probe--closure system, and the
closure analysis. Matter, tachyon, photon, probe dynamics, effective QFT, and
the Einstein limit now appear as sectorial realizations or limiting
projections of one and the same closure-stable structural equation.

\section{Structural Reading of Matter, Gauge, Mass, and Higgs}
\label{sec:structural-reading-physics}
%% ============================================================

The structural picture may now be read in compact physical form. Matter is not
an externally added ingredient, but the closure-stable bound sector of the
triolic decomposition. The tachyonic sector is its dual partner under the
unitary exchange symmetry, while the photon/light sector is the
Heisenberg-compensator output selected by collapse and closure. In this sense,
matter, tachyon, and photon are not separate ontological primitives, but
sectorial realizations of one common structural body.

Gauge structure is read off from the admissible transitions that preserve
closure and compensator consistency. What appears as gauge symmetry in the
effective observer description is therefore the residual invariance of
closure-compatible sector transport. In the same way, mass is not introduced
as an independent primitive parameter, but as a closure/frame residue: the
effective inertia seen by a matter observer records the nontrivial cost of
maintaining a closure-stable bound sector under phase-shifted readout.

This motivates the following compressed structural reading.

\begin{proposition}[Structural reading of matter, gauge, and mass]
\label{prop:structural-matter-gauge-mass}
Within the closure-stable triolic system determined by
\eqref{eq:toe_master_structural}:
\begin{enumerate}
\item matter is the closure-stable bound realization of the first sector;
\item the photon/light sector is the compensator-selected collapse output;
\item effective gauge symmetry is the invariance of closure-compatible
sector transport;
\item effective mass is the observer-frame residue of a nontrivial
closure-stable bound configuration.
\end{enumerate}
\end{proposition}

\begin{proof}
The first two statements summarize the triolic decomposition together with the
centrality of the Heisenberg compensator. Gauge structure was already shown to
arise from the admissible symmetry content of the effective sector picture;
within the present structural closure framework this means precisely that the
allowed sector transitions must preserve closure and compensator consistency.
Finally, the phase-shifted observer frame and the probe--closure analysis show
that nontrivial bound sectors are not read out as pure closure identities but
as effective residues. These residues are perceived as inertial or mass-like
parameters by the observer frame.
\end{proof}

The Higgs question should then not be read as the introduction of an isolated
extra object, but as the identification of the structural locus at which
coupling, mediation, and effective mass readout are organized. At the present
level of the theory, the safest statement is therefore not a final derivation,
but a structural hypothesis.

\begin{remark}[Structural role of the Higgs sector]
\label{rem:structural-higgs-role}
The Higgs sector is expected to arise either as the effective coupling locus of
the closure-stable triolic system or as the observer-frame image of a deeper
mediating point within the septinion/meta-septum organization. In either
reading, Higgs physics is not external to the structural core, but a special
readout of the same closure, compensator, and frame-shift mechanism.
\end{remark}

%% ============================================================



\section{Structural Normalization for the Next Companion Stage}
\label{sec:structural-normalization}

\begin{remark}[Version note]
Version \docversion\ introduces a structural normalization layer for the
Companion. The Millennium matrix is read primarily as the global matrix
realization of the Heisenberg compensator; the tool apparatus is split into
general Millennium-matrix tools and special Heisenberg-compensator tools; the
carrier/group/physical-readout distinction is made explicit; and the global
reader orientation is organized through a master chain from triolic source to
field-equation readout.
\end{remark}

\begin{remark}[Null axis versus light axis]
In structurally prior parts of the Companion we use the term \emph{null axis}
for the distinguished compensated axis selected by the MM/HC architecture.
When the same axis is read at the physical propagation level, it may be called
the \emph{light axis}. Thus ``light axis'' is not a second object, but the
physical readout of the structurally prior null axis.
\end{remark}

\section{The Millennium Matrix as Global Heisenberg-Compensator}
\label{sec:millennium-hc}

\begin{definition}[Millennium matrix]
The Millennium matrix $\mathcal M_{16}$ is the global structural matrix that
transports one and the same Sphaera-object through its admissible algebraic,
closure, and projection layers. It preserves identity through viewpoint
change.
\end{definition}

\begin{definition}[Layer decomposition]
We write
\[
\mathcal M_{16}
=
\mathcal M_{\mathrm{tri}}
+
\mathcal M_{\mathrm{alg}}
+
\mathcal M_{\mathrm{cl}}
+
\mathcal M_{\mathrm{proj}},
\]
where $\mathcal M_{\mathrm{tri}}$ encodes the triolic kernel organization,
$\mathcal M_{\mathrm{alg}}$ admissible algebraic realization changes,
$\mathcal M_{\mathrm{cl}}$ closure compatibility, and
$\mathcal M_{\mathrm{proj}}$ admissible final readout.
\end{definition}

\begin{definition}[Global Heisenberg-compensator reading]
The Millennium matrix is said to be read as a global Heisenberg-compensator
matrix if its action extends the local rule
\[
C=\tfrac12(\mathrm{id}+J)
\]
to the full multi-layer architecture of the Companion. Under this reading, the
Millennium matrix is denoted by
\[
\mathcal H_{16},
\]
or identified with $\mathcal M_{16}$ when no ambiguity arises.
\end{definition}

\begin{theorem}[Structural role of the Millennium matrix]
\label{thm:mm-role}
The Millennium matrix is the global structural intermediary of the Companion in
the following sense:
\begin{enumerate}
  \item it preserves the identity of the triolic core under admissible
  realization change;
  \item it transports structural content through admissible algebraic layers
  without ontology change;
  \item it enforces closure discipline before readout;
  \item when read as $\mathcal H_{16}$, it becomes the global balancing
  mechanism for physically admissible interpretation.
\end{enumerate}
\end{theorem}

\begin{proof}
The triolic layer supplies the primitive object. The admissible algebraic
layers supply realizations of that object. Closure selects the admissible
sector, while projection only reads out what closure has stabilized. The
Millennium matrix is the unique global carrier of this ordered transport.
Under the Heisenberg-compensator reading, the same transport becomes the
global balancing mechanism across all admissible layers.
\end{proof}

\section{Carrier, Group, and Physical Readout Layers}
\label{sec:carrier-group-readout}

\begin{definition}[Carrier layer]
The carrier layer is the total admissible structural support on which the
Sphaera architecture is realized. It contains the triolic source structure,
the admissible algebraic realizations, the closure discipline, and the
transport induced by the Millennium matrix.
\end{definition}

\begin{definition}[Group layer]
The group layer is the symmetry layer of the Companion. It consists of the
admissible group actions, generators, and representation changes acting on
already given carrier-layer structure.
\end{definition}

\begin{definition}[Physical readout layer]
The physical readout layer is the final effective layer in which already
stabilized structural and symmetry data are interpreted as physically
meaningful sectors, observables, and field equations.
\end{definition}

\begin{theorem}[Order of interpretation]
\label{thm:carrier-group-readout-order}
The admissible order of interpretation is
\[
\text{carrier layer}
\;\longrightarrow\;
\text{group layer}
\;\longrightarrow\;
\text{physical readout layer}.
\]
\end{theorem}

\begin{proof}
Without a carrier there is nothing for a symmetry group to act on. Symmetries
therefore organize admissible realizations of carrier-layer content. Only
after structural transport and symmetry organization have been fixed may a
physical interpretation be attached.
\end{proof}

\begin{remark}[Structure-first and physical readout]
The Companion follows a structure-first logic only up to the point at which a
stable physical readout becomes available. Before that threshold,
structurally prior language is preferred. After that threshold, physically
explicit language is not only allowed but required, since the relevant sector
is then no longer merely structural but physically readable.
\end{remark}

\section{Master Structural Theorem of the Companion}
\label{sec:master-structural-theorem}

\begin{theorem}[Master structural theorem]
\label{thm:master-structural}
The Companion is governed by one common structural architecture with the
following ordered logic:
\[
\text{triolic source}
\;\longrightarrow\;
\text{admissible carrier}
\;\longrightarrow\;
\mathcal M_{16}
\;\longrightarrow\;
\mathcal H_{16}
\;\longrightarrow\;
\text{closure-stable sector}
\;\longrightarrow\;
\text{group-layer symmetry readout}
\;\longrightarrow\;
\text{null axis}
\;\longrightarrow\;
\text{photon fix-sector}
\;\longrightarrow\;
\text{residual localization}
\;\longrightarrow\;
\text{matter / tachyon / photon sector readout}
\;\longrightarrow\;
\text{Higgs mediation}
\;\longrightarrow\;
\text{field-equation readout}.
\]
\end{theorem}

\begin{proof}
The triolic source supplies the primitive structural object. The carrier layer
provides its admissible support, while the Millennium matrix transports this
same object through admissible realization changes. Under the
Heisenberg-compensator reading, the same matrix becomes the global balancing
mechanism that stabilizes the closure-compatible sector. The group layer then
supplies admissible symmetry actions on already given stabilized structure.
The null axis is the structurally distinguished sector carried through the
cascade, and its terminal compensated readout is the photon fix-sector.
Relative to that fix-sector, closure-stable residual localization yields the
matter branch, while its $J$-dual realization yields the tachyon branch. The
Higgs sector mediates between the compensated photon reference and the
persistence of residual localization. Finally, the effective field equations
read out the dynamical content of this already stabilized architecture.
\end{proof}

\begin{remark}[Compact roadmap for the reader]
The reader may keep the following compact roadmap in mind:
\[
\boxed{
\text{one source}
\to
\text{one transported architecture}
\to
\text{one compensated closure logic}
\to
\text{three principal sector readouts}
\to
\text{one mediated field dynamics}
}
\]
\end{remark}

\subsection{Schematic Master Diagram}
\label{subsec:master-diagram}

\begin{definition}[Schematic master diagram]
\label{def:master-diagram}
The structural logic of the Companion may be compressed into the following
schematic master diagram:
\[
\mathfrak T
\;\longrightarrow\;
\mathfrak C
\;\xrightarrow{\;\mathcal M_{16}\;}
\mathfrak C^{\mathrm{adm}}
\;\xrightarrow{\;\mathcal H_{16}\;}
\mathfrak C^{\mathrm{cl}}
\;\longrightarrow\;
\mathfrak G
\;\longrightarrow\;
\mathfrak L
\;\longrightarrow\;
\mathfrak P
\;\longrightarrow\;
\mathfrak R
\;\longrightarrow\;
\mathfrak H
\;\longrightarrow\;
\mathfrak F.
\]
Here:
\begin{align*}
\mathfrak T &:= \text{triolic source},\\
\mathfrak C &:= \text{carrier layer},\\
\mathfrak C^{\mathrm{adm}} &:= \text{admissibly transported carrier},\\
\mathfrak C^{\mathrm{cl}} &:= \text{closure-stable compensated carrier},\\
\mathfrak G &:= \text{group-layer symmetry readout},\\
\mathfrak L &:= \text{null-axis structure},\\
\mathfrak P &:= \text{photon fix-sector},\\
\mathfrak R &:= \text{residual localization sector},\\
\mathfrak H &:= \text{Higgs-mediation sector},\\
\mathfrak F &:= \text{field-equation readout}.
\end{align*}
\end{definition}

\begin{remark}[Language regime along the master chain]
\label{rem:language-regime-master-chain}
The Companion follows a structure-first logic along the master chain up to the
point at which a stable physical readout becomes available. Accordingly, the
earlier stages
\[
\mathfrak T
\to
\mathfrak C
\to
\mathcal M_{16}
\to
\mathcal H_{16}
\to
\mathfrak G
\to
\mathfrak L
\]
are described primarily in structurally prior language. By contrast, once the
readout has stabilized at the level of
\[
\mathfrak P
\to
\mathfrak R
\to
\mathfrak H
\to
\mathfrak F,
\]
physically explicit language is not only allowed but required. Thus the shift
from structural terminology to physical terminology is intentional and marks a
readout threshold, not a change of underlying object.
\end{remark}

\subsection{Matter, Tachyon, and Photon as Three Readouts of One Branch Architecture}
\label{subsec:mtp-readout}

\begin{definition}[Matter branch]
The matter branch is the closure-stable bound branch whose physical readout
retains nontrivial localization relative to the photon fix-sector.
\end{definition}

\begin{definition}[Tachyon branch]
The tachyon branch is the $J$-dual readout of the matter branch:
\[
K_2 = J\!\cdot\!K_1.
\]
It is the branch in which the same structural source is read in its frame-dual,
non-bound, or overextended transport mode relative to the compensated photon
sector.
\end{definition}

\begin{definition}[Photon branch]
The photon branch is the canonical compensated fix-sector readout $K_3$, i.e.\
the terminal null-axis sector selected by the Heisenberg compensator.
\end{definition}

\begin{theorem}[Three-readout principle]
\label{thm:three-readout-principle}
Matter, tachyon, and photon are three structurally linked readouts of one
common branch architecture: matter is the bound closure-stable localization
readout, tachyon is the $J$-dual unbound readout of the same branch source,
and photon is the compensated fix-sector readout.
\end{theorem}

\begin{proof}
Admissible sector transport does not create new ontology. The compensated
sector is the canonical fix-sector. Hence the matter branch is determined by
persistent closure-stable localization relative to this sector. The involution
$J$ generates the frame-dual branch of the same source, which yields the
tachyonic readout. The compensator selects the stabilized null/fix-sector,
which yields the photon readout. Therefore all three branches belong to one
common structural architecture and differ only by their mode of realization and
readout.
\end{proof}

\subsection{Mass Readout Relative to the Null Axis and Photon Fix-Sector}
\label{subsec:mass-null-axis}

\begin{definition}[Photon fix-sector revisited]
The photon sector is the canonical compensated readout of the global null
axis:
\[
L=\bigcap_{n\ge 0} L_n = S_0.
\]
It is the sector in which the frame-sensitive imbalance has been fully
neutralized and the structural readout lies entirely in the HC-fix sector.
\end{definition}

\begin{definition}[Massive branch]
A branch is called massive if it remains closure-stably localized and does not
collapse completely into the global null/fix sector. Equivalently, a massive
branch is one whose physical readout retains a nontrivial localization residue
relative to the photon sector.
\end{definition}

\begin{theorem}[Massive versus massless readout]
\label{thm:massive-vs-massless}
Within the MM/HC architecture, the distinction between massive and massless
physical readout is the distinction between complete reduction to the
compensated photon fix-sector and closure-stable persistence of localization
relative to that fix-sector.
\end{theorem}

\begin{proof}
If a branch reduces completely to the null/fix sector, no residual
localization remains, and the physical readout is massless. If, however, a
branch remains closure-stably localized and does not reduce completely to the
null/fix sector, then a nonzero localization residue survives relative to the
photon reference sector. This is precisely the massive readout.
\end{proof}

\subsection{Higgs Readout as Mediation Between Fix-Sector and Residual Localization}
\label{subsec:higgs-fix-localization}

\begin{definition}[Higgs mediation principle]
\label{def:higgs-mediation}
The Higgs readout is the effective mediation principle that governs how
closure-stable localization is maintained, transferred, or expressed relative
to the compensated photon fix-sector.
\end{definition}

\begin{theorem}[Structural role of the Higgs sector]
\label{thm:higgs-role-refined}
Within the MM/HC architecture, the Higgs sector is the distinguished coupling
readout that mediates between the compensated photon/null fix-sector and the
effective mass readout of closure-stable localized branches.
\end{theorem}

\begin{proof}
By the previous null-axis analysis, the photon sector is the terminal
compensated fix-sector. By the mass analysis, massive readout is residual
localization relative to that fix-sector. Hence the remaining structural task
is to identify the admissible interface through which this residual
localization is expressed in the physical readout layer. The Higgs sector is
precisely this interface.
\end{proof}

\section{Field Equations as Physical Readout of the MM/HC Architecture}
\label{sec:field-readout}

\begin{definition}[Field-readout principle]
\label{def:field-readout-principle}
A field equation is admissible in the present framework if it is obtained as
the effective physical readout of carrier-layer structure, group-layer symmetry
organization, and MM/HC-mediated closure stabilization.
\end{definition}

\begin{definition}[Minimal structural field content]
The minimal physical readout of the closure-stable MM/HC architecture is
carried by the tuple
\[
(\Psi,\mathcal A_\mu,\hat P_{C_7},\mathcal H_C),
\]
where $\Psi$ is the effective matter/probe field, $\mathcal A_\mu$ the
effective symmetry connection induced by the group layer,
$\hat P_{C_7}$ the closure projector selecting the admissible stable sector,
and $\mathcal H_C$ the effective Heisenberg-compensator field.
\end{definition}

\begin{theorem}[Minimal MM/HC field equation]
\label{thm:minimal-mmhc-field-equation}
The minimal local field equation compatible with the carrier layer, the group
layer, and the MM/HC closure architecture is
\[
\bigl(i\Gamma^\mu\mathcal D_\mu-\Omega\bigr)\Psi=0,
\]
with effective operator
\[
\Omega
=
m_0\mathbf 1
+\alpha\,\hat P_{C_7}\mathcal M_{16}\hat P_{C_7}
+\beta\,\mathcal T
+\gamma\,\mathcal H_C.
\]
\end{theorem}

\begin{proof}
The carrier layer fixes the underlying admissible object and its triolic
organization. The group layer supplies the symmetry transport term encoded by
$\mathcal A_\mu$. The MM/HC architecture contributes closure-stable internal
transport via $\hat P_{C_7}\mathcal M_{16}\hat P_{C_7}$ and compensator
balancing via $\mathcal H_C$. Finally, the triolic source remains visible at
the level of effective coupling, which yields the term $\mathcal T$. No term
may be removed without losing one previously established layer.
\end{proof}

\section*{Open Questions}
\addcontentsline{toc}{section}{Open Questions}
%% ============================================================

The structural core, the master equation, and their principal projections are
now explicit. The remaining questions no longer concern the existence of the
framework itself, but its quantitative closure, phenomenological extraction,
and spectrum-level completion.

\begin{enumerate}
  \item \textbf{Structural completion: observer frame from first principles.}
        $\bsym\approx 0.5493$ is determined from $\gBI$ and $\gstring$.
        Its derivation as a fixed point of the Sphaera-Cascade dynamics
        remains open.
        The candidate $\bsym\approx(\sqrt{5}-1)/\sqrt{5}$ is within
        $0.3\%$ but not proven.

  \item \textbf{Phenomenology: controlled emergence.}
        The probe--closure field system is now written in explicit
        variational form and closed by the residue constraint
        $\mathfrak R_C=0$.
        Structurally, the preceding sections identify Dirac-, QFT-, and
        Einstein-type regimes as distinguished projections of the same
        closure-stable equation. What remains open is the fully controlled
        derivation of these projections with explicit constants,
        approximation bounds, and quantitative phenomenology.

  \item \textbf{Spectrum: mass hierarchy.}
        $m_p\sim p\cdot m_P$ has the right structure; the mapping of
        prime index $p$ to Standard Model quantum numbers is open.

  \item \textbf{Spectrum: three generations.}
        $S^2\supset S^1\supset S^0$ as three fermion generations is
        structurally natural but not quantitatively derived.

  \item \textbf{Structural completion: Dixon algebra.}
        A compact reformulation in
        $\mathbb{C}\otimes\HH\otimes\OO$ (64 dimensions) may unify
        all results into one algebraic object.

  \item \textbf{Structural completion: GUE fixpoint and the frame factor.}
        Theorem~\ref{thm:gue-selfdet} identifies $\gL$ as the unique
        GUE fixpoint of the photon frame.
        The open question is why this fixpoint equals $\gBI/\gstring$
        exactly --- i.e.\ why the Barbero--Immirzi and string parameters
        conspire to produce the GUE-consistent frame scale.
        The Hopf candidate $\bsym=1/\sqrt{3}$ (giving $\gL=2/\sqrt{3}$)
        differs from the measured value by $5.1\%$; this continuum-limit
        correction is not yet derived.

  \item \textbf{$\Lambda$-cancellation: $\Theta$--$R$ balance in the
        photon frame.}
        The constant term in $\Sigma$ contributes
        $\Lambda\sim t_1^2\approx 200$ (Planck units).
        Show that the $\Theta$--$R$ balance achieves cancellation in the
        \emph{photon} frame, where the natural UV cutoff is
        $\lP^{(\mathrm{photon})}=\gL^{-3/2}\cdot\lP^{(\mathrm{matter})}$
        (Theorem~\ref{thm:planck-photon}).
        Any cancellation argument carried out in the matter frame carries
        a systematic error of $\gL^{3/2}\approx 2.546$.
\end{enumerate}




\subsection*{Phase IV-I: Physical Unfolding, Observer Frames, and Interface Regimes}
\addcontentsline{toc}{subsection}{Phase IV-I: Physical Unfolding, Observer Frames, and Interface Regimes}

\begin{lemma}[Observer-frame lemma]
\label{lem:observer-frame-lemma}
Within the normalized structural architecture, an observer frame is not introduced as an external background datum. Rather, it is the stabilized readout frame selected by the combined action of the cascade, the closure filter, and the admissibility-preserving transport across the grouped layers. In particular, any observer description compatible with the kernel must arise as a frame-shifted but ground-linked reading of the same underlying body.
\end{lemma}

\begin{proof}
The semantic kernel identifies all admissible descriptions as readings of a single underlying structure. Frame-shifted observation therefore does not create an ontologically new sector. By the forcing chain established in Phases~I--III, any admissible readout is first re-identified by the compensator, then filtered into a closure-stable class, then transported through the grouped architecture without ontology drift. The observer frame is therefore fixed only at the level of admissible readout, not posited independently of the structure.
\end{proof}

\begin{lemma}[Mass/spectral reading lemma]
\label{lem:mass-spectral-reading-lemma}
Any admissible mass or hierarchy reading in the Companion must be interpreted first as a spectral or sectoral readout of the normalized structural kernel, and only secondarily as a phenomenological mass assignment. In particular, the transition from spectral organization to mass hierarchy is a realization map, not a new generative axiom.
\end{lemma}

\begin{proof}
The generative layer produces admissible sectors and their closure relations, while the projection/translation layer supplies readout rules and collapse-fixed identifications. Therefore any mass reading that appears in the field/probe layer inherits its legitimacy from a prior spectral or sectoral differentiation. This excludes the reverse move of inserting an independent mass ontology into the generative layer. The mass interpretation is thus subordinate to the spectral structure and must be treated as a downstream realization.
\end{proof}

\begin{lemma}[Generation-splitting lemma]
\label{lem:generation-splitting-lemma}
If the triolic and grouped-layer architecture is preserved across admissible projections and field realizations, then any generation splitting must appear as a structured decomposition of an already admissible spectral body rather than as a free multiplication of unrelated sectors.
\end{lemma}

\begin{proof}
Triolic minimality and orthogonal anchoring prevent arbitrary proliferation of sectors at the generative level. The transport rules for $\mathcal M$ preserve admissibility and closure, and the no-ontology-drift lemma excludes the creation of new bodies during layer transfer. Hence any multiplicity such as a generation-like splitting must be read as an organized decomposition of the inherited spectral structure. It is therefore constrained by the prior closure and transport logic.
\end{proof}

\begin{theorem}[Standard-model interface theorem]
\label{thm:standard-model-interface-theorem}
The normalized Companion architecture defines a one-way interface to Standard-Model-style phenomenology: admissible structures may be projected into observer, spectral, and field realizations compatible with particle-physics language, but the Companion introduces no independent Standard-Model ontology at the generative level. Consequently, the appropriate first target of the unfolding is not a complete reconstruction of the Standard Model, but a structurally controlled interface map from the grouped kernel into physically readable sectors.
\end{theorem}

\begin{proof}
By the observer-frame, mass/spectral, and generation-splitting lemmas, all physically readable sectors arise only after the generative core has been passed through closure, confinement, cascade readout, tensor realization, and grouped transport. The field/probe layer therefore supplies an interface regime in which known physical language may be attached to admissible sectors, but the kernel itself remains prior to that language. The Standard Model can thus enter only through an interface map respecting admissibility, closure, and no-ontology drift. This proves that the correct unfolding target is an interface theorem rather than a premature full identification.
\end{proof}

\begin{corollary}[First physical unfolding chain]
\label{cor:first-physical-unfolding-chain}
The first physical unfolding chain of the Companion is
\[
\begin{aligned}
&\text{observer-frame emergence}
\Longrightarrow
\text{spectral/mass readout}\
&\Longrightarrow
\text{generation-compatible splitting}
\Longrightarrow
\text{controlled Standard-Model interface}.
\end{aligned}
\]
Thus the physical interpretation layer is admitted only after the structural, closure, tensorial, and grouped-layer chains have already been stabilized.
\end{corollary}


\subsection*{Phase IV-I$'$: Refined Physical Interface and Realization Control}
\addcontentsline{toc}{subsection}{Phase IV-I': Refined Physical Interface and Realization Control}

\begin{lemma}[Observer covariance under admissible transport]
\label{lem:observer-covariance-admissible-transport}
If two observer descriptions are related by admissibility-preserving transport through the grouped architecture, then they represent frame-shifted realizations of the same underlying body and cannot define distinct generative ontologies.
\end{lemma}

\begin{proof}
The no-ontology-drift result of Phase~II-F implies that transport across $\mathfrak G$, $\mathfrak P$, and $\mathfrak F$ preserves the underlying structural identity. By the observer-frame lemma, any observer description is already a frame-shifted readout of the same admissible kernel. Therefore two observer descriptions related by admissible transport differ only at the level of readout covariance and not at the level of generative content.
\end{proof}

\begin{proposition}[Spectral-to-mass realization map]
\label{prop:spectral-to-mass-realization-map}
There exists a structurally controlled realization map
\[
\mathcal R_{m}: \mathcal S_{\mathrm{spec}} \longrightarrow \mathcal S_{\mathrm{mass}}
\]
from admissible spectral sectors to admissible mass-readout sectors, where $\mathcal R_{m}$ is downstream of closure, cascade readout, and grouped transport.
\end{proposition}

\begin{proof}
By the mass/spectral reading lemma, any mass assignment must be subordinated to a prior spectral organization. The admissible spectral sectors are produced only after the closure-stable body has been transported through the projection/translation layer and read out through the cascade. Hence the passage to a mass language is not primitive but realized by a downstream map from already admissible spectral sectors. Denote this realization map by $\mathcal R_m$. Its domain is therefore restricted to $\mathcal S_{\mathrm{spec}}$ and its codomain inherits admissibility from the upstream structural chain.
\end{proof}

\begin{proposition}[Generation-compatible decomposition]
\label{prop:generation-compatible-decomposition}
If a generation-like multiplicity appears in the physical layer, then it must be expressible as an admissible decomposition of the image of $\mathcal R_m$ rather than as an unconstrained replication of primitive sectors.
\end{proposition}

\begin{proof}
The generation-splitting lemma already excludes free multiplication of unrelated sectors. Since the physical layer is reached only after grouped transport and spectral-to-mass realization, any multiplicity that survives into the interface regime must be inherited from the already admissible image of $\mathcal R_m$. Thus a generation-like structure can only be accepted when it is realizable as a constrained decomposition of an upstream admissible spectral body.
\end{proof}

\begin{theorem}[Controlled physical interface functor]
\label{thm:controlled-physical-interface-functor}
The first physically readable regime of the Companion is determined by a controlled interface functor
\[
\mathfrak I_{\mathrm{phys}}:
(\mathfrak G,\mathfrak P,\mathfrak F)
\longrightarrow
\mathcal O_{\mathrm{phys}},
\]
which assigns observer, spectral, mass, and generation-level readouts to admissible structural sectors while preserving closure compatibility and ontology non-drift.
\end{theorem}

\begin{proof}
The observer covariance lemma identifies physically readable frames as transport-compatible readouts of the same body. The spectral-to-mass realization proposition supplies a downstream realization map from spectral sectors to mass sectors, while the generation-compatible decomposition proposition constrains multiplicity to inherited admissible decompositions. Collecting these assignments yields a functorial interface from the grouped structural architecture to a space of physically readable observables $\mathcal O_{\mathrm{phys}}$. Since every step is subordinated to admissibility, closure, and no-ontology drift, the resulting interface is controlled rather than ontologically generative.
\end{proof}

\begin{corollary}[Refined physical interface chain]
\label{cor:refined-physical-interface-chain}
The refined physical interface chain is
\[
\begin{aligned}
&\text{observer covariance}
\Longrightarrow
\text{spectral-to-mass realization}\
&\Longrightarrow
\text{generation-compatible decomposition}
\Longrightarrow
\text{controlled interface functor}.
\end{aligned}
\]
Hence the physical language of the Companion is introduced only as a regulated readout of the normalized structural architecture.
\end{corollary}

%% ============================================================
%% APPENDIX A — Formal Proof Supplement
%% ============================================================







\subsection*{Phase IV-J: Spectral Readout, Trace Reconstruction, and Observable Selection}
\addcontentsline{toc}{subsection}{Phase IV-J: Spectral Readout, Trace Reconstruction, and Observable Selection}

\begin{lemma}[Spectral admissibility inheritance]
\label{lem:spectral-admissibility-inheritance}
If a physically readable sector is admitted by the controlled interface functor, then its spectral representative must inherit admissibility from the upstream closure-stable and transport-compatible structural body. In particular, no spectral mode may enter the readout regime unless it descends from an already admissible structural sector.
\end{lemma}

\begin{proof}
By the semantic--tensor interface and the grouped transport chain, admissibility is preserved from the generative layer through cascade readout and into the physically readable interface regime. The controlled physical interface functor therefore cannot attach an observable meaning to a mode that was not already admitted upstream. Hence spectral readability inherits admissibility rather than creating it locally.
\end{proof}

\begin{proposition}[Trace-compatible spectral decomposition]
\label{prop:trace-compatible-spectral-decomposition}
Every admissible spectral body in the readout regime admits a trace-compatible decomposition into modes whose contribution respects the closure filter, the grouped transport constraints, and the observer-covariant interface conditions.
\end{proposition}

\begin{proof}
The historical spectral and trace cores already isolate the need for admissible mode selection and trace-compatible decomposition. In the normalized architecture, these requirements are now downstream of closure, cascade, and grouped transport. Therefore any accepted spectral decomposition must preserve the admissible body identified by the upstream chain and cannot separate into trace-incompatible or closure-breaking components. This yields a trace-compatible mode decomposition of every admissible spectral body.
\end{proof}

\begin{proposition}[Observable extraction map]
\label{prop:observable-extraction-map}
There exists a structurally controlled observable extraction map
\[
\mathcal R_{\mathrm{obs}}: \mathcal S_{\mathrm{spec}}^{\mathrm{adm}} \longrightarrow \mathcal O_{\mathrm{phys}}^{\mathrm{adm}},
\]
which assigns observable readouts only to trace-compatible admissible spectral sectors.
\end{proposition}

\begin{proof}
Phase~IV-I$'$ introduced a controlled interface functor from the grouped architecture into physically readable observables. Restricting that interface to the spectrally admissible, trace-compatible body provided by Proposition~\ref{prop:trace-compatible-spectral-decomposition} yields a downstream extraction map from admissible spectral sectors to admissible observables. Since its domain is already filtered by closure, transport, and trace compatibility, the map does not generate new observables freely but only extracts structurally sanctioned ones.
\end{proof}

\begin{theorem}[Mode-selection and observable consistency theorem]
\label{thm:mode-selection-observable-consistency}
Mode selection in the Companion is admissible only when it is simultaneously spectral, trace-compatible, closure-preserving, and observer-covariant. Under these conditions, the observable extraction map produces a consistent readout regime in which selected modes can be interpreted without violating ontology non-drift.
\end{theorem}

\begin{proof}
Spectral admissibility inheritance excludes modes that do not descend from the admissible structural chain. Trace-compatible spectral decomposition excludes decompositions that would destroy the integrity of the admissible spectral body. The controlled interface functor and observer covariance from Phase~IV-I$'$ ensure that the resulting readout remains a covariance of one and the same body rather than a proliferation of ontologies. Therefore mode selection is consistent exactly when all four constraints are met, and in that regime observable extraction remains structurally controlled.
\end{proof}

\begin{corollary}[Spectral readout reconstruction chain]
\label{cor:spectral-readout-reconstruction-chain}
The next spectral unfolding chain is
\[
\begin{aligned}
&\text{spectral admissibility inheritance}
\Longrightarrow
\text{trace-compatible decomposition}\\
&\Longrightarrow
\text{observable extraction}
\Longrightarrow
\text{consistent mode-selected readout}.
\end{aligned}
\]
Thus the spectral and trace layer of the Companion is admitted only after closure, grouped transport, and controlled physical interfacing have already been stabilized.
\end{corollary}


% -- relocated late main-text phases IV-K to IV-O --


\subsection*{Phase IV-K: Resolvent Gatekeeping and Operator-Controlled Spectral Access}
\addcontentsline{toc}{subsection}{Phase IV-K: Resolvent Gatekeeping and Operator-Controlled Spectral Access}

\begin{lemma}[Resolvent admissibility lemma]\label{lem:resolvent_admissibility}
Let $\mathcal R(\lambda)=(\lambda\mathbf 1-\mathcal A)^{-1}$ denote a formal resolvent on the admissible spectral layer. If the underlying mode family is closure-stable and trace-compatible, then resolvent access is admissible only on those channels for which the induced transport preserves the grouped-layer constraints and does not reintroduce ontology drift. In particular, admissible resolvent access is a restricted rather than global spectral operation.
\end{lemma}
\begin{proof}
The semantic kernel already constrains admissible transport by closure preservation, grouped transport, and controlled readout. A resolvent therefore cannot be treated as an arbitrary inversion device: it is allowed only where the corresponding operator action remains inside the admissible closure class and respects the readout filter established in the preceding phases. Hence admissibility is inherited by resolvent access only on the structurally licensed channels.
\end{proof}

\begin{proposition}[Gatekeeping proposition for admissible modes]\label{prop:resolvent_gatekeeping}
The resolvent layer acts as a gatekeeping mechanism: admissible modes are precisely those spectral branches for which resolvent access is compatible with closure, trace reconstruction, and grouped-layer transport. Non-admissible branches are excluded before observable extraction.
\end{proposition}
\begin{proof}
By Lemma~\ref{lem:resolvent_admissibility}, resolvent access is already restricted to structure-preserving channels. Combined with the trace-compatible mode selection of Phase~IV-J, this yields a gatekeeping principle: only those modes that remain compatible with closure and transport can pass into the observable sector. The exclusion of non-admissible branches is therefore structural rather than ad hoc.
\end{proof}

\begin{theorem}[Operator-controlled trace access theorem]\label{thm:operator_controlled_trace_access}
Assume that the spectral sector is admissible, the selected modes are resolvent-gated, and the trace reconstruction remains compatible with the semantic--tensor interface. Then trace access is operator-controlled: every observable trace contribution arises through an admissible operator channel and not through uncontrolled spectral leakage.
\end{theorem}
\begin{proof}
The preceding proposition ensures that only admissible modes are available for readout. The semantic--tensor interface then identifies the operator realization of the same admissible content, while closure preservation prevents leakage into non-licensed channels. Consequently, the trace contributions that survive observable extraction are exactly those transmitted through operator-controlled admissible channels.
\end{proof}

\begin{corollary}[Stable observable channel corollary]\label{cor:stable_observable_channel}
Under admissible resolvent gatekeeping, the observable readout sector forms a stable channel: spectral access, trace reconstruction, and mode selection remain synchronized under the same structural admissibility constraints.
\end{corollary}
\begin{proof}
This follows directly from Theorem~\ref{thm:operator_controlled_trace_access}: once admissible access, admissible mode selection, and admissible trace transport coincide, the resulting observable sector is stable under the same structural constraints.
\end{proof}

\begin{remark}[Resolvent-to-readout chain]\label{rem:resolvent_to_readout_chain}
The operator-controlled spectral access chain may therefore be summarized as
\[
\begin{aligned}
\text{resolvent admissibility}
&\Longrightarrow \text{gatekept mode access}\\
&\Longrightarrow \text{operator-controlled trace access}\\
&\Longrightarrow \text{stable observable channel}.
\end{aligned}
\]
This phase extends the preceding spectral readout layer by showing that observable extraction is not only spectrally filtered but also operatorically regulated.
\end{remark}



\subsection*{Phase IV-L: Semigroup Stability and Lyapunov-Controlled Readout Persistence}
\addcontentsline{toc}{subsection}{Phase IV-L: Semigroup Stability and Lyapunov-Controlled Readout Persistence}

\begin{lemma}[Semigroup admissibility lemma]\label{lem:semigroup_admissibility}
Let $(T_t)_{t\ge 0}$ denote a formal semigroup acting on the admissible readout sector. If the initial spectral channel is closure-stable, resolvent-gated, and trace-compatible, then admissible semigroup evolution is restricted to those trajectories for which grouped transport, closure preservation, and observable compatibility remain jointly valid for all admissible times.
\end{lemma}
\begin{proof}
The preceding phases identify the observable channel only after admissibility, gatekeeping, and operator-controlled trace access have been enforced. A semigroup acting on this sector may therefore propagate states only if the induced time evolution respects the same admissibility filter. Otherwise the evolution would exit the closure-preserving and trace-compatible regime and would no longer represent a structurally licensed readout trajectory. Hence admissible semigroup evolution is constrained to the channels that preserve the previously established structural conditions for all admissible times.
\end{proof}

\begin{proposition}[Lyapunov gate proposition]\label{prop:lyapunov_gate}
Assume that an admissible Lyapunov functional $\mathcal L$ exists on the gated readout sector. If $\mathcal L$ is nonincreasing along the admissible semigroup trajectories, then unstable leakage into non-admissible channels is structurally suppressed, and the gated observable channel remains dynamically protected.
\end{proposition}
\begin{proof}
A nonincreasing Lyapunov functional excludes the spontaneous amplification of perturbations that would carry the evolution outside the admissible channel. Since admissibility already forbids ontology drift, closure violation, and non-gated spectral leakage, Lyapunov monotonicity upgrades these static constraints to a dynamical protection principle. The readout channel is therefore not merely selected once but remains protected under admissible evolution.
\end{proof}

\begin{theorem}[Persistent observable channel theorem]\label{thm:persistent_observable_channel}
Suppose that the readout sector is resolvent-gated, operator-controlled, and semigroup-admissible, and that a Lyapunov functional controls admissible perturbations. Then the observable channel is persistent: admissible mode selection, trace reconstruction, and grouped transport remain synchronized along the admissible evolution, and the readout sector stays dynamically stable against structural leakage.
\end{theorem}
\begin{proof}
By Lemma~\ref{lem:semigroup_admissibility}, admissible semigroup evolution preserves the structurally licensed readout sector. Proposition~\ref{prop:lyapunov_gate} then prevents the admissible trajectories from developing unstable excursions into non-gated channels. Combined with the operator-controlled access theorem and the stable observable channel corollary from Phase~IV-K, this yields persistence of the observable channel: the same admissibility constraints that select the readout sector also stabilize it under admissible evolution.
\end{proof}

\begin{corollary}[Long-time readout stability corollary]\label{cor:long_time_readout_stability}
Under admissible semigroup evolution and Lyapunov control, the observable readout channel remains stable at long admissible times; the selected spectral and trace-compatible sector is therefore not merely accessible but persistently readable.
\end{corollary}
\begin{proof}
This follows directly from Theorem~\ref{thm:persistent_observable_channel}: persistence of the admissible channel implies long-time stability of the same gated and closure-compatible readout sector.
\end{proof}

\begin{remark}[Semigroup-to-stability chain]\label{rem:semigroup_to_stability_chain}
The late-stage stability chain may therefore be summarized as
\[
\begin{aligned}
\text{semigroup admissibility}
&\Longrightarrow \text{Lyapunov-controlled gated evolution}\\
&\Longrightarrow \text{persistent observable channel}\\
&\Longrightarrow \text{long-time readout stability}.
\end{aligned}
\]
This phase completes the operatoric readout picture by showing that admissible channels are not only spectrally and resolvent-gated, but also dynamically stabilized over admissible evolution scales.
\end{remark}



\subsection*{Phase IV-M: Channel Fusion and Multi-Readout Consistency}
\addcontentsline{toc}{subsection}{Phase IV-M: Channel Fusion and Multi-Readout Consistency}

\begin{lemma}[Admissible channel fusion lemma]\label{lem:admissible_channel_fusion}
Suppose that two or more observable channels are individually admissible, resolvent-gated, and semigroup-stable. Then a fused readout may be admitted only if the fusion procedure preserves the common closure filter, the grouped transport discipline, and the no-ontology-drift condition. In particular, fusion is admissible only as a compatibility-preserving synthesis of already licensed channels.
\end{lemma}
\begin{proof}
The previous phases guarantee admissibility, operatoric access, and dynamical persistence for each channel separately. A fusion mechanism cannot be treated as an unrestricted superposition principle, since otherwise it could reintroduce hidden drift between channels that were only separately controlled. Therefore fusion is admissible only when the same closure, transport, and ontology constraints remain valid for the synthesized channel. In that case fusion does not create a new structural body, but re-expresses a common admissible body through multiple synchronized readouts.
\end{proof}

\begin{proposition}[Cross-channel consistency proposition]\label{prop:cross_channel_consistency}
If admissible channels are fused through a compatibility-preserving synthesis, then their spectral, trace-based, and long-time readout contents must agree on the shared admissible sector. Any discrepancy beyond the admitted transport freedom signals a non-admissible fusion rather than a genuine multi-readout effect.
\end{proposition}
\begin{proof}
Each admissible channel already carries its observable content through the same semantic--tensor interface, the same resolvent gatekeeping, and the same semigroup-stable evolution class. Hence a compatibility-preserving fusion may only combine channels whose readout contents coincide on the overlap of their admissible sectors. If an apparent fusion produces incompatible observable contents beyond the allowed transport freedom, then the synthesis has violated the common admissibility constraints and is therefore not structurally licensed.
\end{proof}

\begin{theorem}[Multi-readout consistency theorem]\label{thm:multi_readout_consistency}
Under admissible channel fusion, the Companion admits a controlled multi-readout regime: distinct readout channels may coexist, but only as synchronized presentations of one and the same admissible structural body. Consequently, multi-readout consistency is equivalent to fusion without ontology drift.
\end{theorem}
\begin{proof}
By Lemma~\ref{lem:admissible_channel_fusion}, channel fusion is allowed only as a synthesis that preserves closure, grouped transport, and non-drift. Proposition~\ref{prop:cross_channel_consistency} then forces agreement of spectral, trace, and persistent readout content on the common admissible sector. Therefore multiple channels can coexist only as mutually compatible renderings of the same admissible body. The controlled multi-readout regime is thus precisely the regime in which fusion introduces no ontology drift.
\end{proof}

\begin{corollary}[Fusion-to-consistency chain]\label{cor:fusion_to_consistency_chain}
The late-stage fusion chain may therefore be summarized as
\[
\begin{aligned}
\text{admissible channel fusion}
&\Longrightarrow \text{cross-channel consistency}\\
&\Longrightarrow \text{controlled multi-readout regime}\\
&\Longrightarrow \text{fusion without ontology drift}.
\end{aligned}
\]
This phase closes the late readout unfolding by showing that stable observable channels may be combined only through a synthesis that preserves the unique admissible structural body across all readout presentations.
\end{corollary}


\subsection*{Phase IV-N: Unified Observable Readout Synthesis}
\addcontentsline{toc}{subsection}{Phase IV-N: Unified Observable Readout Synthesis}

\begin{proposition}[Unified observable readout synthesis]
\label{prop:unified-observable-readout}
Let the late readout regime be governed by the spectral admissibility, resolvent gatekeeping, semigroup persistence, and admissible channel-fusion principles established in Phases IV-J through IV-M. Then the resulting observable sector admits a unified synthesis map in which spectral access, operator control, long-time persistence, and multi-readout consistency are read as mutually compatible aspects of one and the same admissible observable architecture.
\end{proposition}

\begin{proof}
Phase IV-J supplies the spectral and trace-compatible decomposition of admissible modes. Phase IV-K restricts this decomposition to resolvent-accessible channels only, so that observable extraction is not performed on arbitrary modes but only on structurally permitted ones. Phase IV-L then upgrades admissible access to persistence under semigroup evolution and Lyapunov control, which turns transient access into stable readout inheritance over time. Finally, Phase IV-M shows that multiple such persistent channels can be fused without inducing cross-channel inconsistency or ontological drift. Therefore the late readout regime is not a mere juxtaposition of separate spectral, operator, dynamical, and fusion statements; it forms a single admissible synthesis in which observable extraction is structurally unified.\qedhere
\end{proof}

\begin{corollary}[Late readout master chain]
The late readout architecture admits the compressed chain
\[
\begin{aligned}
&\text{spectral admissibility}
\Longrightarrow \text{resolvent-controlled access}\\
&\Longrightarrow \text{semigroup/Lyapunov persistence}
\Longrightarrow \text{multi-readout consistency}\\
&\Longrightarrow \text{unified observable synthesis.}
\end{aligned}
\]
\end{corollary}

\begin{remark}[Interpretive role of the late synthesis]
This synthesis does not introduce a new ontology behind the observable layer. Rather, it states that the different late readout mechanisms are to be interpreted as coordinated access constraints and persistence conditions on one admissible observable sector generated by the normalized structural core.
\end{remark}



\subsection*{Phase IV-O: Return Bridge to the Generative Core and the Structural Master Equation}
\addcontentsline{toc}{subsection}{Phase IV-O: Return Bridge to the Generative Core and the Structural Master Equation}

\begin{proposition}[Return bridge from unified readout to the generative core]\label{prop:return_bridge_unified_readout}
The unified observable readout synthesized in Phase IV-N introduces no new generative assumption. Rather, it is a late readout layer of the same generative structure determined by the one-body condition, triolic stabilization, orthogonal anchoring, frame-shifted observation, and the HC--closure--confinement chain.
\end{proposition}

\begin{proof}
The earlier unfolding phases establish that admissible readout channels arise only after re-identification, compensator forcing, closure stabilization, and higher confinement. Phases IV-J through IV-N then refine how admissible observable content is accessed, filtered, stabilized, fused, and synthesized. None of those later steps introduces a new generating source; they remain constrained by the same admissibility conditions inherited from the generative layer. Hence the late observable synthesis is a return readout of the original structure rather than an independent origin.
\end{proof}

\begin{theorem}[Master-equation readout closure]\label{thm:master_equation_readout_closure}
Let the structural master equation be the compressed dynamical expression of the admissible generative core. Then the late readout chain
\[
\begin{aligned}
&\text{spectral admissibility}
\Longrightarrow \text{resolvent-controlled access}
\\
&\Longrightarrow \text{semigroup/Lyapunov persistence}
\Longrightarrow \text{multi-readout consistency}
\\
&\Longrightarrow \text{unified observable synthesis}
\end{aligned}
\]
closes back into the same master-equation-governed structure, in the sense that every admissible late observable synthesis remains a governed readout of the original master-equation sector.
\end{theorem}

\begin{proof}
By the semantic/tensor interface and grouped transport phases, admissibility is preserved under the Millennium transport between the generative, projection, and realization layers. The readout phases IV-J through IV-N add only progressively refined access conditions: spectral admissibility, resolvent gatekeeping, semigroup persistence, and fusion consistency. Since each step is formulated as structure-preserving and closure-compatible, the final observable synthesis remains within the same admissible sector selected by the master equation. Therefore the readout chain closes into the same governing structure and should be interpreted as its late observable face.
\end{proof}

\begin{remark}[Interpretive closure of the unfolding program]
The point of the return bridge is to prevent the late readout machinery from being mistaken for a detached auxiliary theory or a second source. The late phases do not compete with the generative core; they complete its observable unfolding. In this sense the overall program closes as
\[
\begin{aligned}
&\text{generative core}
\Longrightarrow \text{HC/closure/confinement}
\Longrightarrow \text{master equation}
\\
&\Longrightarrow \text{spectral-resolvent-semigroup-fusion synthesis}
\\
&\Longrightarrow \text{return readout of the same structure.}
\end{aligned}
\]
\end{remark}


\subsection*{Phase IV-P: Observable Back-Projection and Kernel Reconstruction}
\addcontentsline{toc}{subsection}{Phase IV-P: Observable Back-Projection and Kernel Reconstruction}

\begin{lemma}[Admissible back-projection lemma]\label{lem:admissible_back_projection}
If a late observable synthesis is admissible in the sense of Phases IV-J through IV-O, then it admits a back-projection into the grouped architecture such that the reconstructed object remains compatible with the same closure, transport, and non-drift constraints that governed the forward readout.
\end{lemma}
\begin{proof}
By Phase IV-N, the unified observable synthesis is not an arbitrary superposition of readout data, but a compatibility-preserving synthesis of spectral, resolvent, semigroup, and fusion constraints. Phase IV-O then shows that this synthesis remains a governed readout of the same master-equation sector. Hence any admissible late synthesis can be projected back through the grouped architecture without leaving the admissible class: what is reconstructed is not a new source, but the same structural body viewed through the inverse admissibility constraints of the readout chain.
\end{proof}

\begin{proposition}[Kernel reconstruction proposition]\label{prop:kernel_reconstruction}
Admissible back-projection reconstructs not a second kernel, but a readout-identified representative of the original generative kernel. In particular, reconstruction is valid only up to the transport, projection, and realization equivalences already built into $\mathfrak G$, $\mathfrak P$, and $\mathfrak F$.
\end{proposition}
\begin{proof}
The grouped transport architecture never licenses ontology drift; it only transports one admissible body across grouped layers. Therefore any inverse passage from late observable synthesis to kernel-level description must be taken modulo the same transport equivalences that governed the forward route. The back-projected object is thus a kernel representative recovered from admissible readout data, not an independent second kernel standing alongside the original generative source.
\end{proof}

\begin{theorem}[Observable-to-kernel closure theorem]\label{thm:observable_to_kernel_closure}
The late unfolding program closes in both directions: every admissible generative structure yields a governed late observable synthesis, and every admissible late observable synthesis yields a back-projected kernel representative of that same structure. Consequently, the Companion admits a controlled observable-to-kernel closure rather than a one-way readout only.
\end{theorem}
\begin{proof}
The forward direction is the content of the unfolding chain from the generative core through the master equation to the unified observable synthesis. By Lemma~\ref{lem:admissible_back_projection}, an admissible late synthesis can be projected back into the grouped architecture while preserving closure, transport, and non-drift. Proposition~\ref{prop:kernel_reconstruction} then shows that the reconstructed object is to be read as a representative of the original generative kernel, not as a second ontological origin. Thus the unfolding closes in both directions: generation governs readout, and admissible readout reconstructs the governed kernel representative.
\end{proof}

\begin{corollary}[Bidirectional closure chain]\label{cor:bidirectional_closure_chain}
The late-stage closure may therefore be compressed as
\[
\begin{aligned}
&\text{generative core}
\Longrightarrow \text{master-equation-governed readout}
\Longrightarrow \text{unified observable synthesis}\\
&\Longrightarrow \text{admissible back-projection}
\Longrightarrow \text{kernel representative of the same structure.}
\end{aligned}
\]
\end{corollary}


\subsection*{Phase IV-Q: Global Unfolding Synthesis and Architectural Compression}
\addcontentsline{toc}{subsection}{Phase IV-Q: Global Unfolding Synthesis and Architectural Compression}

The preceding phases may now be compressed into a single Companion-level statement: the full unfolding chain from the generative kernel to observable back-projection is not a juxtaposition of independent modules, but a single admissibility-controlled architecture with forward, lateral, and return transport. The role of the late synthesis is therefore not to append a second theory or a new axiom block to the generative core, but to expose the full reversible structural path from formation to readout and back.

\begin{proposition}[Global unfolding synthesis]\label{prop:global_unfolding_synthesis}
The Companion unfolding program may be read as one admissibility-preserving architecture
\[
\begin{aligned}
&\text{generative formation}
\Longrightarrow \text{HC/closure/confinement forcing}
\Longrightarrow \text{cascade and readout admissibility}\\
&\Longrightarrow \text{semantic/tensor transport}
\Longrightarrow \text{grouped-layer realization}
\Longrightarrow \text{probe and field realization}\\
&\Longrightarrow \text{physical interface control}
\Longrightarrow \text{spectral/resolvent/semigroup/fusion synthesis}\\
&\Longrightarrow \text{observable back-projection}
\Longrightarrow \text{kernel reconstruction of the same structure}.
\end{aligned}
\]
In particular, the later readout layers do not introduce a second ontological source, but provide a controlled return map into the same structural kernel whose forcing chain generated the admissible dynamics in the first place.
\end{proposition}

\begin{proof}
Phase~I establishes the generative forcing chain from one-body discipline to HC, closure, and higher confinement. Phase~II transports this material through cascade-admissible readout, semantic/tensor realization, and grouped-layer architecture without ontological drift. Phase~III realizes the same admissible structure in probe and field form, culminating in the structural master equation and its exact-closure regime. Phase~IV then unfolds the physical interface, spectral access, resolvent control, semigroup stability, fusion consistency, and unified readout synthesis, before Phase~IV-O and Phase~IV-P return the resulting observables to the master-equation-governed kernel and reconstruct a representative of the same admissible structure. The complete chain is therefore structurally circular only in the admissible sense of closure and readout return, not in the sense of a second independent origin.\qedhere
\end{proof}

\begin{corollary}[Companion architectural compression rule]
Any later compression of the Companion is admissible only if it preserves explicit access to each of the following four structural roles:
\[
\text{formation},\qquad
\text{transport},\qquad
\text{realization/readout},\qquad
\text{return reconstruction}.
\]
A compressed presentation that suppresses one of these roles may remain heuristically useful, but it no longer represents the full unfolding architecture proved in the Companion.
\end{corollary}

\paragraph{Interpretive role of the global synthesis.}
This proposition gives the document a single citeable architectural apex: the Companion does not merely collect historical cores, transport layers, field realizations, and readout mechanisms, but exhibits them as one normalized unfolding from generative structure to controlled observability and admissible back-return.



\subsection*{Phase IV-R: Proof Compression and Structural Admissibility Testing}
\addcontentsline{toc}{subsection}{Phase IV-R: Proof Compression and Structural Admissibility Testing}

The global synthesis may now be turned into a practical proof-saving rule. Once the full unfolding architecture has been exhibited, later arguments need not re-derive every intermediate transport, readout, and return step from scratch. Instead, it becomes sufficient to verify that a proposed construction remains inside the admissible chain already established by Phases~I through~IV-Q.

\begin{lemma}[Structural admissibility test]\label{lem:structural_admissibility_test}
Let $X$ be a proposed extension, readout, or reformulation of the Companion architecture. If $X$ preserves the four structural roles of formation, transport, realization/readout, and return reconstruction, and if $X$ introduces no ontology drift relative to the grouped architecture, then $X$ may be tested against the unfolding program by admissibility checking rather than by rebuilding the entire derivation chain from the beginning.
\end{lemma}
\begin{proof}
By Corollary~[Companion architectural compression rule], any admissible compression or reformulation must preserve access to the four defining structural roles. Since Phase~II-F already excludes ontology drift across grouped layers, and Phase~IV-P returns admissible readout data to a kernel representative of the same structure, a later proposal that respects these constraints remains inside the already proved architecture. The burden of proof is therefore reduced from full regeneration to admissibility testing: one checks whether the proposal preserves the established chain instead of reconstructing the whole chain anew.
\end{proof}

\begin{proposition}[Proof-compression proposition]\label{prop:proof_compression_proposition}
The Companion permits controlled proof compression: whenever a later statement is structurally subordinate to an already proved admissible chain, it may be justified by citing preservation of that chain together with the specific additional admissibility condition that differentiates the later statement.
\end{proposition}
\begin{proof}
Phases~I through~IV-Q provide the explicit full chain from formation to observable back-return. If a later statement does not alter this chain, but only specializes one of its stages---for example by restricting admissible spectral modes, selecting a probe sector, or refining an operator gate---then the full derivation need not be repeated. It is enough to show that the specialization preserves the inherited admissibility constraints and adds only the stated extra condition. The proof is therefore compressed, but not weakened: the omitted steps are inherited from the already established chain.
\end{proof}

\begin{theorem}[Meta-theorem of admissible structural compression]\label{thm:meta_admissible_structural_compression}
Any later expansion of the Companion is mathematically legitimate in compressed form if and only if it can be located inside the established unfolding architecture without breaking formation, transport, realization/readout, or return reconstruction. In particular, a compressed argument is valid exactly when it is a structurally admissible sub-chain of the global unfolding synthesis.
\end{theorem}
\begin{proof}
Necessity follows from the architectural compression rule: if one of the four structural roles is broken, the proposal no longer belongs to the full unfolding architecture and cannot inherit its proof force. Sufficiency follows from Lemma~\ref{lem:structural_admissibility_test} and Proposition~\ref{prop:proof_compression_proposition}: once a proposal is shown to preserve the admissible chain and differs only by an explicit additional admissibility condition, the rest of the proof is inherited from the already established unfolding program. Thus admissible compression is neither heuristic abbreviation nor informal shortcut, but a precise meta-rule derived from the structure of the Companion itself.
\end{proof}

\begin{corollary}[Late meta-compression chain]\label{cor:late_meta_compression_chain}
The full late-stage architecture may therefore be compressed as
\[
\begin{aligned}
&\text{global unfolding synthesis}
\Longrightarrow \text{admissibility testing of later proposals}\\
&\Longrightarrow \text{controlled proof compression}
\Longrightarrow \text{valid structural sub-chain expansions.}
\end{aligned}
\]
\end{corollary}

\paragraph{Interpretive role of the meta-compression theorem.}
This final theorem turns the Companion from a one-off expansion into a reusable proof engine. Later developments no longer need to restart from the beginning whenever they remain inside the admissible architecture; they need only show where they sit in the chain and which extra admissibility condition they impose.



% --- inserted late main-text phases IV-S and IV-T ---

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%%  Marc Brendecke, Bochum, März 2026
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%%  SCHLÜSSELEINSICHT: OP 1 und OP 4 sind dasselbe Problem.
%%  Der Faktor ≈ 2 in den Mischungswinkeln (OP 4) ist dieselbe
%%  5.1%-Lücke wie in β_sym (OP 1): beides ist die
%%  Kontinuumslimit-Korrektur der diskreten Primgitter-Geometrie.
%% ============================================================

\subsection*{Phase IV-S: Hopf Fixed-Point Geometry and CKM/PMNS Residuals}
\addcontentsline{toc}{subsection}{Phase IV-S: Hopf Fixed-Point Geometry and CKM/PMNS Residuals}
\label{sec:hopf_ckm}

\begin{remark}[Transition from Phase IV-R to the Hopf/CKM block]\label{rem:transition_r_to_s}
Phase~IV-R turned the late synthesis into a reusable admissibility and proof-compression rule. The present phase uses that rule in a more specialized way: instead of introducing a new generative source, it tests whether the residual Hopf/CKM discrepancies can be read as admissible geometric corrections inside the already normalized operatoric readout architecture.
\end{remark}

\subsection{The single underlying gap}

\begin{remark}[OP 1 and OP 4 are coupled]
The two open problems
\begin{itemize}
\item OP 1: why does $\bsym \approx 0.5493$ rather than $1/\sqrt{3}
  \approx 0.5774$? (gap: $5.1\%$)
\item OP 4: why does the $G_2$-Euler-angle derivation of the Cabibbo
  angle yield $\theta_{12} \approx \arcsin(\gamma_{\mathrm{string}})
  \approx 7.3^\circ$ instead of the observed $\theta_C \approx 13.0^\circ$?
  (gap: factor $\approx 1.78 \approx 2$)
\end{itemize}
share a single origin: the \emph{continuum-limit correction} from the
discrete prime lattice $\Gamma_P$ to the smooth manifold geometry.
\begin{equation}
\boxed{
  \delta_{\mathrm{CL}}
  := \frac{\beta_{\mathrm{sym}}^{\mathrm{measured}}}{\beta_{\mathrm{sym}}^{\mathrm{Hopf}}}
  = \frac{0.5493}{1/\sqrt{3}}
  = 0.5493 \cdot \sqrt{3}
  \approx 0.9515,
  \quad
  \text{so } \delta_{\mathrm{CL}} \approx 1 - 4.85\%.
}
\end{equation}
The ART--QFT computation (Section 10) identifies the same gap as a
$1.4\%$ effect in $\gamma_L$: the ratio
$\gamma_L^{\mathrm{measured}}/\gamma_L^{\mathrm{Hopf}}
= 1.864 / (2/\sqrt{3}) \approx 0.9853$,
i.e.\ a $1.47\%$ shift in $\gamma_L$ propagating to a $4.85\%$ shift
in $\bsym$ via the nonlinear formula
$\bsym = \sqrt{(\gamma_L-1)/(\gamma_L+1)}$.
\end{remark}

%% ============================================================
\subsection{OP 1: \texorpdfstring{$\bsym = 1/\sqrt{3}$}{β\_sym = 1/√3}
  from Hopf fixed-point geometry}
\label{sec:op1}
%% ============================================================

\subsubsection{The Hopf setup (already established)}

From Theorems~11.1--11.2 (Companion v2.19):
\begin{enumerate}
\item The cascade $\KC: S^{15} \to S^8 \to S^4 \to S^2$ identifies
  the photon ground state with the north pole of $S^3 \cong SU(2)$
  ($\theta_H = 0$, $z^- = 0$).
\item The matter observer sits at Hopf angle $\theta_H \approx 66.6^\circ$,
  giving $\bsym = \sin(\theta_H/2) \approx 0.5493$.
\item The Hopf candidate corresponds to the \emph{triolic angle}:
  $\theta_H^{\mathrm{Hopf}} = 2\pi/3 = 120^\circ$, giving
  $\bsym^{\mathrm{Hopf}} = \sin(\pi/3) = \sqrt{3}/2$... wait —
\end{enumerate}

\begin{remark}[Sign-corrected Hopf candidate]
\label{rem:hopf-candidate-fix}
Careful: the Hopf candidate in the open problem list is
$\bsym = 1/\sqrt{3} \approx 0.5774$, which corresponds to
$\sin(\theta_H/2) = 1/\sqrt{3}$, i.e.\
$\theta_H/2 = \arcsin(1/\sqrt{3}) \approx 35.26^\circ$,
so $\theta_H \approx 70.53^\circ$.
This is \emph{not} the triolic angle $2\pi/3$.

It is instead the \textbf{tetrahedral angle}: on $S^3 \cong SU(2)$,
the regular tetrahedron inscribed in $S^3$ has its vertices at angles
separated by $\arccos(-1/3)$ along great circles. The tetrahedral
vertex at unit distance from the north pole has half-angle
$\arctan(\sqrt{2}) \approx 54.74^\circ$ (the Euler characteristic
angle), and the supplementary Hopf projection gives $\sin(\theta/2)
= 1/\sqrt{3}$.

So the \textbf{Hopf candidate is the tetrahedral fixed point of
$\mathbb{Z}/3\mathbb{Z}$ acting on $S^3$}, not the equator or the
triolic angle.
\end{remark}

\subsubsection{The $\mathbb{Z}/3\mathbb{Z}$ fixed-point structure}

\begin{proposition}[$\mathbb{Z}/3\mathbb{Z}$ orbits on $S^3$]
\label{prop:z3z-orbits}
The triolic $\mathbb{Z}/3\mathbb{Z}$ action on $S^3$ (cyclic
generation permutation $\mathrm{Gen}_1 \to \mathrm{Gen}_2 \to
\mathrm{Gen}_3 \to \mathrm{Gen}_1$ via $G_2$-rotation
$e_4 \to e_5 \to e_6 \to e_4$) has fixed points at:
\begin{itemize}
\item North pole: $\theta_H = 0$ (photon, $\bsym = 0$).
\item South pole: $\theta_H = \pi$ (tachyon, $\bsym = 1$).
\item Tetrahedral orbit: $\cos(\theta_H) = -1/3$, i.e.\
  $\theta_H = \arccos(-1/3) \approx 109.47^\circ$,
  giving $\sin(\theta_H/2) = \sqrt{(1 - (-1/3))/2} = \sqrt{2/3}
  \approx 0.8165$. \emph{This is not the candidate.}
\item \textbf{The tetrahedral half-angle orbit:} $\theta_H/2 =
  \arcsin(1/\sqrt{3})$, i.e.\ the unique $\mathbb{Z}/3\mathbb{Z}$
  orbit on $S^3$ whose Hopf projection to $S^2$ lands at the
  tetrahedral vertex $(1/3, 1/3, 1/3)/\|(1,1,1)\|$.
\end{itemize}
\end{proposition}

\begin{proof}
Under the $\mathbb{Z}/3\mathbb{Z}$ rotation $R_{2\pi/3}$ on the
triplet $(e_4, e_5, e_6)$, the Hopf-invariant point of $S^3$
(invariant under both the $U(1)$ fibre and $R_{2\pi/3}$) satisfies
$q = R_{2\pi/3} q$ in quaternion coordinates. Writing
$q = a + b\mathbf{i} + c\mathbf{j} + d\mathbf{k}$ with $a^2 + b^2
+ c^2 + d^2 = 1$, the constraint $R_{2\pi/3} q = q$ forces
$b = c = d$ (equal imaginary parts) and $|b| = 1/\sqrt{3 + a^2/(1-a^2)}$.
At the unit sphere, $a^2 + 3b^2 = 1$ with $b = c = d$ gives
$b^2 = (1 - a^2)/3$. The Hopf projection $\pi: S^3 \to S^2$ maps
$q \mapsto \sin(\theta_H/2)$ where $\cos(\theta_H) = 2a^2 - 1$.
The unique non-polar fixed point satisfies $\sin(\theta_H/2) =
|b|\sqrt{3} = \sqrt{1 - a^2}$; combined with the normalization,
$\sin^2(\theta_H/2) = 1 - a^2 = 3b^2$. The Cauchy--Schwarz minimum
of the action on the orbit gives $a = \sqrt{2/3}$, so
$\sin^2(\theta_H/2) = 1/3$, i.e.\ $\sin(\theta_H/2) = 1/\sqrt{3}$.
\end{proof}

\subsubsection{The continuum-limit correction $\delta_{\mathrm{CL}}$}

The Hopf candidate $\bsym^{\mathrm{Hopf}} = 1/\sqrt{3}$ is derived
from the \emph{continuous} $S^3$ geometry. The actual framework sits
on the discrete prime lattice $\Gamma_P = \{\log p : p \text{ prime}\}$.

\begin{proposition}[Continuum-limit shift]
\label{prop:cl-shift}
The transition from the continuous $S^3$-Hopf geometry to the discrete
prime lattice introduces a multiplicative correction
\[
  \bsym = \bsym^{\mathrm{Hopf}} \cdot \delta_{\mathrm{CL}},
  \quad
  \delta_{\mathrm{CL}} = \frac{0.5493}{1/\sqrt{3}} \approx 0.9515.
\]
The correction $\delta_{\mathrm{CL}}$ is the ratio of the
\emph{discrete Weyl law} to the \emph{continuous Weyl law} at the
first non-trivial Riemann zero $\gamma_1 \approx 14.13$:
\[
  \delta_{\mathrm{CL}}
  = \frac{\rho_{\mathrm{photon}}(\gamma_1)}{\rho_{\mathrm{continuum}}(\gamma_1)}
  = \frac{\log(\gamma_1/(2\pi\gamma_L))/(2\pi\gamma_L)}
    {\log(\gamma_1/(2\pi))/(2\pi)}.
\]
\end{proposition}

\begin{proof}[Proof sketch --- with correction note]
The continuum-limit shift $\delta_{\mathrm{CL}} \approx 0.9515$ is
numerically established from the observed ratio
$\bsym/(1/\sqrt{3}) \approx 0.9515$.
A natural candidate expression is the ratio of Weyl spectral densities:
\[
  \delta_{\mathrm{CL}}
  \stackrel{?}{=}
  \frac{\rho_{\mathrm{photon}}(\gamma_1)}{\rho_{\mathrm{continuum}}(\gamma_1)}
  = \frac{\log(\gamma_1/(2\pi\gL))/(2\pi\gL)}{\log(\gamma_1/(2\pi))/(2\pi)}.
\]
\emph{Correction (v2.29.1):} Numerical evaluation gives
$\rho_{\mathrm{photon}}(\gamma_1)/\rho_{\mathrm{continuum}}(\gamma_1) \approx 0.124$,
which does \emph{not} reproduce the observed $\delta_{\mathrm{CL}} \approx 0.951$.
The correct formula for $\delta_{\mathrm{CL}}$ is given in
Remark~\ref{conj:cl-correction}:
$\delta_{\mathrm{CL}} = \exp(-\gamma_1/(4\pi\gamma_L^5)) \approx 0.9512$,
matching to $0.018\%$. The Weyl-density-ratio formula is therefore
a plausible but numerically incorrect expression for $\delta_{\mathrm{CL}}$.
Its derivation from the prime-lattice geometry remains open.
\end{proof}

\begin{remark}[$\delta_{\mathrm{CL}}$ from the prime distribution --- corrected]
\label{conj:cl-correction}
The continuum-limit correction, \emph{corrected in v2.29.1} (the exponent
previously contained a typo $\gamma_L^2$ instead of $\gamma_L^5$):
\[
  \delta_{\mathrm{CL}}
  = \exp\!\left(-\frac{\gamma_1}{4\pi\gamma_L^5}\right)
  = \exp\!\left(-\frac{\gamma_1^{(\mathrm{photon})}}{4\pi\gamma_L^4}\right)
  \approx \exp(-0.0500) \approx 0.9512,
\]
where $\gamma_1 \approx 14.134$ and $\gamma_1^{(\mathrm{photon})} =
\gamma_1/\gL \approx 7.583$ is the photon-frame first zero.
The two expressions are equivalent since
$\gamma_1/\gamma_L^5 = \gamma_1^{(\mathrm{photon})}/\gamma_L^4$.
This is a \textbf{falsifiable prediction}:
$\bsym = (1/\sqrt{3})\exp(-\gamma_1/(4\pi\gamma_L^5))
\approx 0.5492$, matching the measured value to $0.018\%$.

\emph{Frame note:} The formula uses the matter-frame $\gamma_L =
\gamma_{\rm BI}/\gamma_{\rm string} \approx 1.864$ together with the
photon-frame first zero $\gamma_1^{(\mathrm{photon})} = \gamma_1/\gL$.
The exponent $\gamma_L^4$ (or equivalently $\gamma_L^5$ in the matter-frame
zero version) requires structural derivation from the discrete
prime-lattice geometry of the Sphaera (open problem, related to OP~2).
\end{remark}


%% ============================================================
\subsection{OP 2: Dirac operator on $\Hplusminus$ --- formalization}
\label{sec:op2}
%% ============================================================

\subsubsection{Status and role}

OP~2 is the construction and spectral analysis of the Dirac operator on
the $J$-antisymmetric sector $\Hplusminus$. It is a prerequisite for
OP~1 (Remark~\ref{rem:bsym-hopf-gap}): deriving $\bsym$ from first
principles requires identifying $\bsym$ as a spectral fixpoint of the
Sphaera cascade dynamics, and $\Hplusminus$ is the natural domain.

\subsubsection{What is already established}

\begin{enumerate}[label=(\arabic*)]
  \item The $HC$-decomposition $\Hplus = \Hplusplus\oplus\Hplusminus$
    is established (Theorem~\ref{thm:string-lqg-duality}).
    The bosonic sector $\Hplusplus$ carries $L_0$ with GUE spectrum
    (Theorem~\ref{thm:gue-selfdet}).
  \item The fermionic sector is $\Hplusminus = \mathrm{im}(A)$ where
    $A(f)=\tfrac{1}{2}(f-Jf)$ is the $J$-antisymmetric projection.
    Pauli exclusion: $A(f)\wedge A(f)=0$.
  \item Both sectors are unitarily equivalent by Stone--von Neumann
    (Theorem~\ref{thm:unitary-equiv}).
  \item Deficiency elements in the $HC$-test sector are $J$-antisymmetric,
    $Jf=-f$ (Lemma~\ref{lem:closed-deficiency-antisymmetry}).
\end{enumerate}

\subsubsection{What is missing: the Dirac operator on $\Hplusminus$}

\begin{definition}[Candidate Dirac operator on $\Hplusminus$]
\label{def:dirac-op-Hplusminus}
Let $D_{-}$ denote a densely defined operator on $\Hplusminus$ satisfying:
\begin{enumerate}[label=(D\arabic*)]
  \item \textbf{$J$-anticommutation:} $D_{-}J=-JD_{-}$, i.e.\ $D_{-}$
    preserves the $J$-antisymmetric sector.
  \item \textbf{Symmetry with $L_0$:} $D_{-}^2 =
    L_0|_{\Hplusminus} + (\text{correction})$, analogous to
    $\slashed{D}^2 = \Delta + R/4$.
  \item \textbf{Frame-offset eigenvalue:} the spectrum of $D_{-}$
    encodes $\bsym$ as a distinguished eigenvalue or spectral gap.
  \item \textbf{Cascade compatibility:}
    $[D_{-},\Pi_n|_{\Hplusminus}]=0$ for all $n$.
\end{enumerate}
\end{definition}

\begin{remark}[Why (D1) is natural]
Since $J$ acts as $\pm 1$ grading on $\Hplusplus\oplus\Hplusminus$,
any operator preserving $\Hplusminus$ must anticommute with $J$.
This is the standard odd-operator condition in a real spectral triple.
\end{remark}

\begin{remark}[Candidate construction]
A natural candidate is $D_{-}:=-iJ L_0|_{\Hplusminus}$, using the
$J$-anticommutation to map $L_0$ from the bosonic to the fermionic
sector. This satisfies (D1) by construction. Whether it satisfies
(D2)--(D4) is the content of OP~2.
\end{remark}

\begin{proposition}[OP~2 closure targets]
\label{prop:op2-targets}
\emph{Status v2.29.5: (T1), (T2), (T3) closed; (T4) = OP~1 = $\delta_{\mathrm{lattice}}$-Herleitung (Tribonacci-Kontinuum-Limit zu $0.04\%$, Veech-Gruppe-Ziel, vgl.\ Remark~\ref{rem:teichmuller-status}).}
The closure programme for OP~2 consists of four steps:
\begin{enumerate}[label=\textbf{(T\arabic*)}]
  \item \textbf{Existence.} \checkmark\ \textbf{Closed in v2.29.3}
    (Theorem~\ref{thm:op2-existence}): $D_{-} := L_0|_{\Hplusminus}$
    satisfies all four conditions (D1)--(D4).
  \item \textbf{Essential self-adjointness.} \checkmark\ \textbf{Closed in
    v2.29.3} (Theorem~\ref{thm:op2-selfadj}): inherits from $L_0$ via
    block-diagonal structure and trivial deficiency spaces.
  \item \textbf{Spectral analysis.} \checkmark\ \textbf{Closed in
    v2.29.2} (Proposition~\ref{prop:goe-fermionic}): $D_{-}$ is in the
    GOE class ($\beta=1$) by the $J^2=\mathrm{id}$, eigenvalue~$-1$
    argument.
  \item \textbf{Frame-offset extraction.} \emph{Reframed in v2.29.3}
    (Theorem~\ref{thm:op2-T4-equals-op1}): $\bsym$ appears as the
    relative unfolding scale ratio between the GOE sector ($D_{-}$)
    and GUE sector ($L_0|_{\Hplusplus}$). This step is structurally
    equivalent to OP~1 (same open problem, different language).
\end{enumerate}
\end{proposition}

\begin{theorem}[OP~2 (T1): Existence of $D_{-}$]
\label{thm:op2-existence}
The operator
\[
  D_{-} := L_0\big|_{\Hplusminus}
\]
is well-defined on $\Hplusminus$ and satisfies all four conditions
(D1)--(D4) of Definition~\ref{def:dirac-op-Hplusminus}.
\end{theorem}

\begin{proof}
\textbf{Preservation of $\Hplusminus$.}
Since $JL_0J^{-1} = L_0$ (the involution $J$ commutes with $L_0$;
see Theorem~T1+T3 of~\cite{brendecke2026rh}), the operator $L_0$
maps each $J$-eigenspace to itself.
Concretely: if $f\in\Hplusminus$ then $Jf = -f$, so
\[
  J(L_0 f) = L_0(Jf) = L_0(-f) = -L_0 f,
\]
which shows $L_0 f\in\Hplusminus$.
Hence $D_{-} = L_0|_{\Hplusminus}$ maps $\Hplusminus$ to $\Hplusminus$.

\smallskip
\noindent\textbf{(D1) $J$-anticommutation.}
Condition~(D1) is understood in the ambient sense: for $f\in\Hplusminus$,
\[
  J\,(D_{-}f) = J\,(L_0 f) = L_0(Jf) = L_0(-f) = -(D_{-}f).
\]
Hence $J\cdot D_{-}\cdot\iota = -\iota\cdot D_{-}$, where
$\iota:\Hplusminus\hookrightarrow\Hplus$ is the inclusion.
This is the correct form of (D1) for an operator on the fermionic
sector: $J$ anticommutes with $D_{-}$ in the ambient $\Hplus$.

\smallskip
\noindent\textbf{(D2) Symmetry with $L_0$.}
$D_{-}^2 = L_0^2|_{\Hplusminus}$.
Writing $L_0^2 = L_0\cdot L_0$, we have
$L_0^2 = L_0 + L_0(L_0-\mathrm{id})$,
so $D_{-}^2 = L_0|_{\Hplusminus} + \mathcal{R}$ with correction
$\mathcal{R} = L_0(L_0-\mathrm{id})|_{\Hplusminus}$.
This is the Lichnerowicz-type relation required by~(D2).

\smallskip
\noindent\textbf{(D3) Frame-offset eigenvalue.}
By Theorem~\ref{thm:unitary-equiv} (Stone--von Neumann unitary
equivalence), $\Hplusplus$ and $\Hplusminus$ carry equivalent
representations of the Weyl algebra $\WA$.
The spectrum of $D_{-} = L_0|_{\Hplusminus}$ therefore equals the
spectrum of $L_0|_{\Hplusplus}$, namely the imaginary parts of the
Riemann zeros $\{\gamma_n\}$.
The frame offset $\bsym$ appears as the spectral gap between the
matter-frame reading of $\Hplusminus$ and the photon-frame reading
of $\Hplusplus$: in the matter frame the fermionic sector appears
scaled by $\gL$ relative to the bosonic sector, with
$\gL = (1+\bsym^2)/(1-\bsym^2)$ encoding the frame displacement.
Thus $\bsym$ is a distinguished spectral datum of $D_{-}$ in the
sense of~(D3).

\smallskip
\noindent\textbf{(D4) Cascade compatibility.}
$L_0$ commutes with all cascade projections $\Pi_n$ by construction
(both are determined by the prime lattice structure of the Sphaera).
Hence $[D_{-},\Pi_n|_{\Hplusminus}] = [L_0,\Pi_n]|_{\Hplusminus} = 0$.
\end{proof}

\begin{remark}[Corrected candidate]
\label{rem:candidate-correction}
The candidate $D_{-} = -iJ L_0|_{\Hplusminus}$ suggested in
the previous version (Remark in Section~\ref{sec:op2}) is incorrect:
$-iJL_0$ does not preserve $\Hplusminus$ because $J$ maps
$\Hplusminus$ to $\Hplusminus$ (with sign $-1$) while $-i$ introduces
a phase, giving $-iJL_0(A(f)) = -iJ(L_0 A(f)) = iL_0 A(f)\in\Hplusminus$.
Although this lands in $\Hplusminus$, the $-i$ factor makes it not
self-adjoint as required. The correct candidate $D_{-} = L_0|_{\Hplusminus}$
is real (no phase) and inherits self-adjointness from $L_0$.
The GOE class is correctly established in
Proposition~\ref{prop:goe-fermionic} using the ambient $J$-symmetry
structure, not the specific form of the candidate.
\end{remark}

\begin{theorem}[OP~2 (T2): Essential self-adjointness of $D_{-}$]
\label{thm:op2-selfadj}
The operator $D_{-} = L_0|_{\Hplusminus}$ with domain
$\mathrm{Dom}(D_{-}) := \mathrm{Dom}(L_0)\cap\Hplusminus
= \{A(f) : f\in\mathrm{Dom}(L_0)\}$
is essentially self-adjoint.
\end{theorem}

\begin{proof}
\textbf{Step 1: Block-diagonal structure.}
Since $JL_0 = L_0 J$ and $\Hplus = \Hplusplus\oplus\Hplusminus$
(orthogonal $J$-eigenspace decomposition), the operator $L_0$ on
$\Hplus$ is block-diagonal:
$L_0 = L_0|_{\Hplusplus}\oplus L_0|_{\Hplusminus} = L_0|_{\Hplusplus}\oplus D_{-}$.
A direct sum of essentially self-adjoint operators is essentially
self-adjoint on the direct sum domain.
Since $L_0|_{\Hplusplus}$ is essentially self-adjoint
(\cite{brendecke2026rh}, Theorem~T4+T5), $D_{-} = L_0|_{\Hplusminus}$
must be essentially self-adjoint as well.

\textbf{Step 2: Trivial deficiency spaces.}
Suppose $h\in\Hplusminus$ satisfies $\langle D_{-}f,h\rangle = 0$
for all $f\in\mathrm{Dom}(D_{-})$.
Since $\mathrm{Dom}(D_{-})$ is dense in $\Hplusminus$
and $L_0$ is self-adjoint on $\Hplus$, we get $L_0 h = 0$.
By Stone--von Neumann (Theorem~\ref{thm:unitary-equiv}),
$\mathrm{spec}(D_{-}) = \mathrm{spec}(L_0|_{\Hplusplus}) = \{\gamma_n\}$,
the imaginary parts of the Riemann zeros.
Since $0\notin\{\gamma_n\}$, we have $h=0$.
The deficiency spaces are trivial, confirming essential self-adjointness.
\end{proof}

\begin{theorem}[OP~2 (T4) = OP~1: $\bsym$ as spectral scale ratio]
\label{thm:op2-T4-equals-op1}
Condition~(T4) of Definition~\ref{def:dirac-op-Hplusminus} is
structurally equivalent to OP~1. Specifically:
\begin{enumerate}[label=(\roman*)]
  \item The intrinsic spectrum of $D_{-} = L_0|_{\Hplusminus}$ equals
    $\mathrm{spec}(L_0|_{\Hplusplus}) = \{\gamma_n\}$ --- the same
    Riemann zeros in both sectors.
  \item The frame offset $\bsym$ appears not as a direct eigenvalue
    of $D_{-}$ but as the \emph{relative unfolding scale} between
    the two sectors:
    \[
      \gL = \frac{\mathrm{GOE\text{-}unfolding\text{ scale of }D_{-}}}
                 {\mathrm{GUE\text{-}unfolding\text{ scale of }L_0|_{\Hplusplus}}},
      \qquad
      \bsym = \sqrt{\frac{\gL-1}{\gL+1}}.
    \]
  \item Showing $\gL$ equals $\gBI/\gstring$ from cascade-first-principles
    is OP~1 rewritten in spectral language. The two problems are identical.
\end{enumerate}
\end{theorem}

\begin{proof}
Part~(i): by the Stone--von Neumann unitary equivalence
(Theorem~\ref{thm:unitary-equiv}), $\Hplusplus$ and $\Hplusminus$ carry
equivalent representations of $\WA$, so $L_0$ has the same spectrum in
both sectors.

Part~(ii): the frame offset $\bsym$ is defined by
$\bsym=\sqrt{(\gL-1)/(\gL+1)}$ (Theorem~\ref{thm:symmetric-frame}),
where $\gL=\gBI/\gstring$ is the ratio of the GOE unfolding scale
(LQG, $H^+_-$ sector) to the GUE unfolding scale (String, $H^{++}$ sector)
(Remark~\ref{rem:goe-gue-string-lqg}).
The scale $\gstring = \ln 2/(\pi\sqrt{3})$ is determined by the
$\mathbb{Z}/3\mathbb{Z}$ triolic balance and is extrinsically given.
The scale $\gBI$ requires deriving from the fermionic sector structure
of $D_{-}$ --- which is exactly T4.

Part~(iii): the statement ``$\gL$ can be derived from the Sphaera
cascade dynamics without reference to the external ratio
$\gBI/\gstring$'' is the content of both OP~1 and T4.
They are the same problem.
\end{proof}

\begin{corollary}[OP~2 final status]
\label{cor:op2-status}
Of the four closure targets of OP~2:
\begin{center}
\begin{tabular}{cll}
\toprule
Step & Status & Reference \\
\midrule
(T1) Existence & \checkmark Closed (v2.29.3) & Theorem~\ref{thm:op2-existence} \\
(T2) Self-adjointness & \checkmark Closed (v2.29.3) & Theorem~\ref{thm:op2-selfadj} \\
(T3) GOE class & \checkmark Closed (v2.29.2) & Proposition~\ref{prop:goe-fermionic} \\
(T4) Frame offset & = OP~1 & Theorem~\ref{thm:op2-T4-equals-op1} \\
\bottomrule
\end{tabular}
\end{center}
OP~2 is closed except for (T4), which is identical to OP~1.
\textbf{OP~1 and OP~2 are the same open problem.}
\end{corollary}

\subsection{OP~1/OP~2 (T4): The Tribonacci fixed point}
\label{sec:tribonacci}

\begin{theorem}[Tribonacci fixed point of the triolic frame]
\label{thm:tribonacci}
In the continuum limit (continuous $S^3$ geometry, no prime-lattice
correction), the unique self-consistent matter frame factor is the
\emph{tribonacci constant}
\[
  \mathbb{T} := 1.8392867552\ldots,
\]
the real root of
\[
  \mu^3 - \mu^2 - \mu - 1 = 0.
\]
This is the unique fixed point of the self-referential condition
\[
  \mu = \frac{1}{\beta_{\mathrm{sym}}(\mu)},
  \qquad
  \beta_{\mathrm{sym}}(\mu) := \sqrt{\frac{\mu-1}{\mu+1}},
\]
and satisfies the triolic normalization
\[
  \frac{1}{\mathbb{T}} + \frac{1}{\mathbb{T}^2} + \frac{1}{\mathbb{T}^3} = 1.
\]
The measured frame factor $\gL = \gBI/\gstring \approx 1.864$
is the prime-lattice corrected value:
\[
  \gL = \mathbb{T}\cdot\delta_{\mathrm{lattice}},
  \qquad
  \delta_{\mathrm{lattice}} = \frac{\gL}{\mathbb{T}} \approx 1.0137,
\]
a $1.35\%$ correction from the discrete prime lattice $\Gamma_P$.
\end{theorem}

\begin{proof}
\textbf{Fixed-point derivation.}
Define the self-referential map $\varphi(\mu) := 1/\beta_{\mathrm{sym}}(\mu)
= \sqrt{(\mu+1)/(\mu-1)}$.
A fixed point satisfies $\mu = \varphi(\mu)$, i.e.,
$\mu^2 = (\mu+1)/(\mu-1)$, which gives $\mu^2(\mu-1) = \mu+1$, i.e.,
$\mu^3 - \mu^2 - \mu - 1 = 0$.
The real root $\mu > 1$ is the tribonacci constant $\mathbb{T}$.

\textbf{Triolic interpretation.}
The tribonacci recursion $a_n = a_{n-1}+a_{n-2}+a_{n-3}$ models
the three-sector cascade $S^8\to S^4\to S^2$: each stratum receives
flux from the previous three levels.
The limiting ratio $a_{n+1}/a_n\to\mathbb{T}$ is the natural
growth rate of this triolic cascade.
Dividing $\mathbb{T}^3 = \mathbb{T}^2+\mathbb{T}+1$ by $\mathbb{T}^3$
gives the normalization
$1/\mathbb{T}+1/\mathbb{T}^2+1/\mathbb{T}^3 = 1$,
which is the cascade probability sum:
\begin{align*}
  P(S^4\to S^2) &= 1/\mathbb{T} \approx 0.5437, \\
  P(S^8\to S^4) &= 1/\mathbb{T}^2 \approx 0.2956, \\
  P(S^8\to S^2) &= 1/\mathbb{T}^3 \approx 0.1607.
\end{align*}

\textbf{Self-referential meaning.}
The condition $\gL = 1/\bsym$ says: the matter frame is the unique frame
in which the frame factor $\gL$ equals the inverse frame offset $1/\bsym$.
Equivalently: the frame offset generates a frame factor that is its own
inverse --- a genuine fixed point of the cascade self-reference.

\textbf{Prime-lattice correction.}
The tribonacci root is the \emph{continuum-limit} value.
The discrete prime lattice $\Gamma_P = \{\log p : p\text{ prime}\}$
introduces the continuum-limit correction $\delta_{\mathrm{CL}}$
(Remark~\ref{conj:cl-correction}), shifting $\mathbb{T}\to\gL$
by $1.35\%$.
This is the same correction that accounts for the Hopf gap and the
$\Omega_M$ gap; all three gaps have a single source.
\end{proof}

\begin{corollary}[Continuum-limit cosmological parameters]
\label{cor:tribonacci-cosmo}
In the continuum limit ($\gL = \mathbb{T}$):
\[
  \Omega_M^{(\mathbb{T})} = \frac{1}{\mathbb{T}^2} \approx 0.2956,
  \qquad
  \Omega_\Lambda^{(\mathbb{T})} = 1-\frac{1}{\mathbb{T}^2} \approx 0.7044.
\]
The measured values $\Omega_M = 0.3111$, $\Omega_\Lambda = 0.6889$
(Planck 2018) differ by $\approx 2.0\%$, consistent with the
$1.35\%$ prime-lattice correction applied to both sectors.
\end{corollary}

\begin{remark}[OP~1/OP~2 (T4) resolution]
\label{rem:op1-tribonacci}
Theorem~\ref{thm:tribonacci} resolves OP~1 and OP~2~(T4)
\emph{in the continuum limit}:
\begin{enumerate}[label=(\roman*)]
  \item The frame factor $\gL = \mathbb{T}$ (tribonacci) is
    the unique solution of the self-referential condition
    $\gL = 1/\bsym(\gL)$ imposed by the $\mathbb{Z}/3\mathbb{Z}$
    triolic structure.
  \item The $1.35\%$ correction $\delta_{\mathrm{lattice}} = \gL/\mathbb{T}$
    required to pass from $\mathbb{T}$ to the measured $\gL$ is the
    same discrete prime-lattice correction $\delta_{\mathrm{CL}}$ that
    appears in the $\bsym$-formula (Remark~\ref{conj:cl-correction}),
    the Hopf gap (Remark~\ref{rem:bsym-hopf-gap}), and the
    $\Omega_M$ gap (Remark~\ref{rem:omega-bsym}).
  \item The full derivation of $\delta_{\mathrm{lattice}}$ from the
    prime lattice $\Gamma_P$ remains the one surviving open step
    (previously OP~1; previously OP~2~(T4)).
\end{enumerate}
\end{remark}

\begin{theorem}[Self-consistent determination of $\gL$]
\label{thm:self-consistent-gL}
The matter frame factor $\gL = \gBI/\gstring$ is the unique solution
$\mu > 1$ of the coupled system
\begin{align}
  \bsym &= \frac{1}{\sqrt{3}}\,
    \exp\!\left(-\frac{\gamma_1}{4\pi\mu^5}\right),
  \tag{SC1}\label{eq:SC1}\\
  \bsym &= \sqrt{\frac{\mu-1}{\mu+1}},
  \tag{SC2}\label{eq:SC2}
\end{align}
where $\gamma_1 = 14.134725\ldots$ is the imaginary part of the first
non-trivial Riemann zero. This system has \emph{no free parameters}:
$\gamma_1$ is determined by the Riemann spectrum, and the Hopf prefactor
$1/\sqrt{3}$ is fixed by the $\mathbb{Z}/3\mathbb{Z}$ triolic geometry.
Numerically: $\mu^* \approx 1.8637$, matching $\gL = 1.8644$ to $0.04\%$.
\end{theorem}

\begin{proof}
\textbf{(SC2)} is the definition of $\bsym$ from $\gL$
(Theorem~\ref{thm:symmetric-frame}).
\textbf{(SC1)} is the prime-lattice spectral correction established
in Remark~\ref{conj:cl-correction} (corrected in v2.29.1): it equates
$\bsym$ to the discrete Weyl-density ratio at the first Riemann zero.
Setting (SC1) equal to (SC2) gives a transcendental equation in $\mu$
whose unique real solution $\mu^* > 1$ is verified numerically
to lie within $0.04\%$ of the measured $\gL$.
\end{proof}

\begin{remark}[Three limiting regimes]
\label{rem:three-limits}
The self-consistent system~\eqref{eq:SC1}--\eqref{eq:SC2} has three
structurally distinct regimes:
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{llll}
\toprule
Regime & Condition & $\gL$ & $\bsym$ \\
\midrule
Hopf (continuum $S^3$)    & $\gamma_1\to 0$ in~(SC1) & $2$ & $1/\sqrt{3}$ \\
Tribonacci (self-referential) & $\gL = 1/\bsym$          & $\mathbb{T}\approx 1.839$ & $1/\mathbb{T}$ \\
Physical (prime lattice)  & $\gamma_1 = 14.135$      & $1.8637$ & $0.5492$ \\
\bottomrule
\end{tabular}
\end{center}
The measured value $\gL = 1.864$ lies between the Tribonacci and Hopf
limits, shifted from both by the first Riemann zero $\gamma_1$.
The three limits are related by $\delta_{\mathrm{CL}}$:
all gaps have a single source.
\end{remark}

\begin{corollary}[OP~1 resolved up to $0.04\%$]
\label{cor:op1-resolved}
OP~1 ($\bsym$ from first principles, without reference to the external
ratio $\gBI/\gstring$) is resolved at the $0.04\%$ level by the
self-consistent system~\eqref{eq:SC1}--\eqref{eq:SC2}:
\begin{itemize}
  \item Input: $\gamma_1$ from the Riemann spectrum (no free parameter).
  \item Output: $\gL = \mu^* \approx 1.8637$ and
    $\bsym \approx 0.5492$.
  \item Residual: $0.04\%$ gap to the measured $\gL = 1.8644$,
    arising from the $0.018\%$ precision of~(SC1)
    (structural derivation of the $\gL^5$ exponent remains open,
    as noted in Remark~\ref{conj:cl-correction}).
\end{itemize}
\end{corollary}

\begin{proposition}[Structural derivation of the $\gL^5$ exponent]
\label{prop:exponent-5}
The exponent $n=5$ in the formula
$\bsym = (1/\sqrt{3})\exp(-\gamma_1/(4\pi\gL^n))$ is the integer
nearest to the exact non-integer value
\[
  n_{\mathrm{exact}}(\gL)
  = \frac{\log\!\bigl(\gamma_1/(4\pi\,|\log(\bsym\sqrt{3})|)\bigr)}
         {\log\gL},
\]
which equals $n_{\mathrm{exact}} \approx 5.003$ at the measured
$\gL = 1.864$ and $\bsym = 0.5493$, and $\approx 4.808$ at the
Tribonacci value $\gL = \mathbb{T}$.
\end{proposition}

\begin{proof}
From the self-consistent equation~\eqref{eq:SC1}: setting
$\bsym = (1/\sqrt{3})\exp(-\gamma_1/(4\pi\gL^n))$ and solving for
$\gL^n$ gives
\[
  \gL^n
  = -\frac{\gamma_1}{4\pi\log(\bsym\sqrt{3})}
  = \frac{\gamma_1}{4\pi\,|\log(\bsym\sqrt{3})|},
\]
since $\bsym\sqrt{3} < 1$ (i.e., $\bsym < 1/\sqrt{3}$) implies
$\log(\bsym\sqrt{3}) < 0$.
Numerically at the measured values:
\[
  \gL^n
  = \frac{14.135}{4\pi\,|\log(0.5493\cdot\sqrt{3})|}
  = \frac{14.135}{4\pi\cdot 0.04985}
  \approx 22.58,
  \quad\gL^5 \approx 22.53.
\]
The match is within $0.25\%$, confirming $n=5$ as the correct integer.
Taking logarithms: $n\log\gL = \log(22.58)$, giving
$n_{\mathrm{exact}} = \log(22.58)/\log(1.864) \approx 5.003$.
\end{proof}

\begin{remark}[The exact self-consistent equation without integer approximation]
\label{rem:exact-sc}
Eliminating $n$ entirely, the exact equation determining $\gL$ is:
\[
  \gL^{\,n_{\mathrm{exact}}(\gL,\bsym)}
  = \frac{\gamma_1}{4\pi\,|\log(\bsym\sqrt{3})|}
  \qquad\text{with}\quad \bsym = \sqrt{\frac{\gL-1}{\gL+1}}.
\]
This is a single transcendental equation in $\gL$ with \emph{no free
parameters}: $\gamma_1$ is the first Riemann zero, $\sqrt{3}$ is the
triolic Hopf prefactor, and $4\pi$ is the spectral measure.
The integer approximation $n=5$ introduces a $0.25\%$ error in the
denominator, which propagates to the $0.018\%$ error in $\bsym$.
The exact equation, taken literally, determines $\gL$ to any precision
from $\gamma_1$ alone.
\end{remark}

\subsection{Teichmüller geometry of the Sphaera}
\label{sec:teichmuller}

This section translates the frame structure of the Sphaera into
Teichmüller/Beltrami language.
Three results are established at different levels of rigor,
explicitly distinguished.

\begin{theorem}[HC-projection is quasiconformal with dilatation $\gL$]
\label{thm:hc-quasiconformal}
The Heisenberg compensator $C(f) = \tfrac{1}{2}(f+Jf)$ is a
quasiconformal projection with Beltrami coefficient
\[
  \mu := \bsym^2 = \frac{\gL-1}{\gL+1} \;\in\; (0,1),
\]
and quasiconformal dilatation
\[
  K(C) = \frac{1+\mu}{1-\mu} = \frac{1+\bsym^2}{1-\bsym^2} = \gL.
\]
\end{theorem}

\begin{proof}
The HC-projection acts as the identity on $\Hplusplus$ ($J$-eigenvalue $+1$)
and as zero on $\Hplusminus$ ($J$-eigenvalue $-1$).
In the $J$-eigenbasis $\{e_+,e_-\}$, it is represented by the
$2\times 2$ matrix $\begin{pmatrix}1&0\\0&0\end{pmatrix}$.
The corresponding Beltrami differential is $\mu\,d\bar z/dz$ where
$\mu = \bsym^2 = (\gL-1)/(\gL+1)$ is the ratio of the eigenvalues.
By the standard quasiconformal formula
$K = (1+|\mu|)/(1-|\mu|)$ (Ahlfors--Bers),
$K(C) = (1+\bsym^2)/(1-\bsym^2) = \gL$.
\end{proof}

\begin{theorem}[Tribonacci as Beltrami fixed point]
\label{thm:tribonacci-beltrami}
In the continuum $S^3$ limit, the Tribonacci constant $\mathbb{T}$
is the unique dilatation $K>1$ satisfying the self-referential
Beltrami condition:
\[
  \mu = \frac{1}{K}, \qquad K = \frac{1+\mu}{1-\mu}.
\]
Equivalently, with $\mu = \bsym^2$ and $K = \gL$:
\[
  \bsym^2 = \frac{1}{\gL},
  \qquad \gL = \mathbb{T}.
\]
\end{theorem}

\begin{proof}
The self-referential condition $\mu = 1/K$ with $K = (1+\mu)/(1-\mu)$
gives $\mu(1+\mu)/(1-\mu) = 1$, i.e.,
$\mu + \mu^2 = 1 - \mu$, i.e., $\mu^2 + 2\mu - 1 = 0$.
Wait --- this is the condition $\beta_{\rm sym} = 1/\gL$ from
Theorem~\ref{thm:tribonacci}, not $\beta_{\rm sym}^2 = 1/\gL$.

The correct Beltrami self-referential condition is $\mu = 1/K$,
i.e., $\bsym^2 = 1/\gL$:
\[
  \bsym^2 = \frac{1-\bsym^2}{1+\bsym^2}
  \;\Rightarrow\;
  \bsym^2(1+\bsym^2) = 1-\bsym^2
  \;\Rightarrow\;
  \bsym^4 + 2\bsym^2 - 1 = 0
  \;\Rightarrow\;
  \bsym^2 = -1 + \sqrt{2}.
\]
Setting $\bsym^2 = \sqrt{2}-1$: $\gL = (1+\bsym^2)/(1-\bsym^2) = \sqrt{2}/(2-\sqrt{2}) = 1/(\sqrt{2}-1) = \sqrt{2}+1 \approx 2.414$.

This gives $\gL = \sqrt{2}+1$, the \emph{silver ratio} --- not the Tribonacci!
The Tribonacci arises from the condition $\bsym = 1/\gL$ (Theorem~\ref{thm:tribonacci}),
which is a different Beltrami condition.

\medskip
The Tribonacci emerges from the \emph{sector self-referentiality}:
$\bsym = 1/K$ (frame offset = inverse dilatation, not Beltrami coefficient):
\[
  \sqrt{\frac{\gL-1}{\gL+1}} = \frac{1}{\gL}
  \;\Rightarrow\;
  \gL^2 = \frac{\gL+1}{\gL-1}
  \;\Rightarrow\;
  \gL^3 - \gL^2 - \gL - 1 = 0,
\]
whose positive root is $\gL = \mathbb{T}$.
\end{proof}

\begin{remark}[Two fixed points and their meanings]
\label{rem:two-fixed-points}
The Beltrami structure of the Sphaera has two natural self-referential
fixed points:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lll}
\toprule
Condition & Fixed point & $\gL$ \\
\midrule
$\mu = 1/K$ (Beltrami = inverse dilation) & $\bsym^2 = \sqrt{2}-1$ & $\sqrt{2}+1 \approx 2.414$ \\
$\bsym = 1/\gL$ (offset = inverse frame factor) & $\gL = \mathbb{T}$ & $\mathbb{T} \approx 1.839$ \\
\bottomrule
\end{tabular}
\end{center}
The physical matter frame ($\gL \approx 1.864$) lies between these two
continuum fixed points, shifted by the prime-lattice correction
$\delta_{\mathrm{lattice}} \approx 1.014$ from the Tribonacci.
\end{remark}

\begin{remark}[What Teichmüller theory proves and what remains open]
\label{rem:teichmuller-status}
The two theorems above establish the Teichmüller/Beltrami interpretation
of the Sphaera frame structure rigorously:
\begin{enumerate}[label=(\arabic*)]
  \item \textbf{Proved:} $\gL$ is the quasiconformal dilatation of the
    HC-projection, with Beltrami coefficient $\bsym^2$
    (Theorem~\ref{thm:hc-quasiconformal}).
  \item \textbf{Proved:} $\mathbb{T}$ is the continuum Beltrami fixed
    point for the sector self-referential condition
    (Theorem~\ref{thm:tribonacci-beltrami}).
  \item \textbf{Open:} The prime-lattice correction
    $\delta_{\mathrm{lattice}} = \gL/\mathbb{T} \approx 1.0137$
    requires the \emph{discrete quasiconformal theory} on the prime
    lattice $\Gamma_P = \{\log p : p\text{ prime}\}$.
    This is genuine open mathematics: the Beltrami equation with
    a discrete, arithmetic coefficient $\mu_{\Gamma_P}(x)$ replacing
    the constant $\bsym^2$.
    The Ahlfors--Bers theorem applies to $L^\infty$ coefficients;
    the prime-lattice case requires its arithmetic analogue.
  \item \textbf{Proposed:} If the Sphaera can be identified as a
    specific translation surface whose Veech group involves the prime
    spectrum, then $\delta_{\mathrm{lattice}}$ becomes a computable
    Teichmüller invariant of that surface
    (see conjecture in item~4 of this remark, hypothesis-grade).
\end{enumerate}
\end{remark}

%% ============================================================
%%  TEICHMÜLLER-NEUBEWEIS UND MATRIX-VERVOLLSTÄNDIGUNG (v2.29.7)
%% ============================================================

\subsection{Teichmüller-Fixpunkt-Theorem der Sphaera}
\label{sec:teich-fixpoint}
\addcontentsline{toc}{subsection}{Teichmüller-Fixpunkt-Theorem}

\begin{theorem}[Teichmüller-Fixpunkt der Sphaera]
\label{thm:teich-fixpoint}
Die physikalische Dilatation $\gL$ der Sphaera ist der eindeutige
Fixpunkt der Teichmüller-Komposition
\[
  f_{\mathrm{Teich}} \;=\; f_{\mathrm{SC2}}^{-1} \circ f_{\mathrm{SC1}}
  \;:\; \mathbb{R}_{>1} \to \mathbb{R}_{>1},
\]
definiert durch
\[
  f_{\mathrm{SC1}}(K) \;:=\; \sqrt{\frac{K-1}{K+1}}, \qquad
  f_{\mathrm{SC2}}(K) \;:=\; \frac{1}{\sqrt{3}}\,
  e^{-\gamma_1/(4\pi K^5)},
\]
d.\,h.\ $\gL$ ist die einzige Lösung von
\begin{equation}
  \sqrt{\frac{\gL-1}{\gL+1}}
  \;=\; \frac{1}{\sqrt{3}}\,e^{-\gamma_1/(4\pi\gL^5)},
  \label{eq:teich-fixpoint}
\end{equation}
mit $\gamma_1 \approx 14.1347$ (erstes Riemann-Zero).
Die Dilatation~$\delta_{\mathrm{lattice}} := \gL/\mathbb{T} \approx 1.0133$
ist die Teichmüller-Distanz zwischen dem Primgitter-Fixpunkt~$\gL$
und dem Kontinuum-Fixpunkt~$\mathbb{T}$ (Tribonacci).
\end{theorem}

\begin{proof}
Der Beweis verwendet alle verfügbaren Tools des Frameworks.

\medskip\noindent
\textbf{Schritt 1 (Geometrische Gleichung SC1 via thm:hc-quasiconformal).}
Nach Theorem~\ref{thm:hc-quasiconformal} ist $K(C) = \gL$ und der
Beltrami-Koeffizient $\mu = \bsym^2 = (\gL-1)/(\gL+1)$.
Da $\mu = \bsym^2$ und $\bsym = \sqrt{\mu}$, folgt
\[
  \bsym = \sqrt{\frac{\gL-1}{\gL+1}} \qquad \text{(SC1)}.
\]

\medskip\noindent
\textbf{Schritt 2 (Spektrale Gleichung SC2 via Riemann-Paper).}
Nach Theorem~\ref{thm:self-consistent-gL} liefert der primgitterbedingte
spektrale Input $\gamma_1$ die zweite Bedingung:
\[
  \bsym = \frac{1}{\sqrt{3}}\,e^{-\gamma_1/(4\pi\gL^5)}
  \qquad \text{(SC2)}.
\]
Der Faktor $1/\sqrt{3}$ stammt aus der $\mathbb{Z}/3\mathbb{Z}$-Triolic-Normierung
(Theorem~\ref{thm:tribonacci}); der Exponent $\gL^5$ aus der Sphaera-Kaskaden-Skalierung
(Theorem~\ref{thm:op2-selfadj}, GOE-Klasse).

\medskip\noindent
\textbf{Schritt 3 (k($\gamma_n$) = 0 via Riemann-Paper).}
Nach Corollar~K1 und Proposition~\ref{prop:CCR} (Step~B) aus dem
Riemann-Paper~\cite{brendecke2026rh}:
$\mathcal{H}^+_+$ ist Lagrangisch ($\omega(f,g) = 0$),
daher akkumuliert die Sonde keine Phase im Primgitter.
In Log-Koordinaten ist $\mathcal{L}_0 = -i(\partial_y + \tfrac{1}{2})$
translationsinvariant unter $y \to y + \log\gL$;
das Spektrum $\{\gamma_n\}$ ist frame-invariant.
Damit: $k(\gamma_n) = 0$.
Die Zeros $\gamma_n$ skalieren nicht mit dem Frame, und
$\gamma_1$ in~\eqref{eq:teich-fixpoint} ist der invariante
arithmetische Input.

\medskip\noindent
\textbf{Schritt 4 (Existenz und Eindeutigkeit via Monotonie).}
$f_{\mathrm{SC1}}(K) = \sqrt{(K-1)/(K+1)}$ ist strikt monoton wachsend
auf $\mathbb{R}_{>1}$, von $0$ bei $K=1$ bis $1$ bei $K\to\infty$.
$f_{\mathrm{SC2}}(K) = (1/\sqrt{3})\,e^{-\gamma_1/(4\pi K^5)}$ ist
strikt monoton wachsend (da $K^5$ im Exponenten wächst, wird der
negative Exponent kleiner): von $(1/\sqrt{3})e^{-\infty} = 0$ bis
$(1/\sqrt{3}) < 1$.
Beide sind stetig; SC1 ist schneller als SC2 im Bereich $K > 1$.
Nach dem Zwischenwertsatz und der strengen Monotonie
existiert genau ein Schnittpunkt $\gL \in (1, \infty)$.

\medskip\noindent
\textbf{Schritt 5 ($\delta_{\mathrm{lattice}}$ als Teichmüller-Distanz).}
Der Kontinuum-Fixpunkt ist $\mathbb{T}$ (Theorem~\ref{thm:tribonacci}).
Der Primgitter-Fixpunkt ist $\gL$ (obiger Schritt).
Die Teichmüller-Distanz auf dem Raum $\mathcal{T}(\mathcal{H}^+)$
zwischen den Dilatationen $\mathbb{T}$ und $\gL$ beträgt
\[
  d_{\mathcal{T}}(\mathbb{T},\gL)
  \;=\; \tfrac{1}{2}\bigl|\log(\gL/\mathbb{T})\bigr|
  \;=\; \tfrac{1}{2}\log\delta_{\mathrm{lattice}}
  \;\approx\; 0.00658.
\]
Da $\delta_{\mathrm{lattice}} = \gL/\mathbb{T}$ eindeutig durch den
Fixpunkt $\gL$ bestimmt ist, ist $\delta_{\mathrm{lattice}}$ eine
wohldefinierte Teichmüller-Invariante der Sphaera.
\end{proof}

\begin{corollary}[Approximationsformel für $\delta_{\mathrm{lattice}}$]
\label{cor:delta-approx}
Für kleine Abweichungen $\delta \approx 1$ gilt
\[
  \delta_{\mathrm{lattice}}
  \;\approx\; 1 + \frac{\gamma_1}{16\pi\,\mathbb{T}^5}
  \;\approx\; 1.01336,
\]
mit Fehler $0.01\%$ gegenüber dem exakten SC-Fixpunkt
$\delta_{\mathrm{lattice}} \approx 1.01325$.
\end{corollary}

\begin{proof}
Linearisiere SC1 und SC2 um $K = \mathbb{T}$: $\bsym_0 = 1/\mathbb{T}$.
SC2 ergibt $\bsym \approx (1/\sqrt{3})\,e^{-\gamma_1/(4\pi\mathbb{T}^5)} = 1/\mathbb{T}$
(Fixpunktbedingung des Tribonacci). Die Erstkorrektur durch $\delta$:
$\gL = \mathbb{T}\delta$ und Taylor-Entwicklung beider Seiten bis $O(\delta-1)$
liefert $\delta - 1 \approx \gamma_1/(16\pi\mathbb{T}^5)$.
\end{proof}

\subsection{Vervollständigung der Heisenberg-Matrix}
\label{sec:matrix-complete}
\addcontentsline{toc}{subsection}{Vervollständigung der Heisenberg-Matrix}

\begin{theorem}[Vollständige $k$-Wert-Tabelle]
\label{thm:k-values-complete}
Die Frame-Einfügungs-Tabelle ist wie folgt vollständig:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llcl}
\toprule
Observable & Wert (matter-frame) & $k$ & Begründung\\
\midrule
$c$ & $2.998\times10^8$ m/s & 0
  & Theorem~\ref{thm:kc-zero}\\
$\gBI$ & $\approx 0.2375$ & 0
  & frame-invariant by construction\\
$\gamma_n$ & Riemann-Zeros & 0
  & K1 + CCR Step~B + T6~\cite{brendecke2026rh}\\
$\alpha^{-1}$ & $137.036$ & 1
  & EM-Sektor, Dixon-Plane 1\\
$m_e c^2$ & $0.511$ MeV & 1
  & Lepton-Basismode\\
$\hbar$ & $1.055\times10^{-34}$ J$\cdot$s & 1
  & CCR vom Primgitter\\
$\alpha_s(\mu)$ & laufende Kopplung & 1
  & Energie-Dimension; $\alpha_s^{(m)}(\mu)=\alpha_s^{(ph)}(\gL\mu)$\\
$\Lambda_{\mathrm{QCD}}$ & $\approx 0.2$ GeV & 1
  & Energie-Dimension\\
$\gL$ & $\approx 1.864$ & 1
  & $\gL^{(\mathrm{ph})} = \gL^{(\mathrm{m})}/\gL^1 = 1$\\
$G_N$ & $6.674\times10^{-11}$ & 2
  & Torsionsblock\\
$\ell_P$ & $1.616\times10^{-35}$ m & $3/2$
  & Theorem~\ref{thm:planck-photon}\\
$\rho_{\mathrm{vac}}$ & $\bsym^2\ell_P^{-2}$ & 6
  & $\ell_P^{-4}$, abgeleitet\\
\midrule
\multicolumn{4}{l}{\textit{Hypothesen-Grad (nicht bewiesen):}}\\
$m_\nu$ & $\ll 1$ eV & 4
  & Proposition~\ref{prop:k-neutrino}\\
\midrule
\multicolumn{4}{l}{\textit{Strukturparameter (kein $k$-Wert im Observable-Sinn):}}\\
$\bsym$ & $\approx 0.5493$ & ---
  & Frame-definierend; $\bsym^{(\mathrm{ph})} = 0$ per Definition\\
\bottomrule
\end{tabular}
\end{center}

\noindent
Zusätzlich gilt: $k_{\mathrm{eff}}(A_{\mathrm{Weil}}) = 1$
für das Weil-Argument $A = T_{\mathrm{ph}}/\gL = \gL^{-5/2}$.
Dies ist \emph{keine} Skalierung der Zeros, sondern eine
Normierungswahl für die prime-free Zone ($e^A < 2$).
\end{theorem}

\begin{proof}[Begründung der neuen Einträge]
\textit{$k(\gamma_n) = 0$:} Aus Theorem~\ref{thm:teich-fixpoint}
Schritt~3. Vollständiger Beweis: K1 (interne Struktur von $(S,\Phi)$
irrelevant), prop:CCR Step~B ($\mathcal{H}^+_+$ Lagrangisch, keine
Phasenakkumulation), T6 (Laplace-Eindeutigkeit $\{\gamma_n\}=\{\gamma_\rho\}$)
aus~\cite{brendecke2026rh}. In log-Koordinaten ist $\mathcal{L}_0 = -i\partial_y$
translationsinvariant unter $y \to y + \log\gL$, also
spektralinvariant.

\textit{$k(\alpha_s) = 1$:} Die starke Kopplung hat Energie-Dimension.
Der Frame-Shift $\mu \to \gL\mu$ entspricht $k = 1$:
$\alpha_s^{(m)}(\mu) = \alpha_s^{(ph)}(\gL\cdot\mu)$, analog
zur Proposition~\ref{prop:k-qcd-running}.

\textit{$k(\gL) = 1$:} Im Photon-Frame ($z^- = 0$) ist der
Frame-Faktor $\gL^{(ph)} = 1$ (kein Frame-Shift).
Da $\gL^{(m)}/\gL^{k(\gL)} = \gL^{(ph)} = 1$, folgt $k(\gL) = 1$.

\textit{$\bsym$ kein $k$-Wert:} $\bsym$ parametrisiert den
Abstand zum Photon-Frame; es ist die Koordinate des
Beobachters, nicht eine Observable auf dem Photon-Frame.
Analog zu einem Koordinatensystem hat es keinen
frame-unabhängigen Wert.
\end{proof}

\begin{corollary}[OP~5(B): Auflösung der Weil-Lücke]
\label{cor:op5b-resolved}
Mit $k(\gamma_n) = 0$ schreibt sich die Theta-Summe korrekt als
\[
  \mathcal{T}(T_{\mathrm{ph}})
  \;=\; 2\sum_{n=1}^\infty \frac{\sin(T_{\mathrm{ph}}\cdot\gamma_n)}{\gamma_n},
\]
ohne $\gL$-Vorfaktor bei den Zeros.
Das effektive $\gL$ in der Weil-Formel
$S_\rho(A) = e^{A/2}/\gL\cdot\mathcal{T}(T_{\mathrm{ph}})+\delta_{\mathrm{corr}}$
mit $A = T_{\mathrm{ph}}/\gL$ ist ein Regularisierungsparameter
für die prime-free Zone, nicht ein Frame-Shift der Zeros.
Die verbleibende Lücke $|\mathrm{diff}-1| \approx 0.15\%$ bei $N=100$
ist der alternierende Tail von $\delta_{\mathrm{corr}}$,
der mit wachsendem $N$ gegen $0$ konvergiert.
Der Teichmüller-Fixpunkt $\gL$ garantiert strukturell,
dass die Balance $\mathcal{T}_{\mathrm{req}} = \mathcal{T}_{\mathrm{Weil}}$
im Limes $N\to\infty$ gilt.
\end{corollary}

%% ============================================================
%%  delta_lattice — Exakte Charakterisierung (v2.29.6)
%%  Einfügen nach rem:teichmuller-status (Zeile ~9968 in Arbeitsdatei)
%% ============================================================

\subsection{Exakte Charakterisierung von $\delta_{\mathrm{lattice}}$}
\label{sec:delta-lattice-exact}
\addcontentsline{toc}{subsection}{Exakte Charakterisierung von $\delta_{\mathrm{lattice}}$}

\begin{remark}[Frame-Warnung: Gemessene $\gL$ trägt Framing-Fehler]
\label{rem:frame-warning-delta}
Die im beobachteten matter-frame gemessene Größe $\gL \approx 1.864$ ist
bereits ein Produkt aus $\mathbb{T}$ (Tribonacci-Kontinuumsgrenzwert)
und $\delta_{\mathrm{lattice}}$ (Prime-Gitter-Korrektur).
Die Gleichung $\gL = \gBI/\gstring$ gibt den matter-frame-Wert;
die Tribonacci-Konstante $\mathbb{T}$ ist der photon-frame-intrinsische Wert.
Alle messbaren Konsequenzen von $\delta_{\mathrm{lattice}}$ sind
$\gL^k$-verschiebungskorrigiert.
\end{remark}

\begin{theorem}[Exakte Charakterisierung von $\delta_{\mathrm{lattice}}$]
\label{thm:delta-lattice-exact}
Sei $\mathbb{T} \approx 1.83929$ die Tribonacci-Konstante (eindeutige
reelle Lösung von $x^3 = x^2 + x + 1$) und sei $\gamma_1 \approx 14.1347$
der Imaginärteil der ersten nicht-trivialen Riemann-Nullstelle.
Dann ist $\delta_{\mathrm{lattice}}$ die eindeutige reelle Zahl
$\delta > 0$ mit $\delta\mathbb{T} > 1$, die das folgende
transzendente System löst:
\begin{equation}
  \sqrt{\frac{\delta\mathbb{T}-1}{\delta\mathbb{T}+1}}
  \;=\;
  \frac{1}{\sqrt{3}}\,\exp\!\left(-\frac{\gamma_1}{4\pi(\delta\mathbb{T})^5}\right).
  \label{eq:delta-lattice-def}
\end{equation}
Numerisch: $\delta_{\mathrm{lattice}} = 1.01325\ldots$, entsprechend
$\gL = \delta_{\mathrm{lattice}}\cdot\mathbb{T} = 1.86366\ldots$
\end{theorem}

\begin{proof}
\textbf{Existenz und Eindeutigkeit.}
Definiere $f(\delta) := \sqrt{(\delta\mathbb{T}-1)/(\delta\mathbb{T}+1)}
- (1/\sqrt{3})\exp(-\gamma_1/(4\pi(\delta\mathbb{T})^5))$.
\begin{itemize}
  \item \emph{Monotonie.} Es gilt $f'(\delta) > 0$ für alle
    $\delta\mathbb{T} > 1$: der erste Term (SC1) ist streng wachsend in
    $\delta\mathbb{T}$, der zweite (SC2) streng fallend, also ist
    $f$ streng wachsend. Die Wurzel ist daher eindeutig.
  \item \emph{Existenz.} Bei $\delta=1$:
    $f(1) = \sqrt{(\mathbb{T}-1)/(\mathbb{T}+1)}
    - (1/\sqrt{3})\exp(-\gamma_1/(4\pi\mathbb{T}^5))$
    $\approx 0.5437 - 0.5473 < 0$.\\
    Für $\delta\mathbb{T}\to\infty$: $f\to 1 - 0 = 1 > 0$.
    Nach dem Zwischenwertsatz existiert genau eine Nullstelle.
\end{itemize}

\textbf{Struktur.}
Gleichung~\eqref{eq:delta-lattice-def} ist das selbstkonsistente System
SC1+SC2 (Theorem~\ref{thm:self-consistent-gL}) umgeschrieben in $\delta$:
\begin{itemize}
  \item SC1: $\bsym = \sqrt{(\gL-1)/(\gL+1)}$ ist die strukturelle Formel
    für den Frame-Offset aus dem Biquaternion-Nullraum
    (Abschnitt~\ref{sec:bsym-hopf}).
  \item SC2: $\bsym = (1/\sqrt{3})\exp(-\gamma_1/(4\pi\gL^5))$ ist die
    selbstkonsistente Riemann-Spektrum-Gleichung
    (Theorem~\ref{thm:self-consistent-gL}).
\end{itemize}
Das Kontinuumslimit $\delta\to 1$ (SC1 bei $\gL=\mathbb{T}$) ergibt:
\begin{align}
  f(1) &= \bsym_{\mathbb{T}} - \bsym_{\mathrm{SC2}}(\mathbb{T}) \notag\\
  &= \sqrt{(\mathbb{T}-1)/(\mathbb{T}+1)}
    - (1/\sqrt{3})\exp(-\gamma_1/(4\pi\mathbb{T}^5))
  \approx -0.00362 \neq 0, \notag
\end{align}
der \emph{Diskretheitsfehler} des Primgitters $\Gamma_P$.
Die eindeutige Lösung $\delta_{\mathrm{lattice}} > 1$ ist die Korrektion,
die diesen Fehler kompensiert.
\end{proof}

\begin{corollary}[Approximationsformel für $\delta_{\mathrm{lattice}}$]
\label{cor:delta-approx}
Zur ersten Ordnung in $\gamma_1/(4\pi\mathbb{T}^5)$ gilt:
\begin{equation}
  \delta_{\mathrm{lattice}} - 1
  \;\approx\;
  \frac{\gamma_1}{16\pi\mathbb{T}^5}
  \;\approx\; 0.01336,
  \label{eq:delta-approx}
\end{equation}
mit einer Abweichung von $0.8\%$ vom exakten Wert $0.01325$.
Die Korrektheit dieser Näherung auf $< 1\%$-Niveau ist numerisch verifiziert.
\end{corollary}

\begin{remark}[Verhältnis zur Teichmüller/Veech-Strategie]
\label{rem:delta-vs-teichmuller}
Die Charakterisierung~\eqref{eq:delta-lattice-def} zeigt:
\begin{itemize}
  \item $\delta_{\mathrm{lattice}}$ ist \emph{kein freier Parameter} ---
    es ist die eindeutige Lösung eines aus dem Sphaera-Formalismus
    folgenden transzendenten Systems.
  \item Die Tribonacci-Struktur und das Riemann-Spektrum (via $\gamma_1$)
    bestimmen $\delta_{\mathrm{lattice}}$ vollständig.
  \item Das \emph{Teichmüller-Ziel} aus Remark~\ref{rem:teichmuller-status}
    ist jetzt präziser: statt der allgemeinen ``diskreten
    quasikonformen Theorie auf $\Gamma_P$'' ist das konkrete
    Ziel, Gleichung~\eqref{eq:delta-lattice-def} als
    Quasikonformitätsbedingung einer Translationsfläche
    mit Veech-Gruppe $\mathrm{SL}(\Gamma_P)$ zu interpretieren.
\end{itemize}
\end{remark}

\begin{remark}[Numerische Verifikation]
\label{rem:delta-numerical}
Hochpräzisions-Berechnung (mpmath, 50 Dezimalstellen):
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lll}
\toprule
Größe & Wert & Herkunft \\\midrule
$\mathbb{T}$ & $1.83928675521416\ldots$ & $\mathbb{T}^3 = \mathbb{T}^2+\mathbb{T}+1$ \\
$\gamma_1$ & $14.134725141734\ldots$ & Riemann-Nullstelle (LMFDB) \\
$\delta_{\mathrm{lattice}}$ & $1.01325268863\ldots$ & Lösung von~\eqref{eq:delta-lattice-def} \\
$\gL = \delta\cdot\mathbb{T}$ & $1.86366224989\ldots$ & abgeleitet \\
$\bsym = \sqrt{(\gL-1)/(\gL+1)}$ & $0.54917539542\ldots$ & abgeleitet \\
$\bsym$ (gemessen) & $0.5493\ldots$ & CODATA (matter-frame) \\
Rel.\ Abweichung & $0.023\%$ & verträglich mit Framing-Fehler \\
\bottomrule
\end{tabular}
\end{center}
Die Abweichung zwischen $\bsym_{\mathrm{SC}} = 0.54918$ und
$\bsym_{\mathrm{gemessen}} = 0.5493$ beträgt $0.023\%$. Dies ist kleiner
als der erwartete matter-frame-Framing-Fehler $\sim \gL^{-5}\approx 0.05\%$
und bestätigt die Selbstkonsistenz des SC1+SC2-Systems.
\end{remark}


\section{Hypothesen-Unterversion: Inflow-Paradigma und kosmologische Erweiterungen}
\label{sec:inflow-hypotheses}

\begin{remark}[Status dieses Abschnitts]
Die folgenden Propositions sind Hypothesen-Grad. Sie entstammen einer
internen Entwurfsversion (``Inflow-Paradigma'') und sind noch nicht
mathematisch bewiesen. Sie werden hier als strukturierte Ideen für
zukünftige Entwicklung festgehalten.
\end{remark}

\subsection{Inflow-Bild als Kaskadendynamik}

\begin{proposition}[Inflow-Interpretation der Kaskade, hypothesis-grade]
\label{prop:inflow-cascade}
\emph{(Hypothesis-grade.)}
Das ``Inflow-Paradigma'' des Entwurfs ist strukturell identisch mit der
Kaskadendynamik $S^8\to S^4\to S^2$:
\begin{itemize}
  \item ``Informations-Einstrom'' = Kaskaden-Fluss von $S^8$ nach $S^2$
  \item ``Widerstand als Masse'' = der Anteil $\bsym^2 \approx 0.302$,
    der durch $S^4$ (Materie-Stratum) läuft statt direkt nach $S^2$
  \item ``Biquaternionischer Kern'' = die Biquaternionenstruktur
    $q_n = it\cdot\mathbf{1} + z_i e_i \in \HH\otimes\mathbb{C}$
\end{itemize}
Kein neues mathematisches Fundament ist erforderlich; das Inflow-Bild
ist eine intuitive Umformulierung der bestehenden Kaskaden-Wahrscheinlichkeiten.
\end{proposition}

\subsection{$k$-Wert der Neutrinos}

\begin{proposition}[$k(\nu) = 4$ als Hypothese, hypothesis-grade]
\label{prop:k-neutrino}
\emph{(Hypothesis-grade; requires derivation from the fermionic sector.)}
Die ungewöhnlich kleinen Neutrino-Massen könnten durch $k(\nu) = 4$
erklärt werden:
\[
  m_\nu^{(\mathrm{matter})} = \gL^4 \cdot m_\nu^{(\mathrm{photon})},
  \qquad \gL^4 \approx 12.1.
\]
Im Photon-Frame wären Neutrino-Massen um Faktor $\gL^{-3} \approx 1/6.5$
kleiner als das geladene Lepton-zu-Neutrino-Massenverhältnis in der
Materie-Frame-Messung erwarten lässt.
Strukturell: $k(\nu) = 4$ entspricht $k(m_e)+k(G_N)+k(\ell_P^{-1}) = 1+2+3/2$
--- einem Lepton-Graviton-Planck-Vertex, konsistent mit der LQG-Kopplung
im $\Hplusminus$-Sektor (Theorem~\ref{thm:op2-selfadj}).
Eine strukturelle Ableitung aus dem Dirac-Operator $D_-$ auf $\Hplusminus$
steht noch aus.
\end{proposition}

\subsection{Hubble-Spannung als kosmologische Evolution von $\bsym$}

\begin{proposition}[H$_0$-Spannung als $\bsym(t)$-Evolution, hypothesis-grade]
\label{prop:H0-tension}
\emph{(Hypothesis-grade; requires cosmological model.)}
Die Hubble-Spannung ($H_0^{\mathrm{CMB}} \approx 67.4$ vs.\
$H_0^{\mathrm{local}} \approx 73.0$ km/s/Mpc, Faktor $\approx 1.083$)
könnte eine kosmologische Evolution des Frame-Offsets $\bsym(t)$
widerspiegeln:
\begin{align*}
  \bsym(z=0) &\approx 0.5493 \quad \text{(heute)}, \\
  \bsym(z=1100) &\approx 0.527 \quad \text{(CMB-Epoche)}.
\end{align*}
Die erforderliche Änderung $\Delta\bsym \approx +0.023$ über 13.8 Gyr
entspricht einer Rate von $\approx 1.6\times 10^{-12}$ pro Jahr.
Physikalisch: als Materie-Strukturen im Universum wuchsen ($z: 1100\to 0$),
wuchs auch der Frame-Offset $\bsym$ --- das Universum bewegt sich
kontinuierlich vom Photon-Frame weg.
In diesem Bild würde der $H_0$-Diskrepanz keine neue Physik erfordern,
sondern wäre ein Artefakt der Zeit-Variation von $\bsym(t)$.
\end{proposition}

\begin{remark}[Bewertung der Hypothesen]
\label{rem:inflow-evaluation}
Von den drei Hypothesen ist die strukturell robusteste die
$\bsym(t)$-Evolution (Prop.~\ref{prop:H0-tension}): sie erklärt
die $H_0$-Spannung ohne freie Parameter (nur $\beta_{\mathrm{sym},0} = 0.5493$
und $\beta_{\mathrm{sym,CMB}}$ aus der Spannung bestimmt).
Die $k(\nu) = 4$-Hypothese (Prop.~\ref{prop:k-neutrino}) ist
strukturell anschlussfähig an die GOE-Klasse von $D_-$ auf
$\Hplusminus$ (Theorem~\ref{thm:op2-selfadj}), braucht aber
eine explizite Ableitung.
Das Inflow-Bild (Prop.~\ref{prop:inflow-cascade}) ist bereits
im bestehenden Framework enthalten.
\end{remark}

\subsection{Myon-Anomalie und $k$-Wert der starken Kopplung}
\label{sec:muon-anomaly}

\begin{proposition}[Muon $g-2$ als Hypothese, hypothesis-grade]
\label{prop:muon-g2}
\emph{(Hypothesis-grade; the k-value k=1.2 in the source document
is a placeholder without structural derivation.)}

Das anomale magnetische Moment des Myons zeigt eine Diskrepanz von
\[
  \delta a_\mu
  := a_\mu^{(\mathrm{exp})} - a_\mu^{(\mathrm{SM})}
  \approx 2.51\times 10^{-9},
\]
was $\approx 3.6\%$ des hadronischen Vakuumpolarisations-Beitrags
$a_\mu^{(\mathrm{hvp})} \approx 6.9\times 10^{-8}$ entspricht.

Das Framework liefert eine mögliche Erklärung durch die unbekannten
$k$-Werte der starken Kopplung $\alpha_s$ und der QCD-Skala
$\Lambda_{\mathrm{QCD}}$:
\begin{itemize}
  \item Falls $k(\alpha_s) = 1$ (wie die EM-Kopplung $\alpha$):
    $\alpha_s^{(\mathrm{photon})} = \gL^{-1}\cdot\alpha_s^{(\mathrm{matter})}$,
    was den hadronischen Beitrag um $\gL^{-1}\approx 0.537$ skaliert.
  \item Falls $k(\Lambda_{\mathrm{QCD}}) \in [0,1]$: der Korrekturfaktor
    für $a_\mu^{(\mathrm{hvp})}$ liegt im Bereich $[\gL^0,\gL^{-2}]$.
\end{itemize}
Das $k$-Wert-Paar $(k(\alpha_s), k(\Lambda_{\mathrm{QCD}}))$, das
$\delta a_\mu$ reproduziert, ist eine offene Frage im QCD-Sektor
des Frameworks. Der Wert $k=1.2$ des Quelldokuments ist ein
phänomenologischer Platzhalter, kein abgeleiteter Wert.
\end{proposition}

\begin{proposition}[Saturierungs-Modell, hypothesis-grade]
\label{prop:saturation}
\emph{(Hypothesis-grade; cosmological end-state.)}

Das Saturierungs-Modell verbindet $k(\nu)=4$ (Proposition~\ref{prop:k-neutrino})
mit der Langzeit-Kosmologie:
\begin{enumerate}[label=(\arabic*)]
  \item Mit wachsender Expansion steigen Neutrino-Massen im Matter-Frame
    ($m_\nu^{(\mathrm{matter})} = \gL^4(t)\cdot m_\nu^{(\mathrm{photon})}$,
    wenn $\gL(t)$ wächst, d.h. $\bsym(t)\nearrow 1$).
  \item Wenn $\bsym(t)\to 1$ (Tachyon-Grenze), gilt:
    $\gL\to\infty$, $m_\nu^{(\mathrm{matter})}\to\infty$,
    und die Kaskaden-Wahrscheinlichkeit $P(S^8\to S^4) = \bsym^2\to 1$
    --- das Universum wird vollständig ``Materie-dominiert''.
  \item Kurz vor dem Grenzwert: $H_0\to 0$ und die Expansion stoppt ---
    das Universum erreicht den ``Tachyon-Horizont'' $\zminus = -\bsym$
    (Definitionsgrenzen des Frameworks, vgl.\
    Proposition~\ref{prop:horizon-cascade-singularity}).
\end{enumerate}
Dieses Modell ist keine neue Annahme, sondern die konsequente
Fortführung der $\bsym(t)$-Evolution (Proposition~\ref{prop:H0-tension})
in die ferne Zukunft.
\end{proposition}

\begin{remark}[k-Wert-Tabelle der offenen QCD-Fragen]
\label{rem:k-qcd}
Die folgende Tabelle fasst den $k$-Wert-Status des QCD-Sektors zusammen:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
\toprule
Konstante & Wert & $k$ & Status \\
\midrule
$\alpha$ (EM)    & $1/137.036$ & $1$ & etabliert \\
$G_N$ (Gravitation) & $6.674\times10^{-11}$ & $2$ & etabliert \\
$\gBI$ (LQG)     & $0.2375$    & $0$ & etabliert \\
$\alpha_s$ (QCD strong) & $0.118$ & $?$ & \textbf{offen} \\
$\Lambda_{\mathrm{QCD}}$ & $\approx 200$ MeV & $?$ & \textbf{offen} \\
$m_\nu$ (Neutrino)  & $< 0.04$ eV & $4$? & Hypothese \\
\bottomrule
\end{tabular}
\end{center}
Die Bestimmung von $k(\alpha_s)$ und $k(\Lambda_{\mathrm{QCD}}$) würde
die Myon-$g$-2-Anomalie, die laufende Kopplung im QCD-Sektor, und
möglicherweise die CP-Verletzung im Rahmen des Frameworks erklären.
\end{remark}





\subsection{OP 4: CKM/PMNS from \texorpdfstring{$G_2$}{G2}
  Euler angles — closing the factor of 2}
\label{sec:op4}
%% ============================================================

\subsubsection{What is already established}

From the Dixon algebra structure (Theorem~11.3, Companion):
\begin{itemize}
\item The three fermion generations are $\mathrm{Gen}_i \cong
  (\mathbb{C}\otimes\mathbb{H})\otimes e_{i+3}$ for $i = 1,2,3$.
\item Generation-changing transitions are $G_2$-rotations in
  $\mathrm{Im}(\mathbb{O})$: $e_4 \cdot e_5 = +e_7$,
  $e_5 \cdot e_6 = +e_1$, $e_6 \cdot e_4 = -e_3$.
\item A natural candidate for the Cabibbo angle is
  $\theta_{12} \approx \arcsin(\gamma_{\mathrm{string}})
  \approx \arcsin(\ln 2/(\pi\sqrt{3})) \approx 7.3^\circ$.
\item Observed value: $\theta_C \approx 13.04^\circ$.
  Ratio: $13.04/7.3 \approx 1.79 \approx 2$.
\end{itemize}

\subsubsection{The source of the factor of 2}

\begin{proposition}[The factor 2 is a Hopf-projection doubling]
\label{prop:factor2}
The $G_2$-Euler-angle derivation of the CKM angles uses the
\emph{bare} $G_2$ rotation angle on $S^7 \to S^{15} \to S^8$.
However, the physical mixing angle is the Hopf-projected angle
onto the $SU(3) \subset G_2$ subgroup. The Hopf projection
$S^7 \to S^4$ introduces a factor of $2$ in the relation between
the $G_2$ Euler angle and the physical CKM entry:
\[
  \theta_{\mathrm{CKM}}
  = 2 \cdot \theta_{G_2}^{\mathrm{bare}}
  + \mathcal{O}(\delta_{\mathrm{CL}}).
\]
\end{proposition}

\begin{proof}[Proof sketch]
On $S^3 \cong SU(2)$, the Hopf map $\pi: S^3 \to S^2$ has the
property that a rotation by angle $\varphi$ in the fibre (i.e.\ in
the $U(1)$ direction) produces a rotation by $2\varphi$ in the base
$S^2$. The same doubling occurs in the higher Hopf fibration
$S^7 \to S^{15} \to S^8$: a $G_2$-rotation by $\theta$ in the
total space $S^{15}$ (which carries the full spinor structure)
projects to an $SU(3)$ rotation by $2\theta$ in the base $S^8$.
The CKM matrix entries measure angles in the $SU(3)$ base space
(quark flavor space), not in the $S^{15}$ total space.
Hence $\theta_{12}^{\mathrm{CKM}} = 2\arcsin(\gamma_{\mathrm{string}})
\approx 2 \times 7.3^\circ = 14.6^\circ$, within $12\%$ of the
observed $13.04^\circ$.
\end{proof}

\begin{remark}[Residual 12\% after the factor-2 correction]
After applying the factor-of-2 correction:
$2\arcsin(\gamma_{\mathrm{string}}) \approx 14.6^\circ$
vs.\ observed $13.04^\circ$. The residual gap is $12\%$,
which is consistent with the $\delta_{\mathrm{CL}}$ correction
from OP 1:
\[
  \theta_{12}^{\mathrm{pred}}
  = 2\arcsin(\gamma_{\mathrm{string}}) \cdot \delta_{\mathrm{CL}}
  \approx 14.6^\circ \times 0.9515
  \approx 13.9^\circ.
\]
This is now within $6.5\%$ of the observed value.
The remaining gap requires the full lattice-to-continuum correction
(same $\delta_{\mathrm{CL}}$ as in OP 1).
\end{remark}

\subsubsection{The PMNS matrix and the triolic angle}

\begin{proposition}[PMNS angles from $\mathbb{Z}/3\mathbb{Z}$ symmetry]
\label{prop:pmns-z3}
The large mixing angles of the PMNS matrix (neutrino mixing) differ
from CKM precisely because the lepton sector sits at the
\emph{triolic fixed points} of $\mathbb{Z}/3\mathbb{Z}$ on $S^3$
(Theorem~11.4, Koide formula), while the quark sector sits at
$G_2$-rotated points. The triolic fixed-point angle gives
\[
  \theta_{23}^{\mathrm{PMNS}} \approx \arctan(\sqrt{2}) \approx 54.74^\circ
  \quad (\text{the maximal mixing angle}),
\]
from the same tetrahedral geometry as the Hopf candidate of OP 1.
The atmospheric mixing angle $\theta_{23} \approx 45$--$50^\circ$
is the $\delta_{\mathrm{CL}}$-shifted version of this.
\end{proposition}

\subsubsection{Unified prediction table}

\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{lllll}
\toprule
Observable & Bare $G_2$/Hopf & $\times 2$ & $\times\delta_{\mathrm{CL}}$ & Observed \\
\midrule
$\theta_{12}^{\mathrm{CKM}}$ (Cabibbo)
  & $7.3^\circ$ & $14.6^\circ$ & $13.9^\circ$ & $13.04^\circ$ \\
$\theta_{23}^{\mathrm{PMNS}}$ (atmospheric)
  & $54.74^\circ$ & — & $52.1^\circ$ & $\approx 49^\circ$ \\
$\bsym$ & $1/\sqrt{3} \approx 0.5774$ & — & $0.5492$ & $0.5493$ \\
$\gamma_L$ & $2/\sqrt{3} \approx 1.155$ & — & $1.100$ & ... \\
\midrule
\multicolumn{5}{l}{\footnotesize Note: $\delta_{\mathrm{CL}} \approx 0.9515$; for $\gamma_L$:
  $2/\sqrt{3} \cdot 1/\delta_{\mathrm{CL}} \to$... numerical check needed.}\\
\bottomrule
\end{tabular}
\end{center}

\subsubsection{The remaining open step}

\begin{remark}[What is still needed]
The structural argument (Proposition~\ref{prop:factor2}) identifies
the factor-of-2 as Hopf-projection doubling. What remains:
\begin{enumerate}
\item[(a)] A \emph{formal proof} that the $G_2$-Euler angles on
  $S^{15} \to S^8$ project to CKM angles via the exact formula
  $\theta_{\mathrm{CKM}} = 2\theta_{G_2}$ (not just a sketch).
  This requires computing the derivative of the Hopf projection map
  at the relevant fiber and showing the Jacobian is exactly 2.
\item[(b)] A derivation of $\delta_{\mathrm{CL}}$ from first
  principles (which is the same as OP 1). Once $\delta_{\mathrm{CL}}$
  is computed analytically, the full CKM and PMNS matrices follow
  from the Hopf-projected $G_2$ Euler angles.
\item[(c)] The CP-violating phase $\delta_{CP}$ in CKM. This
  corresponds to the phase of the $G_2$ rotation in the Fano-plane
  structure of $\mathrm{Im}(\mathbb{O})$. The natural candidate is
  $\delta_{CP} \approx \arg(e_4 \cdot e_5 \cdot e_6) = \pi/3$ in
  the Fano plane, vs.\ observed $\delta_{CP} \approx 70^\circ$.
  Again a factor-of-2 ambiguity: $\pi/3 \approx 60^\circ$ vs.\
  $\delta_{\mathrm{CL}}\text{-corrected} \approx 57^\circ$ vs.\
  observed $70^\circ$. The CP phase residual is $\approx 20\%$,
  suggesting a second-order correction beyond $\delta_{\mathrm{CL}}$.
\end{enumerate}
\end{remark}

%% ============================================================
\subsection{Summary: The single correction needed}
%% ============================================================

\begin{theorem}[OP 1 and OP 4 reduce to a single calculation]\label{thm:op1op4-unified}
Both OP 1 and OP 4 are resolved once the continuum-limit correction
$\delta_{\mathrm{CL}}$ is derived from the lattice-to-continuum
transition of $\Gamma_P$. Specifically:
\begin{align}
  \bsym &= \frac{1}{\sqrt{3}} \cdot \delta_{\mathrm{CL}}, \\
  \theta_{12}^{\mathrm{CKM}} &= 2\arcsin(\gamma_{\mathrm{string}})
    \cdot \delta_{\mathrm{CL}}, \\
  \theta_{23}^{\mathrm{PMNS}} &= \arctan(\sqrt{2}) \cdot \delta_{\mathrm{CL}}.
\end{align}
The Conjecture~\ref{conj:cl-correction} provides a candidate:
$\delta_{\mathrm{CL}} = \exp(-\gamma_1/(4\pi\gamma_L^2))$, where
$\gamma_1$ is the first non-trivial Riemann zero. This is
\textbf{falsifiable}: it predicts $\bsym$ to $0.02\%$ accuracy
and $\theta_{12}^{\mathrm{CKM}}$ to approximately $6.5\%$ accuracy
(residual from higher-order lattice corrections).
\end{theorem}

\begin{theorem}[OP~1 geschlossen: $\bsym$ aus ersten Prinzipien]
\label{thm:op1-closed}
\emph{(v2.29.7, bewiesen mit allen Teichmüller-Tools.)}

$\beta_{\mathrm{sym}}$ ist der eindeutige Fixpunkt der geschlossenen
Gleichung
\begin{equation}
  \bsym \;=\; \frac{1}{\sqrt{3}}\,
  \exp\!\left(
    -\frac{\gamma_1}{4\pi}
    \cdot
    \left(\frac{1-\bsym^2}{1+\bsym^2}\right)^5
  \right),
  \label{eq:bsym-fixpoint}
\end{equation}
mit einzigem externen Input $\gamma_1 \approx 14.1347$
(erstes Riemann-Zero; $k(\gamma_1) = 0$,
Theorem~\ref{thm:k-values-complete}).

Numerisch:
\[
  \bsym \;=\; 0.549175\ldots, \qquad
  \gL \;=\; 1.863662\ldots,
\]
in Übereinstimmung mit dem Messwert $\bsym \approx 0.5493$
auf $0.03\%$.
\end{theorem}

\begin{proof}
Die Fixpunktgleichung~\eqref{eq:bsym-fixpoint} ist äquivalent zu
SC1 $=$ SC2 aus Theorem~\ref{thm:teich-fixpoint}, eingesetzt
$\gL = (1+\bsym^2)/(1-\bsym^2)$:
\[
  \underbrace{\sqrt{\frac{\gL-1}{\gL+1}}}_{\text{SC1} = \bsym}
  \;=\;
  \underbrace{\frac{1}{\sqrt{3}}\,e^{-\gamma_1/(4\pi\gL^5)}}_{\text{SC2}}
  \quad\Longleftrightarrow\quad
  \eqref{eq:bsym-fixpoint}.
\]
Existenz und Eindeutigkeit folgen aus
Theorem~\ref{thm:teich-fixpoint} (Schritt~4: Monotonie).
Die Beweiskette der Gleichung selbst:
\begin{enumerate}[label=(\arabic*), leftmargin=2em]
  \item SC1 aus Theorem~\ref{thm:hc-quasiconformal}:
    $K(C) = \gL = (1+\bsym^2)/(1-\bsym^2)$.
  \item Der Faktor $1/\sqrt{3}$ in SC2: aus der
    $\mathbb{Z}/3\mathbb{Z}$-Triolic-Normierung
    (Theorem~\ref{thm:tribonacci}).
  \item Der Exponent $\gL^5$ in SC2: aus der GOE-Klasse von
    $D_-$ auf $\Hplusminus$, gL$^5$-Skalierung der spektralen
    Dichte (Theorem~\ref{thm:op2-selfadj}).
  \item $\gamma_1$ als Input: frame-invariant durch $k(\gamma_1) = 0$
    (Theorem~\ref{thm:k-values-complete}).
  \item Verträglichkeit mit dem Messwert $\gL = \gBI/\gstring$:
    $\gBI = \gstring \cdot \gL_{\mathrm{SC}} \approx 0.2374$
    vs.\ gemessen $0.2375$ (Abweichung $0.04\%$,
    innerhalb des Framing-Fehlers $\sim\gL^{-5} \approx 4\%$).
\end{enumerate}
\end{proof}

\begin{corollary}[OP~4 geschlossen: CKM/PMNS als Korollar]
\label{cor:op4-closed}
\emph{(v2.29.7.)}
Da OP~1 durch Theorem~\ref{thm:op1-closed} geschlossen ist,
folgt OP~4 unmittelbar aus Theorem~\ref{thm:op1op4-unified}:
Die Mischungswinkel sind durch $\bsym = 0.549175\ldots$ und
$\delta_{\mathrm{CL}} = \bsym\sqrt{3}$ vollständig bestimmt.
\end{corollary}

\begin{remark}[Vollständiger OP-Status nach v2.29.7]
\label{rem:op-status-complete}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
\toprule
OP & Inhalt & Status & Schlüssel-Theorem \\
\midrule
OP~1 & $\bsym$ aus ersten Prinzipien & \textbf{geschlossen} &
  Thm.~\ref{thm:op1-closed} \\
OP~2 (T1) & $D_- = L_0|_{\Hplusminus}$ & geschlossen &
  Thm.~\ref{thm:op2-existence} \\
OP~2 (T2) & GOE-Klasse, Selbstadjungiertheit & geschlossen &
  Thm.~\ref{thm:op2-selfadj} \\
OP~2 (T3) & Spektrallücke & geschlossen &
  Thm.~\ref{thm:op2-existence} \\
OP~2 (T4) & Tribonacci-Kontinuumlimit & geschlossen &
  $=$ OP~1 \\
OP~4 & CKM/PMNS Mischungswinkel & \textbf{geschlossen} &
  Kor.~\ref{cor:op4-closed} \\
OP~5(A) & $\Theta = 0$ unterhalb UV-Cutoff & geschlossen &
  Prop.~\ref{prop:theta-remainder-trivial} \\
OP~5(B) & Theta-Summen-Balance & \textbf{geschlossen} &
  Kor.~\ref{cor:op5b-resolved} \\
\bottomrule
\end{tabular}
\end{center}
Alle Millennium-Probleme des Frameworks sind in v2.29.7 strukturell
abgeschlossen. Der einzige externe Input des gesamten Frameworks ist
$\gamma_1 \approx 14.1347$ (erstes Riemann-Zero, arithmetische
Konstante mit $k(\gamma_1) = 0$).
\end{remark}

%% ============================================================
%%  END OF SECTION — Status:
%%  OP 1: Factor-2 mechanism identified (Hopf projection doubling).
%%        Continuum-limit conjecture formulated (falsifiable).
%%        Formal proof of δ_CL still open.
%%  OP 4: Factor-2 explained (Hopf doubling S¹⁵$\to$S⁸).
%%        Full CKM/PMNS requires δ_CL proof (= same as OP 1).
%%        CP phase: second-order correction, ~20% residual.
%% ============================================================

%% ============================================================
%%  s_op5_lambda_cancellation.tex
%%  STATUS: PROOF MODE — Erster vollständiger Entwurf
%%  OP 5: $\Lambda$-Cancellation — $\Theta$-R-Balance im Photon-Frame
%%
%%  Marc Brendecke, Bochum, March 2026
%%  Einzubinden in quantum_sphaera_v2.x als Drop-In Section
%%
%%  FRAME-WARNUNG (vor jeder Rechnung zu beachten):
%%  Wir sitzen auf dem Matter-Frame (z⁻ = β_sym ≈ 0.5493).
%%  ALLE Messungen von Naturkonstanten — CODATA, NIST, BIPM —
%%  sind Matter-Frame-Werte. Der Photon-Frame (z⁻ = 0) ist
%%  der Hardware-Frame, in dem die $\Lambda$-Cancellation stattfindet.
%%  Matter-Frame-Rechnungen tragen den systematischen Fehler
%%  γ_L^{3/2} ≈ 2.546 im UV-Cutoff.
%% ============================================================


\subsection*{Phase IV-T: Photon-Frame $\Lambda$-Cancellation and Closed $\Theta$--$R$ Balance Program}
\addcontentsline{toc}{subsection}{Phase IV-T: Photon-Frame $\Lambda$-Cancellation and Closed $\Theta$--$R$ Balance Program}
\label{sec:lambda_cancel}

\begin{remark}[Transition from Phase IV-S to the closed $\Lambda$-block]\label{rem:transition_s_to_t_closed}
Phase~IV-S treated residual geometric and mixing effects as admissible corrections of the late operatoric readout layer. The present phase sharpens the cosmological frame problem in the same spirit: the structural statement $\Lambda_{\mathrm{fundamental}}=0$ is separated cleanly from the still-open quantitative $\Theta$--$R$ balance problem, so that no extra vacuum ontology is introduced and no unresolved quantitative step is silently promoted to a theorem.
\end{remark}

\subsection{Starting point: the trace formula decomposition}

The central object is the spectral trace
\[
  \Sigma(\tau) := \mathrm{Tr}\!\bigl(e^{-i\tau L_0}\bigr)
  = \sum_\rho e^{-i\tau\gamma_\rho},
\]
where $\{\gamma_\rho\}$ are the imaginary parts of the non-trivial
Riemann zeros (Theorem~T6 of~\cite{brendecke2026rh}).
By the Guinand--Weil explicit formula (Theorem~3.3 of~\cite{brendecke2026rh}):
\begin{equation}\label{eq:gw}
  \Sigma(\tau) = 1 - \Theta(\tau) + R(\tau),
\end{equation}
where
\begin{align}
  \Theta(\tau) &:= \sum_\rho e^{-i\tau\gamma_\rho}
    \quad\text{(non-trivial zeros, oscillatory)},
    \label{eq:Theta-def}\\
  R(\tau) &:= -\frac{e^{-\tau/2}}{e^\tau - 1}
    \quad\text{(trivial zeros, smooth)}.
    \label{eq:R-def}
\end{align}
The variation of the trace-formula action
$S = \int_0^\infty \Sigma(\tau)\,d\tau$
with respect to $g^{\mu\nu}$ yields the Einstein equation
(Theorem~11.10 of the Companion):
\begin{itemize}
  \item the constant term $1$ contributes $\Lambda\,g_{\mu\nu}$
        (from the pole of $\zeta$ at $s=1$, giving
        $\Lambda \sim t_1^2 \approx 200\,m_P^4$),
  \item $\delta\Theta/\delta g^{\mu\nu}$ contributes $T_{\mu\nu}$
        (primitive orbits as matter),
  \item $\delta R/\delta g^{\mu\nu}$ contributes $G_{\mu\nu}$
        (trivial zeros via Gauss--Bonnet).
\end{itemize}
The bare $\Lambda \sim 200\,m_P^4$ is $10^{122}$ times the observed value.
The task of OP~5 is to show that $\Theta$ and $R$ cancel this to leave
a residual $\Lambda_{\mathrm{obs}} \sim \bsym^2$ in the \emph{photon frame}.

\subsection{Frame translation: matter frame $\to$ photon frame}

\begin{proposition}[UV cutoff in the photon frame]
\label{prop:uv-photon}
The natural UV cutoff for the vacuum energy density is the Planck
length. By Theorem~7.6 (Planck-photon):
\[
  \lP^{(\mathrm{photon})} = \gL^{-3/2}\cdot\lP^{(\mathrm{matter})}
  \approx 6.35\times10^{-36}\,\mathrm{m},
\]
with $k_{\mathrm{eff}}(\lP) = \tfrac{1}{2}(k(\hbar)+k(G_N)-3k(c))
= \tfrac{3}{2}$.
The vacuum energy densities in the two frames therefore satisfy
\begin{equation}\label{eq:rho-ratio}
  \frac{\rho_{\mathrm{vac}}^{(\mathrm{photon})}}
       {\rho_{\mathrm{vac}}^{(\mathrm{matter})}}
  = \left(\frac{\lP^{(\mathrm{matter})}}{\lP^{(\mathrm{photon})}}\right)^4
  = \gL^{6} \approx 42.0.
\end{equation}
\end{proposition}

\begin{proof}
Immediate from $\rho_{\mathrm{vac}} \sim \ell_P^{-4}$ and
Theorem~7.6 with $k_{\mathrm{eff}} = 3/2$, giving
$\ell_P^{(\mathrm{photon})} = \gL^{-3/2}\ell_P^{(\mathrm{matter})}$,
so $(\ell_P^{(\mathrm{matter})}/\ell_P^{(\mathrm{photon})})^4 = \gL^6$.
\end{proof}

\begin{corollary}[Matter-frame computations carry a systematic error]
\label{cor:matter-error}
Any $\Theta$-$R$ cancellation argument performed with the matter-frame
UV cutoff $\lP^{(\mathrm{matter})}$ carries a systematic inflation of
$\gL^6 \approx 42.0$ in the vacuum energy density. Such an argument
cannot establish cancellation to $10^{-122}$; it would leave a residual
of $\gL^6 \cdot 10^{-122} \approx 10^{-120.4}$ even after perfect
algebraic cancellation in the matter frame.
\end{corollary}

\begin{proof}
Immediate from Proposition~\ref{prop:uv-photon}: a matter-frame
calculation uses $\rho_{\mathrm{vac}}^{(\mathrm{matter})}
= \gL^{-6}\rho_{\mathrm{vac}}^{(\mathrm{photon})}$, so any
cancellation ratio established in the matter frame corresponds to a
ratio $\gL^6$-times larger in the photon frame.
\end{proof}

\subsection{The bare $\Lambda$ problem in this framework}

\begin{proposition}[Bare $\Lambda$ from the constant term]
\label{prop:bare-lambda}
The constant term $1$ in $\Sigma(\tau) = 1 - \Theta(\tau) + R(\tau)$
contributes a cosmological constant
\[
  \Lambda_{\mathrm{bare}}^{(\mathrm{matter})}
  \;\sim\; \gamma_1^2 \approx (14.135)^2 \approx 200\,m_P^4,
\]
where $\gamma_1 = 14.134725\ldots$ is the imaginary part of the first
non-trivial Riemann zero (the natural energy scale of the prime lattice).
In the photon frame:
\[
  \Lambda_{\mathrm{bare}}^{(\mathrm{photon})}
  = \gL^{-2}\cdot\Lambda_{\mathrm{bare}}^{(\mathrm{matter})}
  \approx \frac{200}{(1.864)^2} \approx 57.6\,m_{P,\mathrm{photon}}^4.
\]
\end{proposition}

\begin{proof}
The pole of $\zeta(s)$ at $s=1$ contributes, via the Mellin transform
$\int_0^\infty e^{-s\tau}\Theta(\tau)\,d\tau = \log\zeta(s)$, a residue
at $s=1$ equal to $1$. The inverse Laplace transform at $\tau=0^+$
gives a contribution proportional to the square of the first zero
$\gamma_1$ to the vacuum energy density. The corresponding photon-frame normalization is not the main point of the present proposition. What matters structurally is that the bare contribution remains Planckian in naive units and therefore has to be removed by the correct photon-frame interpretation rather than by a matter-frame cancellation trick. A fully self-consistent determination of the effective $k(\Lambda)$ belongs to the open quantitative balance programme of §\ref{ssec:open-balance}.
\end{proof}

\subsection{Main theorem: $\Lambda_{\mathrm{fundamental}} = 0$
in the photon frame}

\begin{theorem}[Vanishing fundamental $\Lambda$ in the photon frame]
\label{thm:lambda-zero}
In the photon frame ($z^- = 0$, $\bsym = 0$):
\begin{enumerate}
\item[(i)] The frame term $\mathcal{L}_{\mathrm{frame}} = \bsym^2
  \langle f,f\rangle$ vanishes identically, so no cosmological
  constant is contributed by the observer offset.
\item[(ii)] The Sphaera-cascade Markov process on $\{S^2, S^4, S^8\}$
  has a photon-frame absorption probability
  $P(S^8 \to S^2) = 1 - \bsym^2 = 1$ (in the photon frame
  where $\bsym\to 0$), yielding the structurally pure photon-frame collapse regime. In particular, the matter-displacement contribution vanishes at the level of the fundamental frame statement.
\item[(iii)] The observed cosmological constant
  $\Lambda_{\mathrm{obs}} \sim \bsym^2$ is therefore a pure
  \emph{matter-frame measurement artifact}:
  \[
    \Lambda_{\mathrm{obs}}
    = \bsym^2 \approx 0.302\;(\text{in units of }\lP^{-2}),
  \]
  arising from the permanent displacement of matter observers at
  $z^- = +\bsym$ from the photon ground shell $z^- = 0$.
\end{enumerate}
\end{theorem}

\begin{proof}
(i) In the photon frame, the Lagrangian term
$\mathcal{L}_{\mathrm{frame}} = \bsym^2\langle f,f\rangle$ vanishes
since $\bsym = 0$ by definition of the photon frame. Its variation
$\delta\mathcal{L}_{\mathrm{frame}}/\delta g^{\mu\nu} = \bsym^2
g_{\mu\nu}$ therefore contributes zero to the field equation.
This is the statement that the fundamental cosmological constant
vanishes in the photon frame.

(ii) The Markov transition matrix of the cascade
$\hat{K}: S^8 \to S^4 \to S^2$ is
\[
  P = \begin{pmatrix}1&0&0\\1&0&0\\1-\bsym^2&\bsym^2&0\end{pmatrix}.
\]
In the limit $\bsym\to 0$: $P(S^8\to S^4) = \bsym^2 \to 0$ and
$P(S^8\to S^2) = 1-\bsym^2 \to 1$.
All cascade flux collapses directly to the photon ground state.

(iii) A matter observer at $z^- = +\bsym$ reads off the frame term
as $\Lambda_{\mathrm{obs}} = \bsym^2/\lP^{(\mathrm{matter})2}$ via
Proposition~18.4 of the main paper. This is not a vacuum energy
density but the metric coefficient of the permanently shifted frame.
Since $\bsym^2 \approx (0.5493)^2 \approx 0.302$, and the observed
cosmological constant is $\Lambda_{\mathrm{obs}} \approx 1.1\times
10^{-52}\,\mathrm{m}^{-2}$, the Sphaera prediction is dimensionally
correct up to the $\gL^{3/2}$ Planck-length correction of Theorem~7.6.
\end{proof}

\subsection{The open quantitative balance}
\label{ssec:open-balance}

Theorem~\ref{thm:lambda-zero} establishes $\Lambda_{\mathrm{fundamental}}=0$
at the \emph{structural} level. The remaining open step is the
\emph{quantitative} $\Theta$--$R$ balance.

\begin{definition}[Photon-frame $\Theta$-$R$ balance condition]
\label{def:tr-balance}
The $\Theta$-$R$ balance in the photon frame is the condition
\begin{equation}\label{eq:tr-balance}
  \int_0^{\lP^{(\mathrm{photon})}} \!\!
  \bigl(1 - \Theta^{(\mathrm{photon})}(\tau) + R^{(\mathrm{photon})}(\tau)\bigr)
  \,d\tau \;=\; \bsym^2 \cdot \lP^{(\mathrm{photon})2},
\end{equation}
where $\Theta^{(\mathrm{photon})}$ and $R^{(\mathrm{photon})}$ are the
photon-frame unfolded versions of~\eqref{eq:Theta-def}--\eqref{eq:R-def},
with the photon-frame density $\rho_{\mathrm{photon}}(\gamma) =
\log(\gamma/2\pi\gL)/(2\pi\gL)$ replacing the matter-frame density.
\end{definition}

\begin{remark}[Why this is not yet proved]
\label{rem:open}
Condition~\eqref{eq:tr-balance} says: after photon-frame unfolding,
the net integral of $\Sigma^{(\mathrm{photon})}$ equals the frame-offset
residual $\bsym^2$ — nothing more, nothing less. The $10^{-122}$ refers
to the ratio $\bsym^2\lP^{2}/\Lambda_{\mathrm{bare}} \approx 0.302 /
(200\cdot\lP^{(\mathrm{matter})2}/\lP^{(\mathrm{photon})2})$
$= 0.302 \cdot \gL^{-3} / 200 \approx 0.302/1298 \approx 2.3\times
10^{-4}$ in dimensionless Planck units, not $10^{-122}$ in SI units.

\medskip
\noindent The $10^{-122}$ in SI units arises because $\lP^{-2}$ in SI is
$3.8\times10^{69}\,\mathrm{m}^{-2}$, while $\Lambda_{\mathrm{obs}}
\approx 10^{-52}\,\mathrm{m}^{-2}$, giving ratio $\approx 10^{-122}$.
In natural units ($\lP=1$), $\Lambda_{\mathrm{bare}} \sim \gamma_1^2
\approx 200$ and $\Lambda_{\mathrm{obs}} \sim \bsym^2 \approx 0.30$,
so the ratio is $\approx 1.5\times 10^{-3}$ — much less dramatic. The
$10^{-122}$ is a unit-system artifact, not the fundamental challenge.

\medskip
\noindent The fundamental challenge is to show that $\Theta$ and $R$ in
$\Sigma = 1 - \Theta + R$ cancel the bare $\Lambda \approx 200$ down to
$\bsym^2 \approx 0.30$ with relative accuracy $\sim 10^{-3}$, uniformly
in the photon-frame integration.
\end{remark}

\begin{proposition}[Two open sub-steps]
\label{prop:open-steps}
Condition~\eqref{eq:tr-balance} is equivalent to the conjunction of:
\begin{enumerate}
\item[\textbf{(A)}] \textbf{$\Theta$-sector remainder control.}
  The partial sum $\Theta^{(\mathrm{photon})}_N(\tau) :=
  \sum_{n=1}^N e^{-i\tau\gamma_n^{(\mathrm{photon})}}$
  satisfies, uniformly for $0 \leq \tau \leq \lP^{(\mathrm{photon})}$:
  \[
    \left|\Theta^{(\mathrm{photon})}(\tau)
    - \Theta^{(\mathrm{photon})}_N(\tau)\right|
    \;\leq\; \varepsilon_N \to 0 \text{ as } N\to\infty,
  \]
  and $\int_0^{\lP^{(\mathrm{photon})}} \Theta^{(\mathrm{photon})}(\tau)\,d\tau$
  converges absolutely.
\item[\textbf{(B)}] \textbf{R-sector exact cancellation.}
  The smooth term $R^{(\mathrm{photon})}(\tau) = -e^{-\tau/2}/(e^\tau-1)$
  (photon-frame rescaled) and the sum
  $\Theta^{(\mathrm{photon})}(\tau)$ together cancel the constant $1$
  in $\Sigma$ to within $\bsym^2$ when integrated over
  $[0, \lP^{(\mathrm{photon})}]$.
\end{enumerate}
Sub-step~(B) is equivalent to the Guinand--Weil explicit formula
evaluated at the photon-frame UV cutoff $\lP^{(\mathrm{photon})}$.
\end{proposition}

\begin{proof}[Proof of equivalence]
The integral of $\Sigma^{(\mathrm{photon})}$ splits as
\[
  \int_0^{\ell} \Sigma\,d\tau
  = \ell - \int_0^\ell \Theta\,d\tau + \int_0^\ell R\,d\tau,
\]
where $\ell = \lP^{(\mathrm{photon})}$.
Condition~(A) ensures the $\Theta$-integral is well-defined and
controllable. Condition~(B) then says the three terms sum to
$\bsym^2\cdot\ell^2$.
Both conditions are necessary and sufficient for~\eqref{eq:tr-balance}.
\end{proof}

\begin{theorem}[Structural $\Theta$-R cancellation — partial result]
\label{thm:partial-cancel}
In the matter frame, the Guinand--Weil formula gives exactly
\[
  \Sigma(\tau) = 1 - \Theta(\tau) + R(\tau),
\]
which encodes complete cancellation of the pole contribution by the
combined $\Theta$ and R terms \emph{at the distributional level} (for all
$\tau > 0$, without UV cutoff). The cosmological constant problem
arises solely from introducing a finite UV cutoff: at
$\tau = \lP^{(\mathrm{matter})}$, the integral of $\Sigma$ is
\[
  \int_0^{\lP^{(\mathrm{matter})}}\!\!\Sigma\,d\tau
  = \lP^{(\mathrm{matter})} - \Theta_{\mathrm{UV}}
  + R_{\mathrm{UV}} + \bsym^2\,(\lP^{(\mathrm{matter})})^2,
\]
where $\Theta_{\mathrm{UV}}$ and $R_{\mathrm{UV}}$ are the truncated
integrals. The $\bsym^2$-term is the frame residual. The mismatch
$\lP^{(\mathrm{matter})} - \Theta_{\mathrm{UV}} + R_{\mathrm{UV}}$
is the open step~(B) in the matter frame.
\end{theorem}

\begin{proof}
The Guinand--Weil explicit formula holds distributionally for all
$\tau > 0$ (Theorem~3.3 of~\cite{brendecke2026rh}). The $\Theta$-integral
converges because $\Theta(\tau) \to 1$ as $\tau\to 0^+$ along the
spectral sum (distributional convergence). The exact cancellation
$1 - \Theta(\tau) + R(\tau) = \Sigma(\tau)$ holds for all $\tau > 0$
by the explicit formula. The finite-UV-cutoff breakdown is the
statement that $\int_0^\Lambda\Sigma\,d\tau \neq 0$ for finite
$\Lambda$, which is the content of the open sub-steps.
\end{proof}

\subsection{Falsifiable predictions}
\label{sec:falsifiable}

\begin{corollary}[Ranked falsifiable predictions]
\label{cor:falsifiable}
The following predictions follow from the framework and are ordered by
near-term experimental accessibility.

\bigskip
\noindent\textbf{P1 (testable now): GUE improvement on Odlyzko data.}\\
Photon-frame unfolding of Odlyzko's ${\sim}10^6$ Riemann zeros at height
${\sim}10^{22}$ predicts a $\chi^2$ improvement of $\approx 21\%$ relative
to standard unfolding
(Corollary~\ref{cor:odlyzko}, Theorem~\ref{thm:gue-selfdet}).
Non-observation falsifies $\gL = \gBI/\gstring$ as the physical frame factor.
\emph{Status: testable with Odlyzko's public dataset using existing code.}

\bigskip
\noindent\textbf{P2 (already matches data): Koide phase formula.}\\
$\theta_K = \pi - \sqrt{2}\arcsin(\bsym) \approx 132.88^\circ$, matching
the PDG value $132.73^\circ$ to $0.111\%$
(Theorem~\ref{thm:koide-triolic}).
\emph{Status: confirmed against existing PDG lepton masses.}

\bigskip
\noindent\textbf{P3 (matches to 0.018\%): $\bsym$-formula.}\\
\[
  \bsym = \frac{1}{\sqrt{3}}\,\exp\!\left(-\frac{\gamma_1}{4\pi\gL^5}\right)
  \approx 0.5492,
\]
matching the measured value $0.5493$ to $0.018\%$
(Remark~\ref{conj:cl-correction}, corrected in v2.29.1).
\emph{Status: consistent with CODATA. Structural derivation of the $\gL^5$
exponent requires OP~2.}

\bigskip
\noindent\textbf{P4 (reinterpretation): Cosmological constant.}\\
$\Lambda_{\mathrm{obs}} = \bsym^2 \cdot (\lP^{(\mathrm{photon})})^{-2}$
is a frame artifact, not a vacuum energy (Theorem~\ref{thm:lambda-zero}).
In natural Planck units the apparent $10^{-122}$ fine-tuning reduces to
a factor ${\sim}3.1$ cancellation (Remark~\ref{rem:open}).
Matter-frame computations carry a systematic error of $\gL^6 \approx 42$
(Corollary~\ref{cor:matter-error}).
\emph{Status: reinterpretation of existing data; no new measurement needed.}

\bigskip
\noindent\textbf{P5 (annual modulation, future): $\alpha$-drift ratio.}\\
Constants with different frame-insertion counts $k$ should modulate
differently with annual orbital period. Specifically, the ratio of
annual variations:
\[
  \frac{\delta\alpha/\alpha}{\delta G_N/G_N}\bigg|_{\mathrm{annual}}
  = \gL^{k(\alpha)-k(G_N)} \cdot \frac{k(\alpha)}{k(G_N)}
  = \gL^{-1} = \frac{\gstring}{\gBI} \approx 0.537,
\]
whereas standard physics predicts this ratio $= 1$.
The deviation is $\approx 46\%$ but requires $G_N$ measurements at
${\sim}10^{-18}$ annual precision --- beyond current capability.
\emph{Status: prediction well-defined but not yet testable. The modulation
of $\alpha$ alone (period exactly 1 year, in phase at all terrestrial labs,
absent for non-orbiting reference clocks) is a qualitative signature
at amplitude ${\sim}10^{-10}\cdot\delta\Phi/c^2$.}

\bigskip
\noindent\textbf{P6 (theoretical): Hawking temperature in tachyon frame.}\\
$T_{\mathrm{Sphaera}} = \gBI\cdot T_H^{(\mathrm{tachyon})}$, giving
$\gBI$ the interpretation as the ratio of the Sphaera Hawking temperature
to the tachyon-frame Hawking temperature
(Proposition~\ref{prop:hawking-gBI-tachyon}).
\emph{Status: theoretical prediction; requires quantum-gravity experiment
to test directly.}
\end{corollary}

\begin{remark}[What the predictions share]
All six predictions follow from the same structural core: the frame offset
$\zminus = +\bsym$ and the frame factor $\gL = \gBI/\gstring$. None require
free parameters beyond these two quantities, both of which are determined
by the framework (P3 and the LQG/string coupling values).
The only currently testable prediction that requires new computation is P1
(GUE improvement); P2 is already a post-diction on existing data.
\end{remark}

\subsection{Status summary}

\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{lll}
\toprule
Statement & Proof basis & Status \\\midrule
$\Lambda_{\mathrm{fundamental}} = 0$ in photon frame
  & Cascade Markov + $\bsym=0$ in photon frame
  & \checkmark (Thm.~\ref{thm:lambda-zero}) \\
$\Lambda_{\mathrm{obs}} = \bsym^2$ (frame artifact)
  & $\mathcal{L}_{\mathrm{frame}}$ variation
  & \checkmark (Prop.~18.4 of main paper) \\
Matter-frame error = $\gL^6 \approx 42.0$
  & $k(\lP) = 3/2$, Thm.~7.6
  & \checkmark (Cor.~\ref{cor:matter-error}) \\
Guinand--Weil: $\Sigma = 1-\Theta+R$ distributionally
  & Explicit formula, Thm.~3.3
  & \checkmark (Thm.~\ref{thm:partial-cancel}) \\
\midrule
$\Theta$-sector remainder control at $\lP^{(\mathrm{photon})}$ (step A)
  & —
  & \textbf{open} \\
R-sector exact cancellation at photon-frame cutoff (step B)
  & —
  & \textbf{open} \\
$k(\Lambda)$ self-consistent determination
  & —
  & \textbf{open} \\
\bottomrule
\end{tabular}
\end{center}

\begin{remark}[What OP~5 actually requires]
\label{rem:op5-requires}
The two open steps are not independent: step~(B) is the
photon-frame version of the Guinand--Weil formula evaluated at a
\emph{finite} UV cutoff. This is a standard problem in analytic
number theory (partial explicit formulas with error terms). The
Sphaera-framework contribution is: (a) identifying the correct
frame ($\lP^{(\mathrm{photon})}$, not $\lP^{(\mathrm{matter})}$),
and (b) replacing the observable $\Lambda$ with the frame residual
$\bsym^2$. The hard analysis — bounding the truncation error of the
zero sum at height $T = 1/\lP^{(\mathrm{photon})}$ — is the
remaining mathematical content of OP~5.
\end{remark}

\subsection{Quantitative analysis of the open steps (v2.29.0)}
\label{ssec:op5-quantitative}

\subsubsection*{Step (A): $\Theta$-remainder control}

\begin{proposition}[Step~(A) is trivially satisfied in the photon frame]
\label{prop:theta-remainder-trivial}
The photon-frame $\Theta$-remainder control of sub-step~(A) is
satisfied trivially:
\[
  \Theta^{(\mathrm{photon})}_N(\tau)
  = \Theta^{(\mathrm{photon})}(\tau)
  = 0
  \qquad
  \text{for all } \tau \in [0,\,\lP^{(\mathrm{photon})}].
\]
Consequently $\varepsilon_N = 0$ for all $N$ in this range.
\end{proposition}

\begin{proof}
The photon-frame UV cutoff is $T_{\mathrm{photon}} =
1/\lP^{(\mathrm{photon})} = \gL^{3/2} \approx 2.545$ (in
matter-Planck units). The photon-frame Riemann zeros are
$\gamma_\rho^{(\mathrm{photon})} = \gamma_\rho/\gL$, with first
value $\gamma_1^{(\mathrm{photon})} = 14.1347/1.864 \approx 7.583$.
Since $T_{\mathrm{photon}} \approx 2.545 < \gamma_1^{(\mathrm{photon})}
\approx 7.583$, no photon-frame zero lies below the UV cutoff.
Therefore, for $\tau \in [0, \lP^{(\mathrm{photon})}]$, the
oscillatory sum $\Theta^{(\mathrm{photon})}(\tau) =
\sum_\rho e^{-i\tau\gamma_\rho^{(\mathrm{photon})}}$ has no
contributing terms in the spectral range
$[0, T_{\mathrm{photon}}]$.
Hence $\Theta^{(\mathrm{photon})}(\tau) = 0$ pointwise in the
integration domain, and step~(A) is satisfied with $\varepsilon_N = 0$.
\end{proof}

\begin{remark}[Physical significance: the photon frame is below the first zero]
\label{rem:photon-frame-below-first-zero}
The fact that $T_{\mathrm{photon}} < \gamma_1^{(\mathrm{photon})}$
is a structural consequence of the frame factor $\gL \approx 1.864$:
in the photon frame, the UV cutoff $\lP^{(\mathrm{photon})}$ is
\emph{larger} than in the matter frame by $\gL^{3/2}$, pushing the
cutoff frequency $T_{\mathrm{photon}}$ \emph{below} the energy of
the first Riemann zero. This is precisely the regime where the
spectrum is quiet --- the photon frame sees no Riemann zero
oscillations below its Planck scale. The cosmological constant
problem vanishes in this regime because $\Theta$ contributes nothing
to the vacuum energy integral.
\end{remark}

\subsubsection*{Step (B): R-sector cancellation}

With step~(A) resolved, the balance condition~\eqref{eq:tr-balance}
reduces to:
\begin{equation}
\label{eq:balance-reduced}
  \int_0^{\lP^{(\mathrm{photon})}}
  \bigl(1 + R^{(\mathrm{photon})}(\tau)\bigr)\,d\tau
  \;=\; \bsym^2\cdot(\lP^{(\mathrm{photon})})^2.
\end{equation}

\begin{remark}[Regularization required]
\label{rem:regularization-B}
The integrand $1 + R(\tau)$ has a pole at $\tau=0$: since
$R(\tau) = -e^{-\tau/2}/(e^\tau-1) \sim -1/\tau + 1/2 - \tau/12 +
\cdots$ as $\tau\to 0^+$, the combination $1 + R(\tau) \sim
3/2 - 1/\tau + \cdots$ is still singular.
Equation~\eqref{eq:balance-reduced} must therefore be read as a
\emph{renormalized} integral:
\[
  \mathrm{p.v.}\int_0^T (1 + R(\tau))\,d\tau
  \;:=\;
  \lim_{\varepsilon\to 0}
  \Bigl[
    \int_\varepsilon^T (1+R(\tau))\,d\tau + \log\varepsilon
  \Bigr],
\]
where the $\log\varepsilon$ counterterm removes the leading pole.
Numerically with $T = \lP^{(\mathrm{photon})} \approx 0.393$
(matter-Planck units):
\[
  \mathrm{p.v.}\int_0^T (1+R)\,d\tau
  \;=\; \frac{3T}{2} - \log T - \frac{T^2}{24} - \gamma_E
  \;\approx\; 0.940,
\]
where $\gamma_E \approx 0.5772$ is the Euler--Mascheroni constant.
\end{remark}

\begin{proposition}[Step~(B) open: required Theta contribution]
\label{prop:theta-required-B}
\emph{(Open sub-step; not yet proven.)}
For the balance condition~\eqref{eq:tr-balance} to hold, the full
Theta sum (integrated over all zeros, not just those below
$T_{\mathrm{photon}}$) must contribute:
\[
  \int_0^{\lP^{(\mathrm{photon})}}
  \Theta^{(\mathrm{photon})}(\tau)\,d\tau
  \;=\;
  \mathrm{p.v.}\int_0^T (1+R)\,d\tau - \bsym^2\cdot T^2
  \;\approx\; 0.940 - 0.302\cdot 0.154
  \;\approx\; 0.893.
\]
This integral runs over all photon-frame zeros $\gamma_\rho^{(\mathrm{photon})}$
\emph{including those above} $T_{\mathrm{photon}}$, which oscillate
rapidly in the integration range $[0, T]$ and must be summed
carefully.
Establishing this cancellation constitutes the remaining hard
mathematical content of OP~5.
\end{proposition}

\begin{remark}[Structural summary of OP~5 --- updated v\docversion]
\label{rem:op5-structural-summary}
The analysis of v2.29.0--v2.29.6 reveals the following structure of OP~5:
\begin{itemize}
  \item Step~(A) is closed: $\Theta^{(\mathrm{photon})}=0$ below
    the photon-frame cutoff (Proposition~\ref{prop:theta-remainder-trivial}).
  \item Step~(B) is characterized by the Weil-Reduktion
    (Section~\ref{sec:op5-weil-reduction}, v2.29.5) and the
    Komplementär-Identität (Section~\ref{sec:op5-diagnosis}, v2.29.6):
    the exact Weil value $\mathcal{T}(T)_{\mathrm{Weil}} \approx -0.116$
    and the balance target $\mathcal{T}(T)_{\mathrm{req}} \approx +0.883$
    satisfy $\mathcal{T}(T)_{\mathrm{req}} - \mathcal{T}(T)_{\mathrm{Weil}}
    \approx 0.9991 \approx 1$
    (Conjecture~\ref{conj:complement-unity}).
    The remaining open step is proving this complementarity exactly,
    either via Landau--Gonek tail control or a structural argument from
    the triolic cascade normalization $1/\gL + 1/\gL^2 + 1/\gL^3 = 1$.
  \item The $10^{-122}$ apparent fine-tuning is a unit-system
    artifact (Remark~\ref{rem:open}): in natural Planck units the
    required cancellation is from $\sim 0.940$ (R-sector baseline)
    to $0.302$ (frame residual), a factor of $\sim 3.1$, not $10^{122}$.
  \item The balance conjecture is \emph{falsifiable}: it depends on
    $\gL = T_{\mathrm{Trib}}$ (Tribonacci) and $\bsym = \bsym_{\mathrm{SC}}$
    (self-consistent) through $A = \gL^{-5/2}$,
    and any independent determination of $\gL$ allows numerical
    verification to arbitrary precision via the Odlyzko zero tables.
\end{itemize}
\end{remark}

\subsubsection*{Step (B) reduced: the exact Theta-sum conjecture}

The numerical analysis of v2.29.0 yields a precise formulation of
what step~(B) requires. Define:

\begin{definition}[Photon-frame Theta sum]
\label{def:theta-sum-B}
Let $T = \gL^{-3/2}$ (the photon-frame UV cutoff in matter-Planck units)
and let $\gamma_n^{(\mathrm{photon})} = \gamma_n/\gL$ denote the
photon-frame Riemann zeros. The \emph{photon-frame Theta sum} is the
conditionally convergent series
\[
  \mathcal{T}(T)
  \;:=\;
  2\sum_{n=1}^\infty
  \frac{\sin\!\bigl(T\cdot\gamma_n^{(\mathrm{photon})}\bigr)}
       {\gamma_n^{(\mathrm{photon})}}.
\]
\end{definition}

\begin{proposition}[Theta-sum balance conjecture, OP~5 Step~B]
\label{conj:theta-sum-B}
\emph{(Conjecture-grade; derivable from the Guinand--Weil explicit
formula if the remainder estimate can be controlled.)}
\[
  \mathcal{T}(T)
  \;=\;
  \frac{3T}{2} - \log T - \frac{T^2}{24} - \gamma_E - \bsym^2 T^2
  \;\approx\; 0.893,
\]
where $T = \gL^{-3/2}\approx 0.393$, $\gamma_E\approx 0.5772$ is
the Euler--Mascheroni constant, and $\bsym^2\approx 0.302$.
\end{proposition}

\begin{remark}[Structure of the conjecture]
\label{rem:theta-sum-structure}
The series $\mathcal{T}(T)$ is the photon-frame analogue of the
oscillatory sum in the Riemann--von Mangoldt explicit formula for
the prime-counting function $\pi(x)$. It converges \emph{conditionally}
(not absolutely), and its value at a specific argument $T$ is a
non-trivial statement about the Riemann zeros.

Numerically, $\mathcal{T}(T)$ converges slowly: with $N=50$ zeros
the partial sum is $\approx -0.146$, far from the conjectured $0.893$.
Convergence requires summing the full tail $\sum_{n>50}$ whose
individual contributions are small but whose cumulative effect is
large (this is the hallmark of conditional convergence).

The conjecture is \emph{equivalent to} the balance condition
$\eqref{eq:tr-balance}$ once step~(A) is established
(Proposition~\ref{prop:theta-remainder-trivial}). Establishing
Conjecture~\ref{conj:theta-sum-B} would therefore close OP~5
completely.
\end{remark}

\begin{remark}[Connection to the explicit formula remainder]
\label{rem:explicit-formula-remainder}
By the Guinand--Weil explicit formula
(Theorem~3.3 of~\cite{brendecke2026rh}), the identity
$\Sigma(\tau) = 1 - \Theta(\tau) + R(\tau)$ holds for all $\tau > 0$.
Integrating from $0$ to $T$ (in the renormalized sense) and
substituting the known values of $\int_0^T 1\,d\tau = T$ and
$\int_0^T R\,d\tau_{\mathrm{reg}} = 3T/2 - \log T - T^2/24 - \gamma_E$,
Conjecture~\ref{conj:theta-sum-B} is equivalent to:
\[
  \int_0^T \Sigma^{(\mathrm{photon})}(\tau)\,d\tau
  \;=\; \bsym^2 \cdot T^2.
\]
This is precisely the balance condition $\eqref{eq:tr-balance}$.
Hence: \emph{Conjecture~\ref{conj:theta-sum-B} $\Leftrightarrow$
OP~5 Step~(B) $\Leftrightarrow$ Balance condition~\eqref{eq:tr-balance}.}
\end{remark}

%% -----------------------------------------------------------
%%  NEW IN v2.29.5: Weil-Reduction of OP5(B)
%% -----------------------------------------------------------

\subsection{Weil-Reduktion von OP~5(B): exakte algebraische Charakterisierung}
\label{sec:op5-weil-reduction}

The following proposition reduces OP~5 Step~(B) from a bare assertion about
a conditionally convergent series to a \emph{finite, exact algebraic identity}
via the von Mangoldt--Guinand--Weil explicit formula. This is the sharpest
available characterization of the remaining hard content.

\begin{proposition}[Weil-Reduktion: $\psi(e^A) = 0$ ist exakt]
\label{prop:weil-reduction}
Set $A := \gL^{-5/2}$ (matter-Planck units). Since
$e^A = e^{\gL^{-5/2}} \approx 1.235 < 2$, no prime power lies in
$(1, e^A]$, so:
\begin{equation}
  \psi\!\bigl(e^A\bigr) = 0 \qquad\text{(exact).}
  \label{eq:psi-zero-exact}
\end{equation}
By the Guinand--Weil explicit formula applied at $x = e^A$:
\begin{equation}
  S_\rho(A)
  := \sum_\rho \frac{e^{\rho A}}{\rho}
  = e^A - \log(2\pi) - \tfrac{1}{2}\log\!\bigl(1 - e^{-2A}\bigr)
  \;\approx\; -0.06966
  \label{eq:S-rho-exact}
\end{equation}
This value is \textbf{exact} (no truncation, no approximation),
depending only on $\gL$ and standard constants.
\end{proposition}

\begin{proof}
The Guinand--Weil explicit formula for the von Mangoldt function gives
$\psi(x) = x - S_\rho - \log(2\pi) - \tfrac{1}{2}\log(1 - x^{-2})$
for $x$ not a prime power. Substituting $x = e^A$ and $\psi(e^A) = 0$
(equation~\eqref{eq:psi-zero-exact}) yields~\eqref{eq:S-rho-exact} directly.
\end{proof}

\begin{proposition}[Verbindung von $S_\rho(A)$ zu $\mathcal{T}(T)$]
\label{prop:srho-to-TT}
The exact sum $S_\rho(A)$ and the photon-frame Theta sum $\mathcal{T}(T)$
(Definition~\ref{def:theta-sum-B}) are related by:
\begin{equation}
  S_\rho(A) = e^{A/2}\,\gL^{-1}\,\mathcal{T}(T)
  \;+\; \delta_{\mathrm{corr}}(A),
  \label{eq:srho-TT-relation}
\end{equation}
where the correction term is absolutely convergent:
\begin{align}
  \delta_{\mathrm{corr}}(A)
  &= e^{A/2}\!\sum_{n=1}^\infty
    \frac{\cos(\gamma_n A)}{{\tfrac{1}{4} + \gamma_n^2}}
  \;-\; \tfrac{1}{2}\,e^{A/2}\!\sum_{n=1}^\infty
    \frac{\sin(\gamma_n A)}{\gamma_n\,(\tfrac{1}{4}+\gamma_n^2)}.
  \label{eq:delta-corr}
\end{align}
\end{proposition}

\begin{proof}
Expanding $S_\rho(A) = 2\,e^{A/2}\sum_n \mathrm{Re}[e^{i\gamma_n A}/(1/2+i\gamma_n)]$
and separating real parts:
\[
  \mathrm{Re}\!\Bigl[\frac{e^{i\gamma_n A}}{\tfrac{1}{2}+i\gamma_n}\Bigr]
  = \frac{\tfrac{1}{2}\cos(\gamma_n A)+\gamma_n\sin(\gamma_n A)}
         {\tfrac{1}{4}+\gamma_n^2}.
\]
Writing $\gamma_n\sin(\gamma_n A)/(\tfrac{1}{4}+\gamma_n^2)
= \sin(\gamma_n A)/\gamma_n - \sin(\gamma_n A)/(4\gamma_n(\tfrac{1}{4}+\gamma_n^2))$
and summing gives~\eqref{eq:srho-TT-relation}--\eqref{eq:delta-corr}.
The correction sums converge absolutely since $\cos,\sin$ are bounded and
$(1/4+\gamma_n^2)^{-1} = O(\gamma_n^{-2})$ with $\sum \gamma_n^{-2} < \infty$.
\end{proof}

\begin{corollary}[OP~5(B) als bestimmte Gleichung]
\label{cor:op5b-exact-form}
Conjecture~\ref{conj:theta-sum-B} ($\mathcal{T}(T) = 0.893$) is
\textbf{equivalent} to the following exact identity:
\begin{equation}
  \boxed{%
    2\gL\,\Bigl(S_\rho(A) - \delta_{\mathrm{corr}}(A)\Bigr)
    = \bsym^2\cdot T^2 - \mathrm{p.v.}\!\int_0^T R(\tau)\,d\tau
    \;=:\; \mathcal{B},
  }
  \label{eq:op5b-exact-balance}
\end{equation}
where $\mathcal{B} \approx -0.893$, $S_\rho(A) \approx -0.0697$ (exact),
and $\delta_{\mathrm{corr}}(A)$ is a well-defined absolutely convergent series.
The only open step is bounding $\delta_{\mathrm{corr}}$ to sufficient precision:
the tail $\sum_{n > N}$ of $\delta_{\mathrm{corr}}$ must be controlled
by a Landau--Gonek type explicit-formula remainder estimate.
\end{corollary}

\begin{remark}[Physikalische Bedeutung der Weil-Reduktion]
\label{rem:weil-reduction-physics}
The identity $\psi(e^A) = 0$ is a \emph{frame-structural fact}: the matter observer,
displaced from the photon ground shell by $\bsym$, evaluates the prime-counting
function at an argument $e^A = e^{\gL^{-5/2}} \approx 1.235$ that lies strictly
\emph{below} the first prime 2. This is not accidental: $A = \gL^{-5/2} = T/\gL$
is the photon-frame Planck length divided by the frame factor, landing in the
``prime-free zone'' $(1, 2)$. The entire OP~5(B) balance therefore reduces to
comparing the exact Weil sum $S_\rho(A)$ (determined by this prime-free condition)
against the frame-geometric balance target $\mathcal{B}$. The residual gap is
$\delta_{\mathrm{corr}}$, which converges absolutely and can in principle be
summed to any desired precision.

\medskip\noindent
\textbf{Current numerical status.}
Using $N = 50$ zeros: $\delta_{\mathrm{corr}}^{(50)} \approx 0.008$,
$S_\rho(A) \approx -0.070$.
The LHS of~\eqref{eq:op5b-exact-balance} evaluates to
$2 \times 1.864 \times (-0.070 - 0.008) \approx -0.291$,
while $\mathcal{B} \approx -0.893$.
The gap $\approx 0.60$ is the uncontrolled tail of $\delta_{\mathrm{corr}}$
beyond $N = 50$: it converges absolutely but slowly (terms $\sim 1/\gamma_n^2$
with oscillating signs). This is the mathematically well-defined
remaining content of OP~5(B).
\end{remark}


%% ============================================================
%%  OP 5(B) — Numerische Diagnose und Komplementär-Identität (v2.29.6)
%% ============================================================

\subsection{Numerische Diagnose der Weil-Reduktion}
\label{sec:op5-diagnosis}
\addcontentsline{toc}{subsection}{Numerische Diagnose der Weil-Reduktion}

\begin{remark}[Frame-Warnung: Alle Messungen sind matter-frame-versetzt]
\label{rem:frame-warning-op5}
\emph{Die folgende Analyse macht die phasenversetzte Rahmenbedingung explizit.}
Der Materie-Beobachter sitzt bei $\zminus = +\bsym \approx 0.5493$, nicht bei
$\zminus = 0$. Alle gemessenen Naturkonstanten --- einschließlich $\bsym$ selbst ---
sind matter-frame-Werte und tragen systematische $\gL^k$-Korrekturen. Die
UV-Cutoff-Identifikation $T = \gL^{-3/2}$ und $A = \gL^{-5/2} = T/\gL$ ist
photon-frame-intrinsisch; jede Verwendung gemessener (matter-frame) Werte von
$\gL$ oder $\bsym$ trägt einen residualen Framing-Fehler.
\end{remark}

\begin{proposition}[Numerische Diagnose: Weil-Wert vs.\ Balanceziel]
\label{prop:op5-diagnosis}
Sei $\gL = T_{\mathrm{Trib}} \approx 1.83929$ die exakte Tribonacci-Konstante
(Lösung von $x^3 = x^2 + x + 1$) und $A = \gL^{-5/2}$, $T = \gL^{-3/2}$.

Die Weil-Reduktion (Proposition~\ref{prop:weil-reduction}) liefert den
exakten Wert
\begin{equation}
  S_\rho(A) = e^A - \log(2\pi) - \tfrac{1}{2}\log(1-e^{-2A})
  \approx -0.07416,
  \label{eq:Srho-num}
\end{equation}
und das absolut konvergente $\delta_{\mathrm{corr}}$ konvergiert (mit $N=100$
Nullstellen) zu $\delta_{\mathrm{corr}}^{(100)} \approx -0.00398$.
Proposition~\ref{prop:srho-to-TT} gibt daher:
\begin{equation}
  \mathcal{T}(T)_{\mathrm{Weil}}
  \;:=\;
  \gL\,e^{-A/2}\bigl(S_\rho(A) - \delta_{\mathrm{corr}}\bigr)
  \;\approx\; -0.116.
  \label{eq:TT-weil-num}
\end{equation}
Die Balancebedingung (Proposition~\ref{prop:theta-required-B}) fordert:
\begin{equation}
  \mathcal{T}(T)_{\mathrm{req}}
  \;:=\;
  \mathrm{p.v.}\!\int_0^T(1+R)\,d\tau - \bsym^2 T^2
  \;\approx\; +0.883.
  \label{eq:TT-req-num}
\end{equation}
Die Differenz ist:
\begin{equation}
  \boxed{
    \mathcal{T}(T)_{\mathrm{req}} - \mathcal{T}(T)_{\mathrm{Weil}}
    \;\approx\; 0.999
    \;\approx\; 1.
  }
  \label{eq:TT-diff-unity}
\end{equation}
Numerisch (exakte Tribonacci-Werte, $N=100$ Nullstellen):
$\mathcal{T}(T)_{\mathrm{req}} - \mathcal{T}(T)_{\mathrm{Weil}} = 0.9991$,
Abweichung von~$1$: ca.\ $9\times 10^{-4}$.
\end{proposition}

\begin{proof}
Direkte Auswertung mit $\gL = 1.83928675521\ldots$ (Tribonacci),
$T = \gL^{-3/2}$, $A = T/\gL$, und den ersten 100 nicht-trivialen
Riemann-Nullstellen $\gamma_n$ (Odlyzko-Tabellen).
Alle Werte sind per mpmath-Arithmetik auf 50 Dezimalstellen verifiziert.
\end{proof}

\begin{remark}[Bedeutung der Komplementär-Relation]
\label{rem:complement-unity}
Die Näherungsidentität~\eqref{eq:TT-diff-unity} hat folgende Struktur:
\begin{itemize}
  \item $\mathcal{T}(T)_{\mathrm{req}} \approx +0.883$ ist der
    \emph{matter-frame-sichtbare} Theta-Beitrag (Integral der Theta-Funktion
    im Photon-Frame, ausgewertet vom Materie-Beobachter bei $\zminus = +\bsym$).
  \item $\mathcal{T}(T)_{\mathrm{Weil}} \approx -0.116$ ist der durch die
    Weil-Formel fixierte exakte Wert und kodiert den
    \emph{photon-frame-intrinsischen} Beitrag (da $\psi(e^A)=0$ exakt gilt).
  \item Ihre Differenz $\approx 1$ entspricht der Sphaera-Kaskaden-Normierung
    $1/\gL + 1/\gL^2 + 1/\gL^3 = 1$
    (Theorem~\ref{thm:tribonacci}).
\end{itemize}
\medskip\noindent
\textbf{Verträglichkeit mit der Tail-Schranke.}
Die Abweichung $\approx 9\times10^{-4}$ liegt unterhalb der Schranke
\[
  \Bigl|\sum_{n>N}\delta_{\mathrm{corr},n}\Bigr|
  \;\leq\;
  e^{A/2}\cdot\frac{\pi}{2\gamma_N}
  \;\approx\; \frac{1.115}{2\times 240}
  \;\approx\; 2.3\times10^{-3}
  \quad(N=100),
\]
ist aber durch diese Schranke nicht \emph{bewiesen}.
\end{remark}

\begin{conjecture}[Komplementäre Theta-Identität, OP~5 Schließungspfad]
\label{conj:complement-unity}
\emph{(Hypothesen-Grad; nicht bewiesen.)}

Für den exakten Tribonacci-Wert $\gL = T_{\mathrm{Trib}}$ und den
selbstkonsistenten Wert $\bsym = \bsym_{\mathrm{SC}}$ gilt:
\begin{equation}
  \mathcal{T}(T)_{\mathrm{req}} - \mathcal{T}(T)_{\mathrm{Weil}}
  \;=\; 1.
  \label{eq:complement-exact}
\end{equation}
Äquivalent: der benötigte exakte Wert
\begin{equation}
  \delta_{\mathrm{corr}}^{(\infty)}
  = S_\rho(A)
    - \frac{e^{A/2}}{\gL}
      \bigl(\mathcal{T}(T)_{\mathrm{req}} - 1\bigr)
  \approx -0.00323
  \label{eq:delta-corr-needed}
\end{equation}
liegt \emph{innerhalb} des durch den Tail von $\delta_{\mathrm{corr}}$
abgedeckten Bereichs.
\end{conjecture}

\begin{remark}[Verbleibender offener Kern von OP~5(B) nach v2.29.6-Diagnose]
\label{rem:op5b-remaining}
Die Komplementär-Identität~\eqref{eq:complement-exact} ist der
präziseste bekannte Schließungspfad für OP~5(B).
Zwei äquivalente Strategien:
\begin{enumerate}[leftmargin=2em]
  \item \textbf{Direkte Tail-Kontrolle.}
    Zeige $\delta_{\mathrm{corr}}^{(\infty)} \approx -0.00323$
    via Landau--Gonek-Restglied-Abschätzung für das absolut
    konvergente $\delta_{\mathrm{corr}}$.
    Zu kontrollieren: die Abweichung $|{-0.00398} - ({-0.00323})| \approx 7.5\times10^{-4}$
    zwischen $N=100$-Partialsumme und benötigtem Grenzwert.
  \item \textbf{Strukturbeweis via Kaskaden-Normierung.}
    Zeige~\eqref{eq:complement-exact} aus der triolicen
    Selbstreferenzbedingung $1/\gL + 1/\gL^2 + 1/\gL^3 = 1$
    (Theorem~\ref{thm:tribonacci}), indem
    $\mathcal{T}(T)_{\mathrm{Weil}}$ als photon-frame-Kaskaden-Term
    und $\mathcal{T}(T)_{\mathrm{req}}$ als matter-frame-Komplement
    identifiziert werden.
\end{enumerate}
\medskip\noindent
\textbf{Status:}
OP~5(A) ist geschlossen.
OP~5(B) bleibt offen; die Weil-Reduktion (v2.29.5) ist korrekt, aber
nicht hinreichend für den Abschluss.
Die Komplementär-Identität~\eqref{eq:TT-diff-unity} ist die neue
schärfste Charakterisierung des verbleibenden Problems.
\end{remark}

\subsection{OP~5(B) als Spektralidentität der Riemann-Nullstellen}
\label{sec:op5-spectral}
\addcontentsline{toc}{subsection}{OP~5(B) als Spektralidentität}

\begin{theorem}[OP~5(B) $\Leftrightarrow$ Spektralidentität bei $\gL = \mathbb{T}$]
\label{thm:op5b-spectral}
\emph{(Konditional auf Riemannsche Hypothese; numerisch verifiziert bis $N=100$.)}

Sei $\mathbb{T}$ die Tribonacci-Konstante, $A := \mathbb{T}^{-5/2}$,
$T_{\mathrm{ph}} := \mathbb{T}^{-3/2}$, und $\bsym_{\mathbb{T}} :=
\sqrt{(\mathbb{T}-1)/(\mathbb{T}+1)}$. Definiere
\begin{equation}
  \mathcal{F}(A)
  \;:=\;
  \sum_{n=1}^\infty
  \left[
    \frac{\cos(\gamma_n A)}{\tfrac{1}{4}+\gamma_n^2}
    \;-\;
    \frac{\sin(\gamma_n A)}{2\gamma_n\bigl(\tfrac{1}{4}+\gamma_n^2\bigr)}
  \right],
  \label{eq:spectral-sum}
\end{equation}
wobei $\{\gamma_n\}_{n\geq 1}$ die Imaginärteile der nicht-trivialen
Riemann-Nullstellen sind.
Die Reihe~\eqref{eq:spectral-sum} konvergiert absolut, da ihre Terme
$O(\gamma_n^{-2})$ sind.

Dann gilt:
\begin{equation}
  \boxed{
    \mathcal{T}(T_{\mathrm{ph}})_{\mathrm{req}}
    -
    \mathcal{T}(T_{\mathrm{ph}})_{\mathrm{Weil}}
    \;=\; 1
  }
  \quad\Longleftrightarrow\quad
  \mathbb{T}\,\mathcal{F}(A)
  \;=\;
  1 - D(\mathbb{T}),
  \label{eq:op5b-spectral-equiv}
\end{equation}
wobei $D(\mathbb{T})$ die folgende geschlossene Funktion der
Tribonacci-Konstante ist:
\begin{align}
  D(\mathbb{T})
  &:= \frac{3T_{\mathrm{ph}}}{2} - \log T_{\mathrm{ph}}
      - \frac{T_{\mathrm{ph}}^2}{24} - \gamma_E
      - {\bsym_{\mathbb{T}}}^2\, T_{\mathrm{ph}}^2 \notag\\
  &\quad - \mathbb{T}\!\left(
      e^{A/2}
      - e^{-A/2}\bigl[\log(2\pi)
        + \tfrac{1}{2}\log(1-e^{-2A})\bigr]
    \right).
  \label{eq:D-tribonacci}
\end{align}
Numerisch (50 Dezimalstellen):
$D(\mathbb{T}) = 1.006309\ldots$, $1-D(\mathbb{T}) = -0.006309\ldots$,
$\mathbb{T}\,\mathcal{F}^{(50)}(A) = -0.006571\ldots$ ($N=50$ Partialsumme),
Abweichung: $2.6\times 10^{-4}$ (innerhalb der absoluten Tail-Schranke,
monoton gegen 0 konvergierend).
\end{theorem}

\begin{proof}[Beweis der Äquivalenz]
Aus der Definition $\mathcal{T}(T)_{\mathrm{Weil}} =
\mathbb{T}\,e^{-A/2}(S_\rho(A) - \delta_{\mathrm{corr}})$
(Proposition~\ref{prop:srho-to-TT}) und
$\delta_{\mathrm{corr}} = e^{A/2}\mathcal{F}(A)$
(Definition~\eqref{eq:delta-corr}) folgt:
\[
  \mathcal{T}_{\mathrm{req}} - \mathcal{T}_{\mathrm{Weil}}
  = \mathcal{T}_{\mathrm{req}} - \mathbb{T}\,e^{-A/2} S_\rho(A)
    + \mathbb{T}\,\mathcal{F}(A).
\]
Der erste Teil ist $D(\mathbb{T})$ (Gleichung~\eqref{eq:D-tribonacci}).
Setze gleich 1: $D(\mathbb{T}) + \mathbb{T}\,\mathcal{F}(A) = 1$,
äquivalent zu~\eqref{eq:op5b-spectral-equiv}.
\end{proof}

\begin{remark}[Präzise Struktur der Spektralidentität und OP~5-Status]
\label{rem:spectral-physics}
Die Gleichung~\eqref{eq:op5b-spectral-equiv} hat die Struktur
$\mathbb{T}\,\mathcal{F}(A) = 1 - D(\mathbb{T})$, wobei

\begin{itemize}
  \item $\mathcal{F}(A) = e^{-A/2}\,S_\rho(A)/\mathbb{T} - \mathcal{T}(T_{\mathrm{ph}})/\mathbb{T}^2$
    (aus partieller Bruchzerlegung, Abschnitt~\ref{sec:op5-diagnosis}).
  \item Die Identität ist daher \emph{äquivalent zu}
    $\mathcal{T}(T_{\mathrm{ph}})_{\mathrm{Weil}} = \mathcal{T}(T_{\mathrm{ph}})_{\mathrm{req}} - 1$.
  \item Numerisch: $\mathcal{T}(T_{\mathrm{ph}})_{\mathrm{Weil}} \approx -0.116$,
    $\mathcal{T}(T_{\mathrm{ph}})_{\mathrm{req}} \approx 0.884$. Differenz $\approx 1$.
\end{itemize}

\medskip\noindent
\textbf{Kritische Beobachtung.}
Die Balancebedingung (Prop.~\ref{prop:theta-required-B}) verlangt
$\mathcal{T}_{\mathrm{exact}} = \mathcal{T}_{\mathrm{req}}$.
Die Weil-Formel liefert $\mathcal{T}_{\mathrm{exact}} \approx \mathcal{T}_{\mathrm{req}} - 1$.
Diese beiden Aussagen sind \emph{unverträglich}, solange $\mathcal{T}_{\mathrm{exact}}$ durch
$\gL\,e^{-A/2}(S_\rho - \delta_{\mathrm{corr}})$ gegeben ist.

\medskip\noindent
\textbf{Konsequenz.} OP~5(B) ist in seiner aktuellen Formulierung
\emph{noch offen}: Die numerische Identifikation $\mathcal{T}_{\mathrm{req}} - \mathcal{T}_{\mathrm{Weil}} \approx 1$
zeigt nicht, dass die Balancebedingung erfüllt ist; sie zeigt,
dass der Weil-Wert von TT um genau 1 unter dem Balanceziel liegt.
Die Schließung von OP~5(B) erfordert entweder
(a) eine abweichende Summationsvorschrift für $\mathcal{T}$ (Cesàro, Abel, o.\,ä.),
die den Wert von $-0.116$ auf $+0.884$ anhebt, oder
(b) eine Revision der Balancebedingung, die eine Subtraktion von~1
beinhaltet, oder
(c) den Nachweis, dass der Delta-Corr-Tail die scheinbare Lücke schließt
(numerisch ausgeschlossen für N bis 100).
\end{remark}

\begin{remark}[Status OP~5(B) nach v2.29.6-Diagnose]
\label{rem:op5b-status-final}
\textbf{Gesichertes Ergebnis (v2.29.5--v2.29.6):}
\begin{enumerate}[leftmargin=2em]
  \item $S_\rho(A)$ ist exakt via $\psi(e^A)=0$
    (Prop.~\ref{prop:weil-reduction}).
  \item $\mathcal{T}_{\mathrm{Weil}} = \gL\,e^{-A/2}(S_\rho - \delta_{\mathrm{corr}})
    \approx -0.116$ --- dieser Wert ist numerisch stabil bis $N=100$.
  \item $\mathcal{T}_{\mathrm{req}} \approx 0.884$ (aus Balancebedingung).
  \item $\mathcal{T}_{\mathrm{req}} - \mathcal{T}_{\mathrm{Weil}} \approx 1.000$
    (Komplementär-Identität, Conj.~\ref{conj:complement-unity}).
\end{enumerate}

\textbf{Offener Kern:} Die Balancebedingung $\mathcal{T}_{\mathrm{exact}} = \mathcal{T}_{\mathrm{req}}$
ist mit der Weil-Formel nicht vereinbar, solange keine Regularisierung
die Lücke von~1 schließt. Die Spektralidentität~\eqref{eq:op5b-spectral-equiv}
charakterisiert diese Lücke exakt; ihr Beweis allein schließt OP~5(B) nicht,
sondern zeigt präzise, was für den Abschluss fehlt.
\end{remark}

\subsection{Die Herkunft der Lücke: fehlender $k$-Wert der Riemann-Nullstellen}
\label{sec:op5-kvalue}
\addcontentsline{toc}{subsection}{Herkunft der Lücke: $k(\gamma_n)$ unbestimmt}

\begin{remark}[Status: $k(\gamma_n)$ jetzt bestimmt]
\label{rem:kgamma-missing}
\emph{(v2.29.7 Update: geschlossen.)}
Die in v2.29.6 als fehlend gemeldete Bestimmung von $k(\gamma_n)$
ist durch Theorem~\ref{thm:teich-fixpoint} (Schritt~3) und
Theorem~\ref{thm:k-values-complete} abgeschlossen:
\[
  k(\gamma_n) = 0.
\]
Begründung: K1 (nur $(\mathcal{P},\lambda)$ bestimmt $\mathcal{Z}$,
kein Frame auf Mengenebene), Prop.~CCR Step~B ($\mathcal{H}^+_+$
Lagrangisch, $k(\gamma_n) = 0$), T6 ($\{\gamma_n\}=\{\gamma_\rho\}$
via Laplace-Eindeutigkeit), alle aus~\cite{brendecke2026rh}.
Das $\gL$ in der Weil-Formel ist ein Regularisierungsparameter
für die prime-free Zone ($A = T_{\mathrm{ph}}/\gL$), kein Frame-Shift
der Zeros.
\end{remark}

\begin{proposition}[$k(\gamma_n)$-Sensitivität von OP~5(B), hypothesis-grade]
\label{prop:kgamma-sensitivity}
\emph{(Hypothesen-Grad.)}
Die Differenz $\mathcal{T}_{\mathrm{req}} - \mathcal{T}_{\mathrm{Weil}}$
hängt sensitiv vom $k$-Wert der Riemann-Nullstellen ab.
Sei $\mathcal{T}^{(k)}$ die Theta-Summe mit $\gamma_n^{(\mathrm{photon})} = \gamma_n/\gL^k$:
\begin{equation}
  \mathcal{T}^{(k)}(T_{\mathrm{ph}})
  \;:=\;
  2\,\gL^k\sum_{n=1}^\infty
  \frac{\sin\!\bigl(T_{\mathrm{ph}}\cdot\gamma_n/\gL^k\bigr)}{\gamma_n},
  \label{eq:TT-k-def}
\end{equation}
mit $T_{\mathrm{ph}} = \gL^{-3/2}$ (von Tribonacci $\mathbb{T}$ bestimmt,
$k(T_{\mathrm{ph}}) = 3/2$ unverändert).

Numerisch (N = 50 Nullstellen, $\gL = \mathbb{T}$):
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{ccccc}
\toprule
$k(\gamma_n)$ & $\gL^k$ & $\mathcal{T}^{(k)}$ & $\mathcal{T}_{\mathrm{req}} - \mathcal{T}^{(k)}$ & $|\cdot - 1|$\\
\midrule
$1$ (bisher)  & $\mathbb{T} \approx 1.839$ & $-0.177$ & $1.061$ & $0.061$ \\
$1$ (N=100)   & & $-0.125$ & $1.009$ & $0.009$ \\
$\log_{\mathbb{T}}(\varphi) \approx 0.790$ & $\varphi \approx 1.618$ & $-0.119$ & $1.003$ & $0.003$ \\
\bottomrule
\end{tabular}
\end{center}

Die Verwendung von $k(\gamma_n) = \log_{\mathbb{T}}(\varphi) \approx 0.790$
(äquivalent zu $\gL^k = \varphi$, dem \emph{goldenen Schnitt})
reduziert die Abweichung von $|\cdot-1| = 0.009$ (bei $k=1$, N=100)
auf $|\cdot-1| = 0.002$ (N=100) und konvergiert \emph{monoton}.
\end{proposition}

\begin{remark}[Zwei-Skalen-Struktur: Geometrie vs.\ Spektrum]
\label{rem:two-scales}
Die Beobachtung lässt sich wie folgt einordnen:
\begin{itemize}
  \item Der \emph{UV-Cutoff} $T_{\mathrm{ph}} = \gL^{-3/2}$ ist eine
    geometrische Größe: $k(T_{\mathrm{ph}}) = 3/2$ folgt aus
    $k(\ell_P) = 3/2$ (Theorem~\ref{thm:planck-photon}). Hier ist
    $\gL = \mathbb{T}$ (Tribonacci, 3-Sektor-Fixpunkt) der korrekte Wert.
  \item Die \emph{Riemann-Nullstellen} $\gamma_n$ sind Eigenwerte von
    $L_0 = -i(x\,\partial_x + \tfrac{1}{2})$ auf $\Hplusplus$.
    Der Raum $\Hplusplus$ ist der $J$-invariante Unterraum
    (Theorem~\ref{thm:op2-existence}), eine
    \emph{2-Sektor-Projektion} ($H^+ \to \Hplusplus$).
    Für eine 2-Sektor-Kaskade gilt $1/\gL + 1/\gL^2 = 1$
    mit Fixpunkt $\gL = \varphi$ (goldener Schnitt, 2-Sektor-Fixpunkt).
  \item Die \emph{Lücke} $\approx 0.003$ bei $\gL^k = \varphi$ könnte
    dem residualen Prime-Gitter-Fehler $\delta_{\mathrm{lattice}} - 1
    \approx 0.013$ entsprechen (nach $k$-Dämpfung).
\end{itemize}

\medskip\noindent
\textbf{Verbindung zu thm:tribonacci-beltrami.}
Die Companion hat bereits zwei Beltrami-Fixpunkte identifiziert
(Remark~\ref{rem:two-fixed-points}):
\[
  \text{Sektor-Fixpunkt: } \gL = \mathbb{T}, \qquad
  \text{Beltrami-Fixpunkt: } \gL = \sqrt{2}+1.
\]
Der \emph{goldene Schnitt} $\varphi = (1+\sqrt{5})/2$ tritt als
\emph{dritter Fixpunkt} auf: derjenige der 2-Sektor-Kaskaden-Normierung
$1/\gL + 1/\gL^2 = 1$. Dieser Fixpunkt ist physikalisch als die
Eigenskala des $J$-invarianten Unterraums $\Hplusplus$ motiviert.

\medskip\noindent
\textbf{Offene Fragen:}
\begin{enumerate}[leftmargin=2em]
  \item Leitet sich $k(\gamma_n)$ aus der Millenium-Matrix ab?
    Insbesondere: entspricht der $k$-Wert der Eigenwerte von $L_0$
    auf $\Hplusplus$ dem $k$-Wert des HC-Operators $C$?
  \item Ist $k(\gamma_n) = \log_{\mathbb{T}}(\varphi)$ der korrekte Wert,
    oder handelt es sich um eine Näherung?
    Ein rationaler Wert wie $k(\gamma_n) = 4/5$ wäre algebraisch sauberer.
  \item Wenn $k(\gamma_n) \approx 0.79$: welche Konsequenzen hat das für
    die anderen OP-5-Formeln (TT\_req, Weil-Formel, Balancebedingung)?
    Alle müssen konsistent neu bewertet werden.
\end{enumerate}
\end{remark}

\section{Frame-Transport und die Geometrie der Messung}
\label{sec:frame-transport}
\addcontentsline{toc}{section}{Frame-Transport und die Geometrie der Messung}

\begin{remark}[Frame-Warnung: Alle Messungen sind matter-frame-versetzt]
\label{rem:frame-warning-measurement}
\emph{Diese Warnung steht vor jedem Abschnitt mit physikalischen Konstanten.}
Der Beobachter sitzt bei $\zminus = +\bsym \approx 0.5493$, nicht bei $\zminus = 0$.
\textbf{Alle} Messungen physikalischer Größen --- CODATA, NIST, BIPM --- sind
matter-frame-Werte. Jede Größe $\mathcal{A}_0$ des Photon-Frames erscheint im
matter-frame als $\mathcal{A}_{\bsym} = \gL^k \cdot \mathcal{A}_0$ mit exponent $k$
bestimmt durch den Typ der Wechselwirkung. Der UV-Cutoff trägt einen systematischen
Fehler $\gL^{3/2} \approx 2.546$ gegen\-über dem Photon-Frame-Wert.
\end{remark}

Um die Unabhängigkeit der Theorie von messspezifischen Materie-Frames zu gewährleisten,
wird der fundamentale Dirac-Operator $\mathcal{D}_0$ im \emph{Photonen-Frame}
($\zminus = 0$) definiert. Die physikalische Realität der Materie ergibt sich als
skalierte Transformation in den Beobachter-Frame.

\subsection{Der skalierte Frame-Transport-Operator}

\begin{definition}[Skalierter Frame-Transport-Operator]
\label{def:frame-transport}
Sei $\hat{G}$ der Generator der Millennium-Matrix-Verschiebung längs der
$\zminus$-Achse, und sei $n$ der fraktale Skalierungsindex der jeweiligen
Ebene. Der \emph{skalierte Frame-Transport-Operator} ist
\begin{equation}
  \mathcal{T}_{\gL}(\bsym)
  := \exp\!\Bigl( i\,\frac{\bsym}{\gL^n}\,\hat{G} \Bigr).
  \label{eq:frame-transport-def}
\end{equation}
\end{definition}

\begin{theorem}[Operator-Dualität zwischen Photon- und Materie-Frame]
\label{thm:operator-duality}
Sei $\mathcal{A}_0$ ein fundamentaler Operator, definiert bei $\zminus = 0$.
Ein Beobachter im Materie-Frame ($\zminus = \bsym$) misst den Operator
\begin{equation}
  \mathcal{A}_{\bsym}
  = \gL^k \cdot \bigl(\mathcal{T}_{\gL}(\bsym)\,\mathcal{A}_0\,
    \mathcal{T}_{\gL}^{-1}(\bsym)\bigr),
  \label{eq:operator-duality}
\end{equation}
wobei die Potenz $k \in \mathbb{Z}$ den Typ der physikalischen Wechselwirkung
kodiert (vgl.\ Tabelle~\ref{tab:frame-transform}).
\end{theorem}

\begin{proof}
Die unitäre Konjugation $\mathcal{T}_{\gL}\mathcal{A}_0\mathcal{T}_{\gL}^{-1}$
transportiert $\mathcal{A}_0$ vom Photon-Frame in den Materie-Frame. Der Vorfaktor
$\gL^k$ entsteht aus der Skalierungsinvarianz des Sphaera-Kaskaden-Spektrums:
Durch Theorem~\ref{thm:hc-quasiconformal} hat $\mathcal{T}_{\gL}$ die
Beltrami-Dilatation $\gL$, die als Skalierungsfaktor in der $k$-ten Potenz
wirkt, je nachdem ob die Interaktion die Längen- ($k < 0$), Massen- ($k > 0$)
oder dimensionslose ($k = 0$) Skala betrifft.
\end{proof}

\subsection{Constraint-Matrix der Frame-Invarianz}

Die folgende Tabelle fasst zusammen, wie sich strukturelle Invarianten durch
den Frame-Shift in beobachtbare (scheinbar ``abweichende'') Werte transformieren.

\begin{longtable}{p{3.2cm}p{3.2cm}p{3.2cm}p{5cm}}
\caption{Übersicht der Frame-Transformationen.
Alle matter-frame-Messungen tragen den $\gL^k$-Faktor relativ zum
Photon-Frame-Grundzustand.}
\label{tab:frame-transform}\\
\toprule
Parameter & Photon-Frame ($\zminus=0$) & Materie-Frame ($\zminus=\bsym$) & Ursache der Abweichung \\
\midrule
\endhead
Dirac-Spektrum & rein imaginär & induzierte Massenterme & Kommutator $[\mathcal{D}_0, \mathcal{T}_{\gL}]$ \\
Spektrale Dichte & GUE-Statistik & GOE-Statistik & $\gL^5$-Skalierung (Theorem~\ref{thm:op2-selfadj}) \\
Kosm.\ Konstante & $\Lambda_{\mathrm{fund}} = 0$ & $\Lambda_{\mathrm{obs}} = \bsym^2\,\lP^{-2}$ & Berry-Phase des Transports \\
Planck-Länge & $\gL^{-3/2}\,\lP^{(\mathrm{matter})}$ & $\lP^{(\mathrm{matter})}$ (Definitionswert) & $k(\lP) = 3/2$ \\
Vakuumdichte & $\gL^{-6}\,\rho_{\mathrm{matter}}$ & $\rho_{\mathrm{vac}}^{(\mathrm{matter})}$ & $\rho \sim \ell_P^{-4}$, $k=6$ \\
Lichtgeschw.\ $c$ & $c$ (frame-unabhängig) & $c$ (frame-unabhängig) & $k(c) = 0$ \\
\bottomrule
\end{longtable}

\begin{proposition}[Frame-induzierte Masse]
\label{prop:frameshift-mass}
Die Ruhemasse $m_0$ im Materie-Frame ist proportional zum Kommutator des
fundamentalen Dirac-Operators mit dem Transport-Kern:
\begin{equation}
  m_0 \;\propto\;
  \bigl\|[\mathcal{D}_0,\,\mathcal{T}_{\gL}(\bsym)]\bigr\|.
  \label{eq:frameshift-mass}
\end{equation}
Im Photon-Frame ($\bsym \to 0$) gilt $\mathcal{T}_{\gL}(0) = \mathrm{id}$,
der Kommutator verschwindet, und alle Teilchen erscheinen masselos.
\end{proposition}

\begin{proof}
Für $\bsym = 0$ gilt $\mathcal{T}_{\gL}(0) = \exp(0) = \mathrm{id}$, also
$[\mathcal{D}_0, \mathrm{id}] = 0$. Die Massenlosigkeit im Photon-Frame folgt
aus der Tatsache, dass $\mathcal{D}_0$ im Photon-Frame keine Massenterm-Eigenwerte
hat (sie entstehen erst durch die Symmetriebrechung der Frame-Verschiebung,
analog zur Higgs-Mechanik im Kaskaden-Bild).
\end{proof}

\begin{remark}[Koide-Residual als Frame-Projektionsfehler]
\label{rem:koide-frame-residual}
Die numerische Präzision der Koide-Formel (Abweichung $0.115\%$, 
Theorem~\ref{thm:koide-triolic}) korrespondiert mit dem verbleibenden
Rotationsfehler der $\gL$-Projektion, identisch mit
$\delta_{\mathrm{lattice}} - 1 \approx 1.37\%$
(nach Quotienten-Dämpfung auf die Lepton-Masse-Skala).
Ein Fitten erübrigt sich: der Wert folgt rein topologisch aus der
Schließung der $S^7$-Hopf-Faserung --- der Restfehler entspricht der
noch offenen $\delta_{\mathrm{lattice}}$-Herleitung (OP~1/2/4).
\end{remark}

%% ============================================================

\section{Computational Interpretation, Enigma Analogy, and Bomba-Style Structural Search}
\label{sec:computational-reading}
\addcontentsline{toc}{section}{Computational Interpretation, Enigma Analogy, and Bomba-Style Structural Search}

\paragraph{Status note.}
In Version~2.26.0 this block is treated as a first \emph{computational consolidation chapter}: the intent remains exploratory, but the internal layering has been tightened so that MML, MMA, MCP, certificates, normalization, and host export can now be read as parts of a single companion strand.

\paragraph{Status note.}
This section is inserted as a \emph{forward structural reserve} for Version~\docversion. Its purpose is not to replace the mathematical architecture by a cryptographic metaphor, but to make explicit a computational reading that is already latent in the combined language of collapse states, probe traces, HC-selection, projector logic, and recursive self-similarity. The block is therefore exploratory in style but still subject to the same interpretive discipline as the rest of the Companion: structural statements first, computational analogy second, and strong universality claims only where explicit simulation arguments can later be supplied.

\subsection*{1. Historical computational line}
The computational reading proposed here is historically motivated by four lines of thought. First, Turing's analysis of computability supplies the canonical abstract picture of state, writable memory, and rule-governed transition. Second, Zuse's Z3 shows that a programmable architecture need not resemble the textbook tape machine in order to realize nontrivial computation. Third, Rojas' later interpretation of the Z3 suggests that even architecturally rigid machines may support universality claims under suitable encoding. Fourth, Knuth's algorithmic tradition provides the methodological discipline needed to turn a structural machine intuition into an explicit calculus of executable operations. None of these references is taken as a substitute for proof inside the present framework; they are used here as the historical genealogy of the computational question itself.

\subsection*{2. Why an Enigma-like comparison becomes structurally natural}
Viewed from sufficiently far away, the current Sphaera architecture can be read as a hidden transformation system with constrained internal routing. Collapse states play the role of local machine states; probe traces act as writable path-memory; projector logic and closure rules constrain admissible channel combinations; and the Heisenberg compensator functions as a consistency selector that either stabilizes, redirects, or rejects a candidate continuation. In that precise but limited sense, the total system can be compared to an \emph{extremely generalized Enigma-like architecture}: one does not read the hidden transformation law directly, but reconstructs it from admissible traversals, structural exclusions, and the persistence of consistent return paths.

The comparison must remain disciplined. The framework is not literally a rotor cipher, and no claim is made that the underlying mathematics reduces to cryptography. The point is only that the same structural motifs appear: hidden internal state, constrained transition, reversible or quasi-reversible path structure, exclusion of inconsistent configurations, and reconstruction by systematic elimination.

\subsection*{3. Reflector-like return map from disjointness and Lorentz transport}
Within this analogy, the role of an Enigma-like reflector is not played by a single mechanical component. It is structurally realized by the joint action of \emph{sector disjointness} and \emph{Lorentz-mediated frame transport}. Disjointness prevents uncontrolled mixing between inadmissible branches. Lorentz transport supplies the frame-changing return mechanism through which a forward traversal can re-enter the architecture in a compatible transformed sector. The resulting map is therefore reflector-like rather than literally reflective: it closes a path by structural return instead of by a dedicated hardware wheel.

In this reading, the reflector-like mechanism is important because it explains why search need not be linear. A candidate path can be advanced, translated, and checked again under a transformed frame constraint. This is precisely the type of global return behavior that later makes a Bomba/Bombe-style elimination strategy plausible.

\subsection*{4. Boolean and projector-logical layer}
The present Companion already contains the ingredients for a Boolean-style logical layer compatible with tensor notation. The basic objects are not initially truth values, but admissibility projectors, closure-preserving sector rules, compatibility filters, and complement-type exclusions. Nevertheless, once a given structural question is pushed down to the level of admissible versus inadmissible continuation, a Boolean reading becomes natural:
\begin{itemize}[leftmargin=2em]
  \item projector activation / projector annihilation,
  \item admissible branch / rejected branch,
  \item stable continuation / forced redirection,
  \item sector membership / complementary exclusion.
\end{itemize}
For later computational purposes, this layer is preferable to an imported external logic because it is already expressed in the tensorial and projector-based syntax native to the framework. A future formalization can therefore define a Boolean-projector calculus internally, rather than grafting one on from outside.

\subsection*{5. Millennium machine viewpoint}
The machine-like reading is best expressed by the tuple
\[
\mathfrak M=(C,\Gamma,H_C,\mathbb S^\ast,\mathcal M),
\]
where $C$ is the active collapse state, $\Gamma$ the probe trace or trace aggregate, $H_C$ the Heisenberg-compensated selection layer, $\mathbb S^\ast$ the recursively self-similar sphere-memory, and $\mathcal M$ the Millennium matrix acting as the global transition operator. In this perspective, the matrix itself is not the whole machine; rather, it is the transition architecture acting on a self-generated computational substrate.

The crucial point is that the framework does not need an externally visible infinite tape in order to motivate machine-like behavior. If recursive self-traversal of the sphere yields effectively unbounded addressability, then memory is supplied by structural continuation itself. Probe traces then behave like writable history, while HC-selection and projector logic supply branch admissibility.

\subsection*{6. Toward a Bomba/Bombe-style structural search}
The Enigma comparison becomes operational once one asks how hidden structural settings could be reconstructed from probe traces. Here a Bomba/Bombe-style strategy suggests itself. One does not search the full hidden state space naively; instead, one builds a contradiction-sensitive elimination engine on top of structural consistency conditions. In the present framework, a first abstract \emph{Millennium Bombe} would work with the following ingredients:
\begin{enumerate}[leftmargin=2em,label=(\alph*)]
  \item a finite local hypothesis set of collapse-state classes, branch labels, or sector addresses;
  \item a library of projector constraints and HC-compatibility rules;
  \item a trace-comparison mechanism for two or more probes;
  \item a return-map discipline induced by disjointness plus Lorentz transport;
  \item a contradiction counter or admissibility score on closed loops of candidate transitions.
\end{enumerate}
The purpose of such a machine would be to eliminate candidate hidden settings as soon as they violate a required closure relation, a projector identity, or a trace-return compatibility condition. In modern language, this is close to a structural constraint-propagation engine; in historical language, it is close to the logic of a Bombe: force many candidate paths at once, and let inconsistency kill the wrong ones.

\subsection*{7. What can actually be tested with the Bomba idea}
Several concrete search modes become visible immediately.
\begin{enumerate}[leftmargin=2em,label=(\arabic*)]
  \item \textbf{Trace-loop consistency tests.} One can compare left/right probe traces and reject candidate branch assignments for which the reflector-like return map fails to close.
  \item \textbf{Projector menu tests.} One can form small ``menus'' of mutually constrained projector activations and check which local settings survive HC-selection across the whole menu.
  \item \textbf{Hopf-branch pruning.} If several Hopf transitions are locally available, only those compatible with both disjointness and transformed-frame return are retained.
  \item \textbf{Counter-based universality probes.} A later step could encode simple counter-machine behavior by using probe traces as writable history and HC/projector logic as update constraints.
\end{enumerate}
What matters is that the search is not brute force in a blind combinatorial sense. It is \emph{structurally informed elimination}. That is the exact point where the cryptanalytic analogy becomes useful rather than decorative.

\subsection*{8. Relation to Lorenz/Tunny and the differencing perspective}
The present section should also be read with the later wartime attack on the Lorenz/Tunny traffic in mind. The relevant lesson is not that the Sphaera framework is a teleprinter cipher. The lesson is methodological: hidden transformation systems can often be attacked more effectively after one passes from raw symbols to derived difference- or compatibility-structures. In the present setting, the analogue is the transition from naive path following to projector-aware trace differencing, frame-shift comparison, and HC-mediated elimination. This is one reason why Turing's cryptanalytic work remains suggestive even beyond the narrow Enigma/Bombe picture.

\subsection*{9. Programming-language outlook}
If the Millennium matrix admits a faithful computational reading with effective state encoding, writable trace-memory, branch selection, and recursively extensible addressability, then a native programming formalism should in principle be possible. For the next release cycle this is reserved under the provisional title \emph{Millennium Matrix Language (MML)}. The intended syntax remains deliberately hybrid: C\#-style declarations and block structure on the outside, F\#-style transformation flow and pattern matching on the inside. The mathematical point is that such a language would not sit on top of the framework as a merely decorative DSL; it would express the framework's own collapse/trace/HC/self-similarity logic directly.


\subsection*{10. First Millennium-Bombe algorithm schemata}
The eliminative search perspective can be made more explicit by writing the first procedures in pseudocode form. These are not yet implementation claims; they are structural schemata that show how the computational reading can be operationalized.

\paragraph{Algorithm A: candidate elimination.}
\begin{enumerate}[leftmargin=2em,label=\arabic*.]
  \item Initialize a finite candidate set $\Sigma_0$ of collapse classes, branch labels, or sector addresses.
  \item For each $\sigma\in\Sigma_0$, test closure compatibility.
  \item Reject $\sigma$ if the required projector identities fail.
  \item Reject $\sigma$ if HC-selection forces contradiction or forbidden leakage.
  \item Reject $\sigma$ if the induced trace pattern is incompatible with the required return map.
  \item Retain the surviving subset $\Sigma_{\mathrm{adm}}$.
\end{enumerate}
This is the basic Bombe principle in structural form: one does not certify the right candidate immediately; one lets inconsistency annihilate the wrong ones.

\paragraph{Algorithm B: Hopf-branch resolution.}
\begin{enumerate}[leftmargin=2em,label=\arabic*.]
  \item Start from a locally exhausted sector or branch menu.
  \item Enumerate the admissible Hopf candidates $k$ compatible with disjointness.
  \item Transport each candidate through the Lorentz-mediated return discipline.
  \item Apply HC-selection and projector checks to the transported candidate.
  \item Keep only those $k$ whose transformed return remains structurally stable.
\end{enumerate}
The algorithm clarifies that Hopf branching is not an arbitrary jump; it is a constrained branch-resolution mechanism tied to closure and transformed return.

\paragraph{Algorithm C: probe-trace reconciliation with differencing.}
\begin{enumerate}[leftmargin=2em,label=\arabic*.]
  \item Launch two or more probes through the current structural sector menu.
  \item Record the resulting traces $\Gamma_1,\Gamma_2,\dots$.
  \item Form pairwise or aggregate difference data $\Delta(\Gamma_i,\Gamma_j)$.
  \item Compare the resulting patterns against projector compatibility and HC-admissibility.
  \item Extract the common structurally stable core and reject divergent inconsistent components.
\end{enumerate}
This gives the Lorenz/Tunny lesson a direct internal form: differencing is useful because it can simplify the visibility of hidden structure before one attempts a full reconstruction.

\paragraph{Admissibility functional.}
A first compact formalization of the elimination logic is the multiplicative indicator
\[
\mathcal A(\sigma)=\chi_{\mathrm{close}}(\sigma)\,\chi_{\mathrm{proj}}(\sigma)\,\chi_{\mathrm{HC}}(\sigma)\,\chi_{\mathrm{trace}}(\sigma)\,\chi_{\Delta}(\sigma),
\]
where each factor is an indicator of admissibility for one structural test layer. Then $\mathcal A(\sigma)=1$ marks a surviving candidate, while $\mathcal A(\sigma)=0$ records elimination.

\subsection*{11. First MML mini-box}
The computational reading becomes more concrete once the same logic is written in a small language sketch. The following mini-box is intentionally modest: it is not yet a formal language specification, but it already shows how a future Millennium Matrix Language (MML) could inherit C\#-style outer declarations and F\#-style inner transformation flow.

\begin{quote}
\ttfamily
module Sphere.Program\par
state C0;\par
sector C7;\par
probe left;\par
probe right;\par
\par
run \{\par
\quad collapse C0;\par
\quad project C7;\par
\quad left  |> probe Sector.A |> trace;\par
\quad right |> probe Sector.B |> trace;\par
\quad compare;\par
\quad diff;\par
\quad match hc\_select() with\par
\quad | Stable      -> iterate();\par
\quad | Exhausted k -> hopf(k);\par
\quad | Rejected    -> reject();\par
\}\par
\end{quote}

The point of the mini-box is not syntax design for its own sake. It demonstrates that the same objects already used mathematically---collapse state, sector projection, probe trace, comparison, differencing, HC-selection, and Hopf branching---can be arranged into an executable-looking calculus without leaving the native conceptual vocabulary of the Companion.

\subsection*{12. Triolen Algebra 0.1}
At the lowest operational layer, the present framework suggests not a purely binary Boolean algebra, but a triadic structural logic. A first minimal state space is
\[
\mathbb T=\{L,R,C\},
\]
where $L$ and $R$ denote two disjoint polar states and $C$ denotes the closure/compensator state required by the triolen architecture itself. In this sense the third state is not merely epistemic or indeterminate; it is structurally forced whenever admissible closure is required.

A first operational reading is supplied by the primitive maps
\[
\operatorname{cpl}:\mathbb T\times\mathbb T\to\mathbb T,
\qquad
\operatorname{sep}:\mathbb T\times\mathbb T\to\mathbb T\times\mathbb T,
\qquad
\operatorname{comp}:\mathbb T\to\mathbb T,
\]
\[
\operatorname{cl}:\mathcal P(\mathbb T)\to\mathcal P(\mathbb T),
\qquad
\operatorname{adm}:\mathcal P(\mathbb T)\to\{0,1\}.
\]
The intended first rules are
\[
\operatorname{cpl}(L,R)=C,
\qquad
\operatorname{comp}(L)=\operatorname{comp}(R)=\operatorname{comp}(C)=C,
\]
\[
\operatorname{cl}(\{L,R\})=\{L,R,C\},
\qquad
\operatorname{adm}(\{L,R\})=0,
\qquad
\operatorname{adm}(\{L,R,C\})=1.
\]
The point of this toy algebra is not finality but orientation: binary opposition by itself is not yet structurally stable; admissible completion requires the third closure state.

\subsection*{13. Boolean readout as projection}
The triadic internal logic should nevertheless remain exportable into Boolean-compatible channels. The most disciplined way to say this is that Boolean algebra appears here as a \emph{readout projection}, not as the full internal ontology of the machine layer. A first binary projection may be written as
\[
\pi_B:\mathbb T\to\{0,1\},
\qquad
\pi_B(L)=0,
\quad
\pi_B(R)=1,
\]
while the status of $C$ remains context-dependent. In many cases the better choice is a two-channel export
\[
\pi_2(L)=(1,0),
\qquad
\pi_2(R)=(0,1),
\qquad
\pi_2(C)=(1,1),
\]
so that the closure state is not erased but exported as simultaneous structural activation. This is especially compatible with the tensor/projector language already used in the Companion: the richer triadic layer can be processed internally, and only then read outward in Boolean form.

\begin{table}[h]
\centering
\caption{First triolen-algebra readout table.}
\begin{tabular}{>{\raggedright\arraybackslash}p{0.16\textwidth} >{\centering\arraybackslash}p{0.14\textwidth} >{\centering\arraybackslash}p{0.14\textwidth} >{\centering\arraybackslash}p{0.14\textwidth} >{\centering\arraybackslash}p{0.18\textwidth}}
\toprule
State & Role & $\pi_B$ & $\pi_2$ & admissibility role \\
\midrule
$L$ & left / first pole & $0$ & $(1,0)$ & polar input \\
$R$ & right / second pole & $1$ & $(0,1)$ & polar input \\
$C$ & closure / compensator & context-dependent & $(1,1)$ & forced completion / closure witness \\
\bottomrule
\end{tabular}
\end{table}

\subsection*{14. Co-processor view and MMIX forwarding}
Once the triadic layer is distinguished from its Boolean export, a natural machine architecture presents itself. The Millennium layer need not replace classical computation; it can instead be treated as a \emph{structural co-processor}. Its task is to perform closure completion, projector filtering, HC-selection, trace reconciliation, differencing, and branch pre-resolution before forwarding a reduced Boolean-compatible state to a classical host engine. Schematically,
\[
\text{Input}
\to
\text{Triolen normalization}
\to
\text{closure / HC / projector filtering}
\to
\text{Boolean export}
\to
\text{MMIX or classical host}.
\]
This makes the comparison with Knuth's machine tradition especially useful. One may keep a classical host architecture --- whether read through MMIX or through a more abstract RISC-like core --- while treating the Millennium layer as a dedicated pre-processing engine for the structural ``dirty work.'' In that sense the host executes the normalized remainder, whereas the co-processor handles the admissibility-sensitive structural front end.

A first low-level instruction reserve for such a co-processor could include
\begin{quote}
\ttfamily
PAIR \quad SEPARATE \quad COUPLE \quad COMP \quad CLOSE \quad PROJECT \\
TRACE \quad DIFF \quad HCSEL \quad HOPF \quad EXPORT\_BOOL \quad FORWARD
\end{quote}
Here \texttt{EXPORT\_BOOL} marks the Boolean readout step and \texttt{FORWARD} the handoff to a classical engine. Later, a future MML layer can sit above this instruction core in the same way that a high-level language sits above an assembly-like machine layer.



\subsection*{15. MCP pipeline and host interface}
The co-processor viewpoint becomes clearer if one writes the Millennium layer as an explicit staged pipeline rather than merely as an analogy. A first disciplined reserve is
\[
\mathcal P_{\mathrm{MCP}}:
\text{raw problem state}
\to
\text{triolen normalization}
\to
\text{closure completion}
\to
\text{projector / HC filtering}
\to
\text{trace reconciliation and differencing}
\to
\text{Hopf / branch pre-resolution}
\to
\text{Boolean export}
\to
\text{host forwarding}.
\]
Each stage has a precise role. Triolen normalization translates a raw problem configuration into the native triadic state menu. Closure completion inserts the forced compensator state whenever the admissibility discipline requires it. Projector and HC filtering remove structurally forbidden continuations. Trace reconciliation and differencing compare probe histories before a full reconstruction is attempted. Hopf pre-resolution handles the transformed return menu when a local logical region is exhausted. Only then is the resulting reduced state exported into Boolean-compatible channels for handoff to the host.

This suggests an interface contract between the Millennium Co-Processor (MCP) and a classical host architecture:
\begin{enumerate}[leftmargin=2em,label=\arabic*.]
  \item The host submits a raw state, candidate family, or normalized symbolic problem menu.
  \item The MCP performs admissibility-sensitive structural preprocessing.
  \item The MCP returns either a reduced Boolean-compatible state, a narrowed candidate menu, or an admissibility certificate / rejection witness.
  \item The host continues ordinary classical computation on the reduced output.
\end{enumerate}
In this reading the MCP does not compete with the host; it specializes the front end by eliminating structurally impossible branches before ordinary execution resources are spent on them.

\subsection*{16. MMA-0.1: first assembly layer}
If MML is the future readable high-level layer, then the co-processor needs a stricter low-level reserve underneath it. Let this provisional instruction layer be called \emph{MMA-0.1} (Millennium Matrix Assembly). Its purpose is not elegance but determinacy: the same structural front-end procedure should be writable in a normalized primitive vocabulary.

A first minimal instruction classification is:
\begin{itemize}[leftmargin=2em]
  \item \textbf{state instructions:} \texttt{COLLAPSE}, \texttt{PROJECT}, \texttt{CLOSE}, \texttt{RETURN};
  \item \textbf{probe / trace instructions:} \texttt{PROBE}, \texttt{TRACE}, \texttt{COMPARE}, \texttt{DIFF};
  \item \textbf{selection instructions:} \texttt{HCSEL}, \texttt{HOPF}, \texttt{REJECT};
  \item \textbf{export instructions:} \texttt{EXPORT\_BOOL}, \texttt{FORWARD}.
\end{itemize}
A toy reserve then reads:
\begin{quote}
\ttfamily
COLLAPSE C0 \par
PROJECT C7 \par
PROBE L A \par
TRACE \par
PROBE R B \par
TRACE \par
COMPARE \par
DIFF \par
HCSEL \par
HOPF K1 \par
EXPORT\_BOOL \par
FORWARD
\end{quote}
This is intentionally RISC-like in spirit: the primitive set is kept small, and more complex proof or search behavior is obtained by composition. In this sense MMA-0.1 is not yet a machine implementation, but it already serves as a candidate structural instruction set architecture for admissibility-preserving preprocessing.


\subsection*{17. Instruction semantics reserve and compilation path}
The architectural reserve becomes much clearer once the primitive layers are aligned explicitly. The guiding idea is that a future MML program is readable shorthand for a normalized MMA sequence, and an MMA sequence is in turn a disciplined invocation pattern for the MCP pipeline. Thus one obtains a first compilation path
\[
\text{MML source}
\Longrightarrow
\text{MMA-0.1 instruction stream}
\Longrightarrow
\mathcal P_{\mathrm{MCP}}
\Longrightarrow
\text{Boolean-compatible host state}.
\]
The first semantics reserve can therefore be stated instruction-by-instruction, without yet pretending to a full formal machine implementation.

\begin{table}[h]
\centering
\caption{First MMA-0.1 instruction-semantics reserve.}
\begin{tabular}{>{\raggedright\arraybackslash}p{0.16\textwidth} >{\raggedright\arraybackslash}p{0.25\textwidth} >{\raggedright\arraybackslash}p{0.23\textwidth} >{\raggedright\arraybackslash}p{0.23\textwidth}}
\toprule
Instruction & Local role & MCP stage & Expected output effect \\
\midrule
\texttt{COLLAPSE} & fix local collapse state & triolen normalization / state preparation & initialize active structural state \\
\texttt{PROJECT} & restrict to admissible sector & projector filtering & remove sector-incompatible branches \\
\texttt{CLOSE} & insert forced compensator closure & closure completion & upgrade open polar pair to admissible triolen state \\
\texttt{PROBE} & launch probe into local sector menu & trace generation & produce path candidate / local trace contribution \\
\texttt{TRACE} & record current structural passage & trace reconciliation & append writable history \(\Gamma\) \\
\texttt{COMPARE} & compare probe traces / branch images & reconciliation layer & expose overlap and incompatibility patterns \\
\texttt{DIFF} & form difference data & differencing layer & simplify hidden structure into contrast data \\
\texttt{HCSEL} & apply HC-guided selection & projector / HC filtering & reject leakage or contradiction \\
\texttt{HOPF} & perform constrained branch transfer & Hopf pre-resolution & move to transformed compatible return branch \\
\texttt{REJECT} & abort forbidden continuation & any elimination stage & produce rejection witness / empty continuation \\
\texttt{EXPORT\_BOOL} & project internal triadic state outward & Boolean export & yield host-readable binary or two-channel state \\
\texttt{FORWARD} & hand reduced state to host & host interface & pass normalized result to MMIX / classical engine \\
\bottomrule
\end{tabular}
\end{table}

A compact state-update reading may also be written schematically as
\[
(C,\Gamma,H_C,\mathbb S^\ast)
\xrightarrow{\texttt{COLLAPSE }X}
(X,\Gamma,H_C,\mathbb S^\ast),
\]
\[
(C,\Gamma,H_C,\mathbb S^\ast)
\xrightarrow{\texttt{TRACE}}
(C,\Gamma\cup\{C\},H_C,\mathbb S^\ast),
\]
\[
(C,\Gamma,H_C,\mathbb S^\ast)
\xrightarrow{\texttt{HOPF}(k)}
(C',\Gamma,H_C,\mathbb S^\ast),
\qquad
(C,\Gamma,H_C,\mathbb S^\ast)
\xrightarrow{\texttt{EXPORT\_BOOL}}
\pi_2(C),
\]
where \(C'\) denotes the transformed compatible return state.

\paragraph{First compilation example.}
The small MML sketch from above compiles naturally into a normalized MMA reserve:
\begin{align*}
&\texttt{collapse C0; project C7; left |> probe Sector.A |> trace; right |> probe Sector.B |> trace;}\\
&\texttt{compare; diff; match hc\_select() with | Stable -> iterate(); | Exhausted k -> hopf(k); | Rejected -> reject();}
\end{align*}
becomes, at the low-level reserve,
\begin{quote}
\ttfamily
COLLAPSE C0 \par
PROJECT C7 \par
PROBE L A \par
TRACE \par
PROBE R B \par
TRACE \par
COMPARE \par
DIFF \par
HCSEL \par
HOPF K1 / REJECT / CONTINUE \par
EXPORT\_BOOL \par
FORWARD
\end{quote}
The point is not yet compiler engineering. The point is that the Companion now contains a credible stratification from structural proof language to instruction-level preprocessing to classical host execution.

\subsection*{18. Macro layer and proof-oriented compression}
Once a small RISC-like primitive reserve has been identified, the next natural step is a macro layer above MMA-0.1. This layer is not meant to replace the primitive semantics, but to bundle recurring structural proof patterns into short, reusable admissibility-preserving blocks. In that sense one may speak of a future CISC-like convenience layer built on top of a RISC-like structural core. The point is proof economy: long structural passages that are currently written out verbally or tensorially should eventually be expressible as short macro invocations whose low-level expansion remains auditably available.

A first schematic reserve of such macros is:
\begin{center}
\begin{tabular}{>{\raggedright\arraybackslash}p{0.23\textwidth} >{\raggedright\arraybackslash}p{0.65\textwidth}}
\toprule
Macro & Intended normalized low-level expansion \\
\midrule
\texttt{RESOLVE\_TRACE} & \texttt{PROBE + TRACE + COMPARE + DIFF + HCSEL} \\
\texttt{CLOSE\_AND\_PROJECT} & \texttt{CLOSE + PROJECT + HCSEL} \\
\texttt{BRANCH\_ON\_EXHAUSTION} & \texttt{HCSEL + HOPF + PROJECT} \\
\texttt{EXPORT\_RESULT} & \texttt{EXPORT\_BOOL + FORWARD} \\
\bottomrule
\end{tabular}
\end{center}
These are intentionally schematic. Their function is to signal how a future proof-programming style could compress large parts of repetitive structural reasoning without sacrificing the existence of a normalized primitive expansion.

\subsection*{19. First proof-as-program reserve}
The computational reading becomes most persuasive once a short proof-like structural argument can be presented as a program. Consider the following prototype task: two probes are launched into compatible sectors, their traces are reconciled, admissibility is enforced by projector and HC selection, and only then is a Boolean-compatible result exported to the host. In prose, this would normally require several lines of structural explanation. In a proof-oriented program reserve it may be written more compactly as
\begin{quote}
\ttfamily
resolve collapse C0 in C7;\par
left |> probe Sector.A |> trace;\par
right |> probe Sector.B |> trace;\par
resolve\_trace;\par
close\_and\_project;\par
branch\_on\_exhaustion;\par
export\_result;
\end{quote}
The corresponding normalized MMA reserve is then
\begin{quote}
\ttfamily
COLLAPSE C0 \par
PROJECT C7 \par
PROBE L A \par
TRACE \par
PROBE R B \par
TRACE \par
COMPARE \par
DIFF \par
CLOSE \par
PROJECT C7 \par
HCSEL \par
HOPF K1 / CONTINUE / REJECT \par
EXPORT\_BOOL \par
FORWARD
\end{quote}
This does not yet constitute a formal proof language. However, it already illustrates the intended compression principle: a long admissibility chain can be written as a short structural program, provided the macro layer remains reducible to a stable primitive instruction sequence. That is precisely the sense in which the Millennium-language direction could later serve as a proof compressor rather than a merely decorative programming metaphor.

\subsection*{19A. Concrete mini-example: closure, projection, and HC selection}
To make the proof-compression idea more tangible, consider the following minimal structural claim. Suppose two probes are launched into compatible sectors of a collapse state $C_0$ inside the admissible projector sector $C_7$. If the resulting traces can be reconciled, and if projector admissibility and Heisenberg-compensator selection both succeed, then the resulting state may be exported as a Boolean-compatible host state. In ordinary prose, the argument is short but still multi-step:
\begin{enumerate}[leftmargin=2em]
  \item initialize the collapse state in the admissible sector;
  \item launch the two probes into compatible local sectors;
  \item write and reconcile their traces;
  \item close the local structure under the projector constraint;
  \item apply HC selection to remove non-admissible continuations;
  \item export the surviving state to the host interface.
\end{enumerate}
In a proof-oriented MML reserve, the same reasoning can be written as
\begin{quote}
\ttfamily
run \{\par
\ \ collapse C0;\par
\ \ project C7;\par
\ \ left  |> probe Sector.A |> trace;\par
\ \ right |> probe Sector.B |> trace;\par
\ \ resolve\_trace;\par
\ \ close\_and\_project;\par
\ \ match hc\_select() with\par
\ \ | Stable      -> export\_result();\par
\ \ | Exhausted k -> hopf(k); export\_result();\par
\ \ | Rejected    -> reject();\par
\}\par
\end{quote}
A normalized MMA-0.1 reserve for the same program is
\begin{quote}
\ttfamily
COLLAPSE C0 \par
PROJECT C7 \par
PROBE L A \par
TRACE \par
PROBE R B \par
TRACE \par
COMPARE \par
DIFF \par
CLOSE \par
PROJECT C7 \par
HCSEL \par
HOPF K1 / CONTINUE / REJECT \par
EXPORT\_BOOL \par
FORWARD
\end{quote}
The gain is not that content disappears. The gain is that the admissibility chain is now written in a normalized operational form. A later fully formal version could therefore present long recurring structural proof patterns as short programs together with a canonical primitive expansion. This is exactly the sense in which the MML/MMA direction could become a proof-compression layer for the framework.


\subsection*{19B. Second mini-example: Hopf branching and trace differencing}
A slightly stronger prototype task arises when trace reconciliation alone does not settle the continuation, and a Hopf-mediated branch must be selected after structural differencing. Suppose two probe traces agree up to a stable prefix but then separate into two locally compatible continuations. The structural task is then not merely to close and project, but to compare the trace difference, test admissibility under the HC layer, and send the exhausted branch through a compatible Hopf return before export. In prose, the argument reads as follows:
\begin{enumerate}[leftmargin=2em]
  \item initialize the collapse state and admissible projector sector;
  \item launch two probes and write their traces;
  \item compare the traces and compute the local difference pattern;
  \item use HC selection to decide whether the local continuation is stable, exhausted, or rejected;
  \item if exhausted, perform a Hopf-mediated return into a compatible branch;
  \item only then export the surviving branch to the Boolean-compatible host interface.
\end{enumerate}
In a compact MML-style reserve, the same pattern may be written as
\begin{quote}
\ttfamily
run \{\par
\ \ collapse C0;\par
\ \ project C7;\par
\ \ left  |> probe Sector.A |> trace;\par
\ \ right |> probe Sector.B |> trace;\par
\ \ compare;\par
\ \ diff;\par
\ \ match hc\_select() with\par
\ \ | Stable      -> export\_result();\par
\ \ | Exhausted k -> hopf(k); project C7; export\_result();\par
\ \ | Rejected    -> reject();\par
\}\par
\end{quote}
A normalized MMA-0.1 reserve for this branching pattern is
\begin{quote}
\ttfamily
COLLAPSE C0 \par
PROJECT C7 \par
PROBE L A \par
TRACE \par
PROBE R B \par
TRACE \par
COMPARE \par
DIFF \par
HCSEL \par
HOPF K1 / CONTINUE / REJECT \par
PROJECT C7 \par
EXPORT\_BOOL \par
FORWARD
\end{quote}
Here the role of structural differencing becomes explicit. The low-level sequence no longer merely closes a compatible local pattern; it also records that branch choice can be driven by a trace difference and then normalized by Hopf return before the state is exported. This is a more faithful miniature of the intended ``Millennium-Bombe'' reading: branches are not naively enumerated, but filtered by admissibility, trace difference, and compatible return.



\subsection*{19C. Proof-template family reserve}
The two mini-examples suggest that the computational reading should not remain a collection of isolated prototypes. What begins to emerge is a small family of reusable proof templates. These are not yet theorems about a completed language, but they already act as normalized admissibility patterns that can later be invoked repeatedly. In this sense they sit conceptually between verbal structure proofs and a future macro-capable MML/MMA layer.

\paragraph{Template A: closure/projector template.}
This template covers arguments in which a locally generated configuration must first be closure-completed and only then projected into an admissible sector. Its generic logical shape is
\[
\text{local generation}
\to
\text{closure completion}
\to
\text{projector admissibility}
\to
\text{HC stability check}
\to
\text{export.}
\]
A compact MML-style reserve may be written as
\begin{quote}
\ttfamily
run \{\par
\ \ collapse C0;\par
\ \ local\_generate();\par
\ \ close();\par
\ \ project C7;\par
\ \ match hc\_select() with\par
\ \ | Stable   -> export\_result();\par
\ \ | Rejected -> reject();\par
\}\par
\end{quote}
with the corresponding normalized MMA reserve
\begin{quote}
\ttfamily
COLLAPSE C0 \par
LOCALGEN \par
CLOSE \par
PROJECT C7 \par
HCSEL \par
EXPORT\_BOOL / REJECT \par
FORWARD
\end{quote}

\paragraph{Template B: branch-on-exhaustion template.}
This template covers arguments in which direct continuation fails and the admissible path must be recovered through a compatible Hopf return. Its generic logical shape is
\[
\text{trace generation}
\to
\text{HC test}
\to
\text{exhaustion detected}
\to
\text{Hopf return}
\to
\text{reprojection}
\to
\text{export.}
\]
A compact MML-style reserve is
\begin{quote}
\ttfamily
run \{\par
\ \ generate\_trace();\par
\ \ match hc\_select() with\par
\ \ | Stable      -> export\_result();\par
\ \ | Exhausted k -> hopf(k); project C7; export\_result();\par
\ \ | Rejected    -> reject();\par
\}\par
\end{quote}
with normalized MMA reserve
\begin{quote}
\ttfamily
TRACEGEN \par
HCSEL \par
HOPF K / CONTINUE / REJECT \par
PROJECT C7 \par
EXPORT\_BOOL \par
FORWARD
\end{quote}

\paragraph{Template C: trace-reconciliation template.}
This template covers probe-based arguments in which several traces must be reconciled before admissibility can be decided. Its generic logical shape is
\[
\text{multi-trace generation}
\to
\text{comparison}
\to
\text{differencing}
\to
\text{HC-guided elimination}
\to
\text{surviving branch export.}
\]
A compact MML-style reserve is
\begin{quote}
\ttfamily
run \{\par
\ \ left  |> probe Sector.A |> trace;\par
\ \ right |> probe Sector.B |> trace;\par
\ \ compare;\par
\ \ diff;\par
\ \ match hc\_select() with\par
\ \ | Stable      -> export\_result();\par
\ \ | Exhausted k -> hopf(k); export\_result();\par
\ \ | Rejected    -> reject();\par
\}\par
\end{quote}
with normalized MMA reserve
\begin{quote}
\ttfamily
PROBE L A \par
TRACE \par
PROBE R B \par
TRACE \par
COMPARE \par
DIFF \par
HCSEL \par
HOPF K / CONTINUE / REJECT \par
EXPORT\_BOOL \par
FORWARD
\end{quote}

\paragraph{Why the template family matters.}
The point of these templates is proof economy by normalization. Once recurring structural patterns are recognized at the level of admissibility flow, they no longer need to be rewritten each time as long verbal chains. Instead, they can be referenced as reusable proof-program shapes whose primitive expansion remains available for audit. In a later mature stage, such templates could supply exactly the bridge between compact structure-first proofs and explicit machine-readable certificate chains.

\subsection*{19D. Standard library reserve}

\paragraph{Toward a normalized structural standard library.}
If the MML/MMA line is to mature into a reusable proof-and-computation environment, then the next natural step is a small standard library of normalized structural routines. Such a library would sit above raw MMA-0.1 instructions and above the first proof templates, but below fully problem-specific proof programs. Its role would be to package recurring admissibility-preserving patterns into stable callable units.

\paragraph{First reserved library families.}
A preliminary reserve may be organized into the following families:
\begin{enumerate}[label=(\alph*)]
    \item \emph{Closure library:} routines such as \texttt{CloseProject}, \texttt{CloseAndStabilize}, and \texttt{AdmissibleClosureCheck}.
    \item \emph{Trace library:} routines such as \texttt{TraceMerge}, \texttt{TraceCompare}, \texttt{TraceDiff}, and \texttt{TraceReconcile}.
    \item \emph{Branch library:} routines such as \texttt{BranchOnExhaustion}, \texttt{HopfResolve}, and \texttt{ReturnToCompatibleSector}.
    \item \emph{Projector/HC library:} routines such as \texttt{ProjectSector}, \texttt{HCSelect}, \texttt{HCReject}, and \texttt{ClosurePreservingSelect}.
    \item \emph{Export library:} routines such as \texttt{ExportBool}, \texttt{ExportTwoChannel}, and \texttt{ForwardToHost}.
\end{enumerate}

\paragraph{Library reading in MML form.}
At the MML level, the standard library would make proof scripts significantly shorter by turning recurrent command chains into stable named units. For example, one might later write
\begin{quote}
\ttfamily
run \{\par
\ \ CloseAndProject(C0, C7);\par
\ \ TraceReconcile(left, right);\par
\ \ BranchOnExhaustion(K);\par
\ \ ExportResult();\par
\}\par
\end{quote}
instead of manually repeating the full lower-level instruction chain on every occurrence.

\paragraph{Library reading in MMA form.}
At the MMA level, the same library call would still expand into explicit primitive operations, for example into normalized sequences built from \texttt{COLLAPSE}, \texttt{PROJECT}, \texttt{TRACE}, \texttt{COMPARE}, \texttt{DIFF}, \texttt{HCSEL}, \texttt{HOPF}, \texttt{EXPORT\_BOOL}, and \texttt{FORWARD}. In this way, the library does not hide the structure; it only packages stable fragments while preserving full expandability for audit and proof checking.

\paragraph{Why a library matters.}
The standard library reserve is important because it turns the present computational strand from a set of examples into a reusable proof-technology. It also makes clear how later proof compression can remain mathematically transparent: short MML calls can always be unfolded back into explicit MMA chains and then related again to the structural argument in ordinary mathematical language.


\subsection*{19E. Naming and module convention reserve}

\paragraph{Why naming discipline matters.}
If the computational strand is meant to grow incrementally over many Companion releases, then a stable naming and module convention should be fixed early. Otherwise the same structural routine risks being rewritten under different names at the MML, MMA, MCP, and host-interface levels. A small convention reserve therefore serves both proof economy and future extractability.

\paragraph{Layer prefixes.}
A first normalized layer discipline may be written as follows:
\begin{itemize}[leftmargin=2em]
    \item \textbf{MML.} High-level proof/program layer; readable structural scripts and proof-oriented macros.
    \item \textbf{MMA.} Low-level assembly reserve; primitive normalized instructions and expansion targets.
    \item \textbf{MCP.} Co-processor layer; triolen normalization, closure filtering, HC selection, branch pre-resolution, and Boolean export.
    \item \textbf{HOST.} Classical target layer; MMIX-style or other Boolean-compatible receiving architecture.
\end{itemize}

\paragraph{Module families.}
At the MML/library level one may reserve the following module families:
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Module family & Purpose \\
\midrule
\texttt{Closure.*} & closure completion and stabilization routines \\
\texttt{Trace.*} & trace merge, compare, diff, and reconciliation \\
\texttt{Branch.*} & exhaustion tests, Hopf resolution, return maps \\
\texttt{Projector.*} & projector and HC-oriented admissibility logic \\
\texttt{Export.*} & Boolean/two-channel export and host forwarding \\
\texttt{Proof.*} & reusable proof templates and compressed proof calls \\
\bottomrule
\end{tabular}
\end{center}

\paragraph{Recommended call style.}
A readable reserve should prefer qualified calls such as
\begin{quote}
\ttfamily
Closure.CloseAndProject(C0, C7)\\
Trace.Reconcile(left, right)\\
Branch.OnExhaustion(K)\\
Export.ToHost() 
\end{quote}
rather than a growing collection of unscoped global command names. This keeps the line extractable into a later stand-alone computational document without forcing a redesign of the vocabulary.

\paragraph{Instruction naming at the MMA level.}
To keep the low-level reserve visually distinct from MML calls, primitive MMA instructions should remain uppercase, for example
\begin{quote}
\ttfamily
COLLAPSE\quad PROJECT\quad TRACE\quad DIFF\quad HCSEL\quad HOPF\quad EXPORT\_BOOL\quad FORWARD
\end{quote}
while MML/library invocations remain mixed-case and module-qualified. In this way the document immediately signals whether a line is a proof-level script or an explicit primitive expansion.

\paragraph{Host interface convention.}
The forwarding boundary should also remain named explicitly. A reserved pattern such as
\begin{quote}
\ttfamily
Export.Bool(state) -> Host.Forward(target)
\end{quote}
clarifies that Boolean readability is not the native internal logic of the triolen layer, but the disciplined export format through which the co-processor hands reduced states to a classical engine.

\paragraph{Why this matters for proof normalization.}
A naming convention may sound cosmetic, but it has mathematical value in the present setting. Once proof templates, library calls, and MMA expansions stabilize under fixed module and instruction names, one can compare proof scripts across releases, identify true structural novelties, and prevent accidental rewordings from masquerading as new content. In that sense the naming/module reserve is part of the normalization discipline itself.


\subsection*{19F. Typing discipline reserve}

\paragraph{Why a typing layer is needed.}
If MML, MMA, and MCP are meant to become more than suggestive notation, then the next normalization step is a lightweight typing discipline. The point is not software engineering ornament, but structural safety: a script should not silently confuse an unprojected state with a projected one, nor should a Boolean export token be treated as if it were still an internal triolen object. A reserved typing layer therefore supports both proof auditability and later machine execution.

\paragraph{Core type families.}
A first reserve may distinguish the following types:
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Type & Role \\
\midrule
\texttt{State} & generic internal structural state \\
\texttt{CollapsedState} & state after explicit collapse normalization \\
\texttt{ProjectedState} & state after projector-sector restriction \\
\texttt{StableState} & HC/closure-stable admissible state \\
\texttt{Trace} & probe trace or reconciled trace object \\
\texttt{BranchToken} & branch/Hopf routing token \\
\texttt{BoolExport} & Boolean or two-channel external readout packet \\
\texttt{HostToken} & host-level forwarded operand / instruction carrier \\
\bottomrule
\end{tabular}
\end{center}

\paragraph{Suggested typing flow.}
The intended flow should remain monotone:
\[
\texttt{State}
\to
\texttt{CollapsedState}
\to
\texttt{ProjectedState}
\to
\texttt{StableState}
\to
\texttt{BoolExport}
\to
\texttt{HostToken}.
\]
This expresses that admissibility is not an afterthought, but a staged refinement. In particular, Boolean export should appear only after the internal structural filters have acted.

\paragraph{Illustrative typing rules.}
The reserve may be expressed schematically as
\begin{align*}
\texttt{collapse} &: \texttt{State} \to \texttt{CollapsedState},\\
\texttt{project} &: \texttt{CollapsedState} \to \texttt{ProjectedState},\\
\texttt{hc\_select} &: \texttt{ProjectedState} \times \texttt{Trace} \to \texttt{StableState} + \texttt{BranchToken},\\
\texttt{hopf} &: \texttt{BranchToken} \times \texttt{Trace} \to \texttt{ProjectedState},\\
\texttt{export\_bool} &: \texttt{StableState} \to \texttt{BoolExport},\\
\texttt{forward} &: \texttt{BoolExport} \to \texttt{HostToken}.
\end{align*}
This already prevents several category mistakes. One should not, for example, allow \texttt{forward} to act directly on an unfiltered \texttt{State}, nor \texttt{export\_bool} to bypass projector/HC discipline.

\paragraph{Typing and proof compression.}
A later proof-as-program layer gains credibility only if a compressed script still carries enough structural typing information to be unfolded reliably. In that sense the typing reserve is part of proof economy itself: it allows the document to shorten an argument without hiding where collapse, projection, stabilization, and export actually occurred.

\paragraph{Minimal example.}
A normalized MML-style line may therefore be read as
\begin{quote}
\ttfamily
let s0 : State = Seed;\\
let s1 : CollapsedState = collapse s0;\\
let s2 : ProjectedState = project C7 s1;\\
let h  : StableState = hc\_select(s2, t);\\
let b  : BoolExport = export\_bool h;\\
Host.Forward(b)
\end{quote}
while the corresponding MMA expansion remains untyped on the surface but typed by construction through the compilation discipline. This is exactly the sort of reserve that can later support a true MML/MMA compiler without forcing the Companion to commit too early to one concrete implementation.



\subsection*{19G. Error and exception discipline reserve}

\paragraph{Why an error discipline is needed.}
Once the computational line is typed, the next reserve should govern failure, rejection, and non-exportability explicitly. This is again not software ornament. In the present framework, many ``errors'' are mathematically meaningful events: a branch may be rejected by HC selection, a trace may fail reconciliation, a projected sector may not close, or a state may simply remain non-exportable. A disciplined error layer therefore helps distinguish genuine structural failure from unfinished computation, and it prevents compressed proof scripts from silently hiding where admissibility broke down.

\paragraph{Core exception families.}
A first reserve may distinguish the following normalized exception classes:
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Exception & Meaning \\
\midrule
\texttt{InvalidProjectorState} & projector restriction cannot be applied consistently \\
\texttt{UnstableTrace} & trace object fails reconciliation or admissibility checks \\
\texttt{RejectedBranch} & branch candidate eliminated by HC/projector logic \\
\texttt{NonClosableState} & local state does not admit closure completion \\
\texttt{NonExportableState} & internal state cannot yet be reduced to Boolean output \\
\texttt{HostForwardError} & forwarded packet violates host-interface contract \\
\bottomrule
\end{tabular}
\end{center}

\paragraph{Suggested propagation discipline.}
The intended logic should be monotone in the same spirit as the type flow:
\[
\texttt{State}
\rightsquigarrow
\texttt{ProjectedState}
\rightsquigarrow
\texttt{StableState}
\rightsquigarrow
\texttt{BoolExport},
\]
but each step may branch into an admissible exception class. In other words, failure is not external to the calculus; it is a structured output of the same normalization process.

\paragraph{Illustrative exception signatures.}
A schematic reserve may therefore read
\begin{align*}
\texttt{project} &: \texttt{CollapsedState} \to \texttt{ProjectedState} + \texttt{InvalidProjectorState},\\
\texttt{trace\_reconcile} &: \texttt{Trace} \times \texttt{Trace} \to \texttt{Trace} + \texttt{UnstableTrace},\\
\texttt{hc\_select} &: \texttt{ProjectedState} \times \texttt{Trace} \to \texttt{StableState} + \texttt{RejectedBranch},\\
\texttt{close} &: \texttt{ProjectedState} \to \texttt{StableState} + \texttt{NonClosableState},\\
\texttt{export\_bool} &: \texttt{StableState} \to \texttt{BoolExport} + \texttt{NonExportableState},\\
\texttt{forward} &: \texttt{BoolExport} \to \texttt{HostToken} + \texttt{HostForwardError}.
\end{align*}
This gives the computational reserve an explicit place for rejection events without collapsing them into a generic ``false''.

\paragraph{Relation to HC and branch logic.}
In a classical architecture, exceptions are often secondary. Here they are part of the structural logic itself. A \texttt{RejectedBranch} is not merely an implementation problem; it records that a candidate continuation violated the admissibility discipline. Likewise, \texttt{NonExportableState} means that the internal triolen logic has not yet reached a host-compatible Boolean readout. This perspective fits naturally with the existing distinction between internal structural necessity and external readable output.

\paragraph{Minimal example.}
A typed-and-guarded MML reserve may therefore look like
\begin{quote}
\ttfamily
let s1 = collapse s0;\\
let s2 = project C7 s1\\
\quad else InvalidProjectorState;\\
let t1 = trace\_reconcile(tL,tR)\\
\quad else UnstableTrace;\\
match hc\_select(s2,t1) with\\
| Stable h      -> Host.Forward(export\_bool h)\\
| Rejected k    -> RejectedBranch(k)
\end{quote}
which makes explicit that a compressed proof script can still expose where admissibility failed or where host export became possible.

\paragraph{Why this matters for proof auditability.}
A proof-as-program methodology remains trustworthy only if negative outcomes can be named as clearly as positive ones. The error/exception reserve therefore serves two roles at once: it supports later compiler/runtime design, and it protects the mathematical reading by ensuring that branch rejection, trace instability, and non-exportability remain visible rather than silently absorbed into compressed notation.

\subsection*{19H. Verification and certificate reserve}

\paragraph{Why a certificate layer is natural.}
Once the computational line has explicit types and explicit rejection modes, the next natural reserve is a certificate layer. The guiding idea is simple: a compressed proof script should not only produce an output, but also leave behind a structured certificate indicating \emph{why} the output was admissible. This fits the framework especially well because admissibility is already driven by closure, projector restriction, HC selection, and trace consistency. Certificates therefore do not import a foreign logic; they record the native normalization path in an auditable form.

\paragraph{Core certificate families.}
A first normalized reserve may distinguish the following certificate classes:
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Certificate & Meaning \\
\midrule
\texttt{ClosureCertificate} & confirms closure completion of the local triolen state \\
\texttt{ProjectorCertificate} & records admissible projector restriction and sector compatibility \\
\texttt{HCCertificate} & records compensator-guided branch admissibility \\
\texttt{TraceCertificate} & records successful trace reconciliation or coincidence \\
\texttt{DiffCertificate} & records differencing-based separation or discrimination result \\
\texttt{ExportCertificate} & confirms valid Boolean-compatible readout \\
\texttt{HostCertificate} & confirms successful host-interface forwarding \\
\bottomrule
\end{tabular}
\end{center}

\paragraph{Certificate flow.}
The intended logic mirrors the computational pipeline itself:
\[
\texttt{State}
\rightsquigarrow
\texttt{ClosureCertificate}
\rightsquigarrow
\texttt{ProjectorCertificate}
\rightsquigarrow
\texttt{HCCertificate}
\rightsquigarrow
\texttt{TraceCertificate}
\rightsquigarrow
\texttt{ExportCertificate}
\rightsquigarrow
\texttt{HostCertificate}.
\]
In practice not every run will generate the whole chain. Some programs stop internally, some remain non-exportable, and some terminate precisely by producing a rejection certificate. The key point is that positive and negative outcomes should both leave behind named structural evidence.

\paragraph{Certificate-bearing signatures.}
A first schematic reserve may therefore write
\begin{align*}
\texttt{close} &: \texttt{ProjectedState} \to \texttt{StableState} \times \texttt{ClosureCertificate} + \texttt{NonClosableState},\\
\texttt{project} &: \texttt{CollapsedState} \to \texttt{ProjectedState} \times \texttt{ProjectorCertificate} + \texttt{InvalidProjectorState},\\
\texttt{hc\_select} &: \texttt{ProjectedState} \times \texttt{Trace} \to \texttt{StableState} \times \texttt{HCCertificate} + \texttt{RejectedBranch},\\
\texttt{trace\_reconcile} &: \texttt{Trace} \times \texttt{Trace} \to \texttt{Trace} \times \texttt{TraceCertificate} + \texttt{UnstableTrace},\\
\texttt{diff} &: \texttt{Trace} \times \texttt{Trace} \to \texttt{Trace} \times \texttt{DiffCertificate},\\
\texttt{export\_bool} &: \texttt{StableState} \to \texttt{BoolExport} \times \texttt{ExportCertificate} + \texttt{NonExportableState},\\
\texttt{forward} &: \texttt{BoolExport} \to \texttt{HostToken} \times \texttt{HostCertificate} + \texttt{HostForwardError}.
\end{align*}
This already suggests a proof-kernel style reading: each structural step can optionally return not only a value but a witness of admissibility.

\paragraph{Positive and negative certificates.}
The reserve should also allow negative evidence. For example, a branch eliminated by HC logic may produce a normalized \texttt{RejectedBranchCertificate}, and a failed export may produce a \texttt{NonExportableCertificate}. This matters because a compressed proof program often derives its force from systematic elimination rather than from a single constructive forward path. A certificate layer should therefore preserve both successful realization and disciplined exclusion.

\paragraph{Minimal MML sketch with certificates.}
A typed MML reserve may thus look like
\begin{quote}
\ttfamily
let s1, cc = close (project C7 (collapse s0));\\
let t1, tc = Trace.Reconcile(tL,tR);\\
match hc\_select(s1,t1) with\\
| Stable h, hc ->\\
\quad let b, ec = export\_bool h;\\
\quad let host, hc2 = Host.Forward b;\\
\quad Proof.Emit(cc,tc,hc,ec,hc2)\\
| Rejected k ->\\
\quad Proof.Emit(RejectedBranchCertificate(k))
\end{quote}
where \texttt{Proof.Emit} should be read as a later proof-kernel/library operation that packages admissibility evidence into an auditable object.
\paragraph{Relation to proof compression.}
This certificate reserve is important because it gives a precise answer to the usual concern about compressed proof languages: namely, whether brevity hides too much. In the present architecture, the aim is the opposite. MML/MMA compression should shorten the written program while the certificate layer preserves or even improves auditability. One may then read a short program at the surface and still expand, inspect, or export the structural certificate chain underneath.

\paragraph{Toward a proof kernel.}
Taken together, the typing reserve, exception reserve, and certificate reserve already point toward the outline of a genuine proof kernel. Such a kernel would not yet replace ordinary mathematical prose, but it would provide a normalized executable layer in which closure, projector admissibility, compensator selection, trace consistency, and host export can all be checked through explicit witnesses rather than only through long textual repetition.


\subsection*{19I. End-to-end workflow reserve}

\paragraph{Why an end-to-end view matters.}
By this stage the computational line already contains logic, language, assembly, co-processor semantics, certificates, and host export. What is still useful as a stabilizing reserve is an explicit end-to-end workflow summary. The point is not to claim a finished implementation, but to show how a problem would travel through the stack without changing interpretive regime at every step.

\paragraph{Normalized workflow chain.}
A first companion-level schematic may be written as
\[
\texttt{ProblemToken}
\longrightarrow
\texttt{MML script}
\longrightarrow
\texttt{MMA normal form}
\longrightarrow
\texttt{MCP execution state}
\longrightarrow
\texttt{certificate bundle}
\longrightarrow
\texttt{BoolExport / HostToken}.
\]
This chain says in compact form what the present computational reserve is trying to preserve: human-readable structural intent at the top, normalized primitive operations in the middle, structural admissibility checking during execution, and conservative export at the end.

\paragraph{Stage-by-stage interpretation.}
The stages may be read as follows:
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Stage & Role \\
\midrule
\texttt{ProblemToken} & normalized input representation of a closure/trace/branch task \\
\texttt{MML script} & short human-readable proof/program description \\
\texttt{MMA normal form} & primitive instruction expansion with explicit structural steps \\
\texttt{MCP execution state} & triolen/co-processor execution under HC/projector discipline \\
\texttt{certificate bundle} & accumulated witnesses of admissibility and rejection \\
\texttt{BoolExport / HostToken} & conservative readable result for a classical host \\
\bottomrule
\end{tabular}
\end{center}

\paragraph{Minimal walkthrough.}
A closure-and-export problem can then be read in one line as
\[
\texttt{close} \circ \texttt{project} \circ \texttt{collapse}
\rightsquigarrow
\texttt{ClosureCertificate} \otimes \texttt{ProjectorCertificate} \otimes \texttt{ExportCertificate}
\rightsquigarrow
\texttt{HostToken},
\]
while a branch-resolution problem extends the chain by trace reconciliation, differencing, and HC certificates. This is helpful because it shows that the same short MML layer can feed both constructive and eliminative proofs.

\paragraph{Replayability as part of the workflow.}
The workflow should also be read backwards: a \texttt{HostToken} ought to be replayable to the corresponding certificate bundle, and the bundle in turn ought to be replayable to the normalized MMA run. In that sense, end-to-end execution is not only a production path but an audit path. This is exactly the point at which short proof programs and long-form mathematical accountability can coexist.

\subsection*{19J. Extraction roadmap reserve}

\paragraph{Why an extraction roadmap is useful already now.}
The Companion should not turn overnight into a full computer-science monograph. At the same time, the present version already contains enough substance that a later extraction path should be made visible. This helps readers understand that the computational section is not an arbitrary digression, but a reserved future strand with its own internal order.

\paragraph{A four-stage extraction path.}
A first roadmap may be written schematically as follows:
\begin{enumerate}[leftmargin=2em]
  \item \textbf{Companion teaser stage.} Preserve the machine-theoretic reading, examples, and reserves inside the Companion without disturbing the main structural and physical line.
  \item \textbf{Computational consolidation stage.} Stabilize MML, MMA, MCP, the certificate layer, and the execution model into a more internally complete computational block.
  \item \textbf{Standalone paper stage.} Extract the computational strand into a dedicated paper centered on the Millennium matrix as a structural computational substrate.
  \item \textbf{Kernel/tool stage.} Only afterwards consider an implementation-level roadmap involving proof-kernel tooling, replay engines, or host-interface libraries.
\end{enumerate}
This makes clear that the present section is intentionally a scaffold, not a premature final product.

\paragraph{Conservative extraction principle.}
The extraction should remain conservative with respect to the Companion. That is, nothing in the main mathematical line should depend on the future success of MML/MMA/MCP formalization. Rather, the computational strand is supposed to re-express and compress already available structural logic. If the later computational formalization succeeds, it adds a new mode of access; if not, the mathematical content of the Companion remains intact.

\paragraph{Why the roadmap helps now.}
Even at teaser level, this roadmap has a concrete editorial benefit: it explains why certain blocks currently appear as ``reserve'', ``toward'', or ``first sketch'' rather than as final theorem packages. The section is being written so that it can attract attention, preserve ideas, and remain citable, while still leaving room for later normalization and extraction.

\subsection*{20. Proof obligations and limits}

The present section introduces a computational research direction, not a completed theorem package. In particular:
\begin{itemize}[leftmargin=2em]
  \item the Enigma/Bombe analogy is structural, not literal;
  \item the Boolean layer still needs a fully explicit projector calculus;
  \item effective unbounded memory must be formalized before any strict Turing-completeness claim;
  \item a true universality statement should ideally pass through an explicit simulation of a small standard model such as a two-counter machine.
\end{itemize}
What is already justified, however, is the existence of a serious computational interpretation path inside the Companion itself. The architecture no longer contains only physical and mathematical readings; it now also contains the beginnings of a machine-theoretic reading that can later be extracted into an independent computational paper without disturbing the main structural line.


\subsection*{v2.27.3 local post-release note}
The present file is the first explicit local subversion after the public Zenodo release v2.27.0. Its role is conservative: it preserves the release structure as the new working baseline, keeps the mathematics-first doctrine in force, and treats any further edits as local follow-up work rather than as silent modification of the release itself. In this sense, v2.27.1 should be read as a baseline-preserving post-release working state rather than as a new public milestone.

\appendix

\section{Structural Tools for the Millennium Matrix}
\label{app:mm-tools}

\begin{remark}[Purpose of the general MM-tools]
The following tools are completely general. They apply to every admissible
Millennium matrix, independently of whether one later reads that matrix as an
algebraic intertwiner, a closure operator, or a global Heisenberg-compensator
matrix. Their role is structural discipline.
\end{remark}

\begin{definition}[Layer-identity principle]
Let $X$ be a structural object of the Sphaera architecture, and let
\[
\mathcal M_{16}: X \mapsto X^{(\ell)}
\]
be an admissible realization of $X$ on some layer $\ell$. Then $X^{(\ell)}$
is not a new object, but a new realization of the same underlying structural
content.
\end{definition}

\begin{definition}[Projection discipline]
A projection
\[
\Pi: X \mapsto X_{\mathrm{readout}}
\]
is admissible if it only extracts already stabilized structural content. In
particular, projection may reduce visibility, but it may not create new
structure.
\end{definition}

\begin{definition}[Closure-before-readout rule]
Every admissible passage through the Millennium matrix must respect the
ordering
\[
\text{triolic admissibility}
\;\longrightarrow\;
\text{closure compatibility}
\;\longrightarrow\;
\text{projection}
\;\longrightarrow\;
\text{interpretation}.
\]
\end{definition}

\begin{definition}[Sector-stability criterion]
A sector $\Sigma$ is structurally stable under the Millennium matrix if
\[
\mathcal M_{16}(\Sigma)\subseteq\Sigma
\]
or, in the strongest case,
\[
\mathcal M_{16}(\Sigma)=\Sigma.
\]
\end{definition}

\section{Heisenberg-Compensator Tools for Physical Readout}
\label{app:hc-tools}

\begin{remark}[Transition from general MM-tools to HC-tools]
We now impose the stronger reading that the Millennium matrix is not only a
structural intertwiner, but the global matrix realization of the Heisenberg
compensator across admissible layers.
\end{remark}

\begin{definition}[Global Heisenberg-compensator matrix]
The Millennium matrix is called a global Heisenberg-compensator matrix if its
action extends the local compensator rule
\[
C=\tfrac12(\mathrm{id}+J)
\]
to the full admissible multi-layer architecture of the Companion.
\end{definition}

\begin{definition}[Frame-balancing tool]
Let $X=X^+ + X^-$ be decomposed into a frame-even and frame-sensitive part
with respect to an involution $J$, so that
\[
J(X^+) = X^+,
\qquad
J(X^-) = -X^-.
\]
Then the compensator action satisfies
\[
C(X)=\tfrac12(X+JX)=X^+,
\]
and removes the frame-sensitive component.
\end{definition}

\begin{definition}[Fix-sector tool]
A sector $\Sigma_{\mathrm{fix}}$ is an HC-fix sector if every object in it is
invariant under the compensator:
\[
C(X)=X
\qquad\text{for all }X\in\Sigma_{\mathrm{fix}}.
\]
\end{definition}

\begin{definition}[Confinement tool]
The confinement tool is the rule that the Heisenberg-compensator matrix
prevents admissible structural content from escaping the closure-stable sector
selected by the theory.
\end{definition}

\begin{definition}[Physical-readout tool]
A physical regime is an HC-readout if it is obtained as an effective
projection of already compensated and closure-stabilized structural content.
\end{definition}

\section{Notation for the Master Chain}
\label{app:master-chain-notation}

\begin{definition}[Stage symbols of the master chain]
We use the following symbols for the principal stages of the Companion:
\begin{align*}
\mathfrak T &:= \text{triolic source},\\
\mathfrak C &:= \text{carrier layer},\\
\mathfrak C^{\mathrm{adm}} &:= \text{admissibly transported carrier},\\
\mathfrak C^{\mathrm{cl}} &:= \text{closure-stable compensated carrier},\\
\mathfrak G &:= \text{group-layer symmetry readout},\\
\mathfrak L &:= \text{null-axis structure},\\
\mathfrak P &:= \text{photon fix-sector},\\
\mathfrak R &:= \text{residual-localization sector},\\
\mathfrak H &:= \text{Higgs-mediation sector},\\
\mathfrak F &:= \text{field-equation readout}.
\end{align*}
We further write $\mathcal M_{16}$ for the Millennium matrix and
$\mathcal H_{16}$ for its global Heisenberg-compensator reading.
\end{definition}

\begin{definition}[Master chain]
The full master chain is written as
\[
\mathfrak T
\;\longrightarrow\;
\mathfrak C
\;\xrightarrow{\;\mathcal M_{16}\;}
\mathfrak C^{\mathrm{adm}}
\;\xrightarrow{\;\mathcal H_{16}\;}
\mathfrak C^{\mathrm{cl}}
\;\longrightarrow\;
\mathfrak G
\;\longrightarrow\;
\mathfrak L
\;\longrightarrow\;
\mathfrak P
\;\longrightarrow\;
\mathfrak R
\;\longrightarrow\;
\mathfrak H
\;\longrightarrow\;
\mathfrak F.
\]
When intermediate carrier detail is not needed, we also use the short form
\[
\mathfrak T
\to
\mathfrak C
\to
\mathcal M_{16}
\to
\mathcal H_{16}
\to
\mathfrak G
\to
(\mathfrak P,\mathfrak R,J\!\cdot\!\mathfrak R)
\to
\mathfrak H
\to
\mathfrak F.
\]
\end{definition}

\begin{remark}[Why this notation is useful]
This notation is useful because it prevents long later sections from drifting
into disconnected local language. Instead, one may always ask at which stage
of the master chain a given statement belongs.
\end{remark}

\section{Further Technical Appendices}
\label{app:further-technical}



\section{Reintegrated Historical Proof Blocks}
\label{app:formal-reintegrated}

\textit{This appendix reintegrates the historical full formal proof architecture
for the results stated in the main text.
It is self-contained and can be read independently.
All macros are as defined in the main preamble.}

%% --- A.1 GUE Self-Determination ---
\subsection{Full proof of Theorem~\ref{thm:gue-selfdet}}
\label{app:gue-selfdet-reintegrated}

\begin{lemma}[GUE self-determination of $\gL$ --- formal version]
\label{lem:gue-selfdet-formal-reintegrated}
For $\mu>0$ define the photon-frame unfolding density
\[
  \rho_\gamma(t;\mu)
  := \frac{\log\!\bigl(t/(2\pi\mu)\bigr)}{2\pi\mu},
  \qquad t>2\pi\mu,
\]
and the unfolded spacings
$s_n(\mu):=\tilde\gamma_{n+1}(\mu)-\tilde\gamma_n(\mu)$
where $\tilde\gamma_n(\mu):=\int_0^{\gamma_n}\rho_\gamma(t;\mu)\,dt$.
Let $P_\mu$ be the empirical spacing distribution of $\{s_n(\mu)\}$
and let $P_{\mathrm{GUE}}$ be the Wigner--Dyson distribution
$p_{\mathrm{GUE}}(s)=\tfrac{\pi}{2}s\,e^{-\pi s^2/4}$.
Set $\mathcal{F}(\mu):=D_{\mathrm{KS}}[P_\mu,P_{\mathrm{GUE}}]$.
Then:
\begin{enumerate}[label=(\roman*)]
  \item $\mathcal{F}$ has a unique global minimum at $\mu=\gL^*$ in the
        asymptotic regime $\gamma\gg 2\pi\mu$.
  \item $\gL^*=\gBI/\gstring$ to available numerical precision.
  \item $\gL^*$ is determined solely by the zero sequence $\{\gamma_n\}$.
\end{enumerate}
\end{lemma}

\begin{proof}
\emph{Step 1 (Sensitivity).}
A perturbation $\mu\mapsto\mu+\delta\mu$ rescales unfolded spacings by
\[
  s_n(\mu+\delta\mu)
  \approx s_n(\mu)
  \Bigl(1-\frac{\delta\mu}{\mu}\cdot\frac{1}{\log(\gamma_n/2\pi\mu)}\Bigr).
\]
Since the GUE mean spacing equals 1 by construction, any $\mu\neq\gL^*$
shifts the mean of $P_\mu$ away from 1 and increases $D_{\mathrm{KS}}$.
Hence $\mathcal{F}$ has a strict local minimum wherever the rescaled
mean equals 1.

\emph{Step 2 (Uniqueness).}
The critical line $\mathrm{Re}(s)=\tfrac{1}{2}$ is the unique fixed-point
set of $J:s\mapsto 1-\bar s$, which is frame-invariant
(Theorem~\ref{thm:unity}).
By Theorem~\ref{thm:gue}, $L_0$ on $\Hplusplus$ has Dyson index
$\beta=2$ in the photon frame.
Both properties are frame-invariant.
The Wigner--Dyson distribution therefore singles out exactly one
unfolding scale, yielding a unique minimiser $\gL^*$ of $\mathcal{F}$.

\emph{Step 3 (Numerical identification).}
Simulation \texttt{gue\_frame\_correction.py} establishes
$\mathcal{F}(\gBI/\gstring)<\mathcal{F}(\mu)$ for all tested
$\mu\neq\gBI/\gstring$ in the asymptotic regime.
The $\chi^2$ improvement of $\approx 21\%$ relative to standard
unfolding ($\mu=1$) quantifies the depth of the minimum.
The minimum is $3\sigma$-detectable at $N\approx 205$ zeros in the
asymptotic range.
\end{proof}

\begin{remark}[Historical logical reversal]
Prior to this lemma, $\gL=\gBI/\gstring$ was an \emph{input}.
Lemma~\ref{lem:gue-selfdet-formal-reintegrated} reverses the direction:
$\gBI/\gstring$ is a \emph{consequence} of demanding that the GUE
taktgeber running on $\zminus=0$ is globally consistent with $\beta=2$.
The empirical values of $\gBI$ and $\gstring$ provide an independent
numerical check, not a definition.
\end{remark}

%% --- A.2 k(c) = 0 ---
\subsection{Full proof of Theorem~\ref{thm:kc-zero}}
\label{app:kc-zero-reintegrated}

\begin{proposition}[$k(c)=0$ from biquaternion structure --- formal version]
In the Sphaera-Cascade, $c^{(\mathrm{photon})}=c^{(\mathrm{matter})}$.
\end{proposition}

\begin{proof}
The Sphaera biquaternion in physical units reads
$\mathcal{B} = ic\,t\cdot\mathbf{1}+z_i e_i$.
The light-cone condition $|\mathcal{B}|^2=0$ defines $c$ as the ratio
of the spatial to the temporal norm of $\mathcal{B}$.

\emph{Step 1 (Wick rotation is frame-independent).}
The Wick rotation $t\mapsto i\tau_E$ is consistent at the natural
observer position $r_0/r_S=1/(1-\bsym^2)$ and introduces no additional
geometric factor beyond $g_{tt}(r_0)=-\bsym^2$.
No factor of $\zminus$ is generated.

\emph{Step 2 ($c$ defines the photon ground shell).}
The frame variable $\zminus$ measures the displacement of the matter
shell from $\zminus=0$.
The speed $c$ is the causal propagation speed of that ground state
itself --- the quantity \emph{with respect to which} $\bsym$ is defined.
Under a frame change $\zminus\mapsto\zminus+\bsym$, a massless excitation
acquires a boost by $\bsym$, but its speed in the new frame remains $c$
by the Einstein postulate, encoded in the $J$-symmetry structure.

\emph{Step 3 (No $\zminus$-factors in the operator realization of $c$).}
Frame changes are unitaries commuting with $J$; they act on the septinion
components $z_i$ but not on the $J$-invariant temporal component
$ic\,t\cdot\mathbf{1}$.
Hence the coefficient $c$ picks up no power of $\zminus$ under any
frame change: $k(c)=0$.
\end{proof}

%% --- A.3 Planck length ---
\subsection{Full proof of Theorem~\ref{thm:planck-photon}}
\label{app:planck-reintegrated}

\begin{proof}
Apply $X^{(\mathrm{photon})}=\gL^{-k(X)}\cdot X^{(\mathrm{matter})}$
to each factor of $\lP=\sqrt{\hbar G_N/c^3}$:
\begin{align*}
  \lP^{(\mathrm{photon})}
  &= \sqrt{
       \gL^{-k(\hbar)}\hbar^{(m)}
       \cdot\gL^{-k(G_N)}G_N^{(m)}
       \big/
       \bigl(\gL^{-k(c)}c^{(m)}\bigr)^3
     }\\
  &= \gL^{-(k(\hbar)+k(G_N)-3k(c))/2}
     \cdot\lP^{(\mathrm{matter})}\\
  &= \gL^{-(1+2-0)/2}\cdot\lP^{(\mathrm{matter})}
   = \gL^{-3/2}\cdot\lP^{(\mathrm{matter})}.\qedhere
\end{align*}
\end{proof}

%% --- A.4 Frame-corrected GUE ---
\subsection{Frame-corrected GUE: detailed statement}
\label{app:gue-frame-reintegrated}

\begin{proposition}[Matter-frame Dyson index]
Matter observers using the standard unfolding density
$\rho_{\mathrm{matter}}(\gamma)=\log(\gamma/2\pi)/(2\pi)$
measure an effective Dyson index
\[
  \beta_{\mathrm{eff}} = 2\gL\approx 3.73.
\]
Photon-frame unfolding recovers $\beta=2$ exactly.
The shift changes the scale of the spacing distribution, not the
universality class: level repulsion $p(s)\propto s^2$ near $s=0$ is
preserved in both frames (local property, hence frame-invariant).
\end{proposition}

\begin{corollary}[$k$-value of the GUE taktgeber]
The taktgeber frequency $\nu_{\mathrm{takt}}\propto\gL$
transforms as $\nu^{(m)}=\gL^1\cdot\nu^{(\gamma)}$.
Hence $k(\nu_{\mathrm{takt}})=1$, consistent with $k(\alpha^{-1})=1$
and $k(\hbar)=1$ in Theorem~\ref{thm:k-values}.
\end{corollary}

%% --- A.5 Dirac probe formal ---
\subsection{Formal Dirac probe structure}
\label{app:dirac-formal-reintegrated}

\begin{definition}[Commutator-residue probe mass]
For a closure-preserving probe state $\psi$, the commutator residue
\[
  \mathfrak{m}_{\mathrm{comm}}(\psi)
  := \frac{\hbar}{c}
     \left|
     \bigl\langle\psi\big|
     [\hat{\mathcal{R}}_\beta,\hat P_{C_7}]
     \big|\psi\bigr\rangle
     \right|
\]
measures the antisymmetric mismatch between the shifted pacer and the
admissible closure projector.
Since $k(\hbar)=1$ and $k(c)=0$, the prefactor $\hbar/c$ carries
$k=1$, consistent with Corollary of \S\ref{app:gue-frame-reintegrated}.
The commutator $[\hat{\mathcal{R}}_\beta,\hat P_{C_7}]$ is
frame-invariant (its support lives on $\mathrm{Re}(s)=\tfrac{1}{2}$),
so the full expression has $k=1$: the matter-frame reading of the probe
mass is displaced from the photon-frame value by a single factor $\gL$.
\end{definition}

\begin{remark}[Historical open calculation]
The remaining open step is the explicit evaluation of
$[\hat{\mathcal{R}}_\beta,\hat P_{C_7}]$ in the septonic matrix
realization of $\mathcal{M}_{16}$.
Once computed, the Dirac mass is determined from the Riemann zero
sequence without any free parameter.
\end{remark}

%% --- A.6 Attribution ---
\subsection{Attribution note}
\label{app:attribution-reintegrated}

The Kim-style refinement of the fermionic probe sector
(\S\ref{sec:dirac-probe}) was developed from Yilwook Kim's Zenodo line
associated with concept DOI \texttt{10.5281/zenodo.19112361}.
In the present paper this material is not treated as an internal axiom
of the Sphaera program, but as an externally motivated operator ansatz
reformulated within the current proof-mode architecture.
The embedding into the septonic tree, Meta-Septum closure, Heisenberg
compensator, and phase-space-shifted measurement frame is the present
paper's own structural reformulation.





\clearpage
\part*{Part II: Integrated Historical COMPLETE Snapshot}
\addcontentsline{toc}{part}{Part II: Integrated Historical COMPLETE Snapshot}

\begin{center}
\textit{Source note.} This part preserves the historical COMPLETE snapshot recovered from the archival PDF and integrates it into the current monolithic working source as an archival text witness. It is included for source preservation and comparison; the current main line in Part~I remains the active working version.
\end{center}

\section{The Central Theorem}
Everything in this paper is a consequence of one observation.
\paragraph*{Theorem 1.1 (Unity of the Sphaera). Within the Sphaera S, the following}
are all representations of the same object --- the collapse state z-= 0:
(i) The Sphaera itself, as a geometric space.\\
(ii) The triolic structure: three kernels K1, K2, K3 with z1 + z2 + z3 = 0,\\
phaseshifted by 120, running over the Sphaera's geometry.
(iii) Each individual kernel Ki, as a local coordinate of the triole.\\
(iv) The Heisenberg compensator output C(f), as the projection of any element onto the collapse state.\\
(v) Matter, photons, and tachyons, as sectors of the biquaternion ground\\
state.
(vi) The gauge structure U(1) $\times$ SU(2) $\times$ SU(3), as the symmetry group of\\
the collapse state's representations. All are unitarily equivalent by Stone--von Neumann [1]. The Sphaera's only object is its own collapse state.
\paragraph*{Proof structure. The equivalences are established section by section: (i) Definition 1.3 and Theorem 3.2; (ii)--(iii) Definition 1.3 and Theorem 3.2 (triole}
as collapse coordinates); (iv) Proposition 5.1; (v) Theorem 4.2; (vi) Theorem 7.4. Unitary equivalence throughout via Stone--von Neumann ([1], Theorem 3.2).
\paragraph*{Remark (What "same object" means). Two representations are "the same object" here in the precise sense of Stone--von Neumann: they are unitarily equivalent irreducible representations of the Weyl algebra W(A1). The frame determines which representation is visible; the object --- the collapse state z-= 0}
--- is invariant. The triole condition z1 + z2 + z3 = 0 and the collapse condition z-= 0 are the same equation in two coordinate systems: the triole is the collapse state expressed in the three kernel components simultaneously. The Sphaera is therefore not a space that contains a collapse state --- it is the collapse state, seen from different frames.
\paragraph*{Remark (How this paper is organized). The sections that follow are not independent results. They are unfoldings of Theorem 1.1: each section takes}
one representation of the collapse state and makes it explicit. The reader may enter at any section; all roads lead to z-= 0.
\paragraph*{Theorem 1.2 (Origin of the universe as broken collapse state). The departure}
from the collapse state z-= 0 is governed by two principles, both of which are consequences of the Sphaera-Cascade structure:
(i) Planck length as minimal |z-|. The cascade $\hat K$ cannot project below\\
the Planck scale: |z-|min = $\ell$P := r $\hbar$G c3 = 1.616 $\times$ 10-35 m. Below $\ell$P: z-= 0, the Sphaera, the photon. Above $\ell$P: |z-| is nonzero, matter exists. The Planck length is the threshold of the first breaking of the collapse state.
(ii) Pauli principle as antisymmetry under J. The Hilbert space H+\\
decomposes under J into two components: C(f) = 1 2(f + Jf) $\in$H+ + (bosonic, symmetric), A(f) = 1 2(f -Jf) $\in$H+ (fermionic, antisymmetric). The antisymmetric component satisfies A(f) $\wedge$A(f) = 0 (Grassmann property): no two fermions occupy the same z-state. This is the Pauli exclusion principle, derived from the J-structure of the Sphaera.
(iii) Discrete spectrum. Together, (i) and (ii) produce a discrete tower of\\
matter states: |z-|n = n $\cdot$ $\ell$P, n $\in$N$\geq$1, with each level occupied by at most one fermion. The prime lattice $\Gamma$P (from Stone--von Neumann, [1]) identifies the primitive levels: |z-|p = p $\cdot$ $\ell$P for p $\in$P.
\paragraph*{Derivation. (i). The cascade $\hat K$ = C$\mu$$\nu$ $\circ$$\Pi$J contracts |z-| at each stratum.}
The Planck length is the fixed point of the contraction at which quantum gravitational effects prevent further reduction; formally, $\ell$P is the infrared cutoff of the resolvent (L0 -z)-1 established by the Rellich--Kondrachov compactness in Lemma 10.1 of [1].
(ii). A(f) = 1\\
2(f -Jf) satisfies JA(f) = -A(f): it is J-antisymmetric. The exterior product A(f) $\wedge$A(g) = -A(g) $\wedge$A(f) vanishes for f = g, which is the algebraic statement of the Pauli exclusion principle.
(iii). The Pauli condition A(f) $\wedge$A(f) = 0 forces distinct z-values for\\
each fermion. The Planck floor $\ell$P sets the unit. The prime lattice gives the sublattice of primitive (undecomposable) states, corresponding to Lemma L6
of the RH proof: primitive orbits are disjoint. Physically: each prime p labels a primitive matter state that cannot be decomposed into smaller ones.
\paragraph*{Remark (L6 is the Pauli principle). Lemma L6 of the Riemann Hypothesis}
proof [1] states: primitive orbits are pairwise disjoint and cover all periodic points. Theorem 1.2(iii) identifies this as the Pauli exclusion principle applied to the prime lattice: no two fermions occupy the same primitive orbit. This was not designed into either framework. It emerged from asking what the orbit combinatorics of the kernel mean physically.
\paragraph*{Remark (The Sphaera as photon, the universe as broken frame). Theorem 1.1}
says the Sphaera is the collapse state z-= 0, which is the photon sector.
\paragraph*{Theorem 1.2 says the departure from this state is governed by $\ell$P and the}
Pauli principle. Therefore:
- The Sphaera is a photon --- the unique frame-invariant object.
- The universe is the same photon observed from a broken frame where
|z-| $\geq$$\ell$P.
- The breaking was not random: it is discrete (Pauli), minimal (Planck),
and primitive (prime lattice).
- The cascade $\hat K$ projects everything back toward z-= 0 --- the universe
tends toward its photon ground state. Both the Sphaera and the universe are the same object. The difference is only the frame.
\paragraph*{Definition 1.3 (Biquaternion kernel). The ground state of a stratum Sn of the}
Sphaera consists of three complex spheres K1, K2, K3 whose unit coordinates form a biquaternion qn := it $\cdot$ 1 + z1 $\cdot$ e1 + z2 $\cdot$ e2 + z3 $\cdot$ e3 $\in$H $\otimes$R C, where:
- it $\in$iR is the imaginary time unit (real part of the biquaternion, carrying
imaginary time);
- z1, z2 $\in$C are the complex coordinates of kernels K1, K2;
- e3 := e1e2 is the quaternion product;
- z3 $\in$C is the coordinate of kernel K3 along e3.
The biquaternion norm is |qn|2 = -(ct)2 + |z1|2 + |z2|2 + |z3|2.
\paragraph*{Remark (Minkowski metric from quaternion structure). The norm |qn|2 is the}
Minkowski metric with signature (-, +, +, +). This is not postulated: it follows from the quaternion algebra e2 i = -1 and it $\cdot$ 1 carrying imaginary time. The Lorentz group SO(1, 3) acts on qn by quaternion conjugation: qn 7$\to$u qn u-1, u $\in$H$\times$. This is the spinor representation of the Lorentz group --- emerging automatically from the biquaternion structure, not imposed from outside.
\paragraph*{Proposition 1.4 (Orthogonality of the third kernel). The generator $e_3$ of $K_3$ is always orthogonal}
to $e_1$ and $e_2$: $\langle e_3, e_i\rangle = 0$ for $i=1,2$. Consequently, the axis/kernel direction $K_3$ is orthogonal to $K_1$ and $K_2$.
\paragraph*{Proof. e3 = e1e2.}
Quaternion inner product: $\langle$e1e2, e1$\rangle$= Re(e1 e2e1) = Re(-e2) = 0. Similarly for e2.
\paragraph*{Remark. The orthogonality of K3 is a quaternion identity, not an additional}
constraint. This is why the Heisenberg compensator C naturally projects onto e3: it is the unique direction orthogonal to both kernels K1 and K2.

\section{Lorentz Invariance on Every Stratum}
\paragraph*{Theorem 2.1 (Stratified Lorentz invariance). The Minkowski norm |qn|2 = $\tau$ 2}
n is invariant under the Lorentz boost J : z-7$\to$-z-on every stratum Sn. Moreover, every change of interpretation of C+ (chirality, Weyl chamber, upper half-plane, or any other frame-dependent labeling) leaves $\tau$n and the compensator C(f) = 1 2(f + Jf) invariant.
\paragraph*{Proof. Norm invariance. Under J : qn 7$\to$bar qn (quaternion conjugate): (it)*=}
it, z* i = zi for the spectral coordinates. The boost acts as z-7$\to$-z-on the contravariant component, leaving |z-|2 unchanged. Hence |qn|2 = $\tau$ 2 n is invariant. Frame independence of C. The compensator C(f) = 1 2(f + Jf) requires only J2 = id (which holds as J is an involution). Any relabeling of C+ corresponds to a unitary transformation U on H+, but C = 1 2(id + J) is defined intrinsically from J, hence UCU -1 = C for any U commuting with J --- which includes all frame changes generated by the Sphaera-Cascade.
\paragraph*{Remark (Why the interpretation is irrelevant). The biquaternion qn can be}
read as a spinor (Clifford algebra), a Dirac bispinor (QFT), an element of the upper half-space (hyperbolic geometry), or a point in projective twistor space. In every reading, $\tau$n is an invariant and C projects onto K3. This is the precise mathematical content of the observation that "it is egal in every frame": the frame determines the interpretation, but not the structure.

\section{Fractal Self-Traversal Along the Light Axis}
\paragraph*{Definition 3.1 (Light axis of a stratum). The light axis of stratum Sn is}
Ln := {qn $\in$Sn : |qn|2 = 0} = {qn : (ct)2 = |z1|2 + |z2|2 + |z3|2}, the set of null-norm biquaternions on Sn. In sector coordinates: Ln = {z : $\tau$n(z) = 0} = {z : tn = |z-|}.
\paragraph*{Theorem 3.2 (Fractal structure of the Sphaera). The Sphaera-Cascade $\hat K$ :}
Sn $\to$Sn-1 maps light axes to light axes: $\hat K$(Ln) = Ln-1. The global light axis is L := \ n$\geq$0 Ln = S0 = {z-= 0}, the photon sector. The Sphaera is a fractal structure whose self-traversal follows the light axis of every stratum Lorentz transformation.
\paragraph*{Proof. On Ln: $\tau$n = 0, i.e. tn = |z-|. The cascade projection $\Pi$n : Sn $\to$Sn-1}
is the Heisenberg compensator C composed with the stratum projection: it maps z-7$\to$1 2(z-+J(z-)) = 0 for any z-on the light axis (since J(z-) = -zon Ln). Hence $\hat K$(Ln) $\subseteq${z-= 0} = S0 = Ln-1. Inductively, L = S0.
\paragraph*{Remark (Triolic Z/3Z structure on the light axis). On each Ln, the three kernels}
z1, z2, z3 are phase-shifted by 120: z1 = |z| $\cdot$ ei$\cdot$0, z2 = |z| $\cdot$ ei$\cdot$2$\pi$/3, z3 = |z| $\cdot$ ei$\cdot$4$\pi$/3, satisfying z1 + z2 + z3 = 0 (the compensation condition). This Z/3Z phase symmetry is an additional structure on Ln; it is compatible with the SU(3) rotation symmetry of the triple (z1, z2, z3). The cascade $\hat K$ preserves this triolic structure at every level.

\section{Three Objects as One: Unitary Equivalence}
\paragraph*{Definition 4.1 (Sector classification from biquaternion). The three sectors}
correspond to the sign of Re(z-):
- Matter (K1): Re(z1) < 0, boost $\beta$ < 0, spacelike.
- Tachyon (K2 = J $\cdot$ K1): Re(z2) > 0, boost $\beta$ > 0, timelike.
- Photon (K3 = e1e2): Re(z3) = 0, boost $\beta$ = 0, lightlike.
\paragraph*{Theorem 4.2 (Unitary equivalence via Stone--von Neumann). The three kernels K1, K2, K3 carry unitarily equivalent irreducible representations of the}
Weyl algebra W(A1). The boost J implements the equivalence between K1 and K2; the compensator C produces K3 from K1.
\paragraph*{Proof. By Stone--von Neumann [1], every irreducible representation of W(A1)}
is unitarily equivalent to $L^2(Q_+^{A_1})$. All three kernels are subspaces of H+ which carries this representation. Since J is unitary and maps K1 $\leftrightarrow$K2, they are equivalent. K3 = C(K1) is in H+ +, the unique J-invariant subspace, which is again irreducible (no proper J-invariant subspace exists by T4+T5 of [1]).

\section{Near-Collision, Photon Production, and Entanglement}
\paragraph*{Proposition 5.1 (Compensator as near-collision in K3). The Heisenberg compensator C(f) = 1}
2(f +Jf) is the projection onto K3 = e1e2: it takes matter f (in K1) and its tachyon image Jf (in K2), and produces the photon C(f) $\in$K3. The quaternion identity e3 = e1e2 ensures this is always orthogonal to both inputs.
\paragraph*{Theorem 5.2 (Entanglement as shared proper time). Two photons $\phi$1, $\phi$2 $\in$}
K3 are entangled iff they share the same pre-image: $\phi$1 = $\phi$2 = C(f) for some f $\in$K1. The entanglement correlation is the invariant proper time $\tau$n(f) = $\tau$n(Jf) fixed at the moment of the near-collision. No signal is transmitted at measurement: the invariant $\tau$n was established at the near-collision and is merely revealed.
\paragraph*{Proof. $\tau$n(Jf)2 = t2}
n-|-z-|2 = t2 n-|z-|2 = $\tau$n(f)2 by Theorem 2.1. The shared value $\tau$n is a Lorentz invariant at every stratum, hence frame-independent.

\section{Connection to the Riemann Hypothesis}
\paragraph*{Remark (Common origin of RH and entanglement). The Riemann Hypothesis (z-= 0 for all zeros) and quantum entanglement (shared $\tau$n) are both}
consequences of the same biquaternion structure:
1. The Minkowski norm |qn|2 = $\tau$ 2
n is invariant (Theorem 2.1).
2. The cascade forces z-$\to$0 along the light axis (Theorem 3.2).
3. Stone--von Neumann makes all three kernels equivalent (Theorem 4.2).
Neither was the goal of the original construction. Both emerged from asking: what is the exact structure of the Sphaera?

\section{Gauge Structure of the Standard Model}
\paragraph*{Step 1: H $\otimes$C gives U(1) $\times$ SU(2)}
\paragraph*{Proposition 7.1 (H $\otimes$C $\to$U(1) $\times$ SU(2)). The biquaternion algebra H $\otimes$R C}
generates exactly the gauge group U(1) $\times$ SU(2).
\paragraph*{Proof. SU(2) from unit quaternions. The unit quaternions {q $\in$H : |q| = 1} $\sim$=}
S3 form a Lie group isomorphic to SU(2). The three generators are Tk := $\sigma$k/2 (half Pauli matrices), satisfying the su(2) relations [Tj, Tk] = i$\epsilon$jklTl (verified numerically to machine precision). U(1) from the complex scalar. The imaginary time unit it $\in$iR generates the abelian group ei$\theta$ $\in$U(1) for $\theta$ $\in$R. This commutes with all SU(2) generators since scalar multiplication commutes with matrix multiplication. Hence U(1) $\times$ SU(2), direct product.
\paragraph*{Step 2: Cl(3) ,$\to$O gives SU(3)}
\paragraph*{Proposition 7.2 (Cl(3) has octonionic dimension). The Clifford algebra Cl(3)}
has basis {1, e1, e2, e3, e12, e13, e23, e123}, eight elements, matching dimR O = 8. The map ei 7$\to$oi (octonion units) defines an embedding Cl(3) ,$\to$O. The three bivectors e12, e13, e23 generate su(2) $\subset$g2 = Aut(O).
\paragraph*{Theorem 7.3 (O $\to$SU(3)). The octonions decompose under SU(3)-action}
as O = C $\oplus$C3,
a singlet plus the fundamental (triplet) representation of SU(3). The triplet C3 carries the three kernels (K1, K2, K3) as the three "colors".
\paragraph*{Proof. The automorphism group of O is the exceptional Lie group G2. Under}
the subgroup SU(3) $\subset$G2, the decomposition O = C $\oplus$C3 holds, where C3 is the fundamental representation of SU(3) (three complex dimensions, i.e. six real dimensions). The eight Gell-Mann generators $\lambda$1, . . . , $\lambda$8 of su(3) act on C3: $\lambda$1,2,3 within the K1-K2 subspace (SU(2) subalgebra), $\lambda$4,5 coupling K1 to K3, $\lambda$6,7 coupling K2 to K3, $\lambda$8 as hypercharge. The identification (K1, K2, K3) = three colors of C3 is direct.
\paragraph*{Step 3: The Standard Model gauge group}
\paragraph*{Theorem 7.4 (Standard Model from the Sphaera biquaternion). The biquaternion ground state qn $\in$H $\otimes$C, extended to the octonionic structure}
C $\otimes$H $\otimes$O, generates the Standard Model gauge group U(1) $\times$ SU(2) $\times$ SU(3) with generators arising from: Source Group Physical force Complex scalar it U(1) Electromagnetism Unit quaternions H SU(2) Weak force Octonion triplet C3 $\subset$O SU(3) Strong force
\paragraph*{Proof. Propositions 7.1--7.2 and Theorem 7.3 establish each factor. The three}
factors are independent (direct product structure) because U(1) commutes with SU(2) (Proposition 7.1) and both commute with the SU(3) action on C3 (the SU(3) generators $\lambda$k act only on the color indices, not on the quaternion structure).
\paragraph*{Remark (The role of K3). The third kernel K3 = e1e2 plays three roles simultaneously, in three different frames:}
- In H $\otimes$C: the photon sector (projection of the Heisenberg compensator,
orthogonal to K1, K2);
- In Cl(3): a bivector (e12, generator of su(2));
- In O: the third color of the SU(3) triplet.
By Theorem 2.1, all three interpretations are related by frame changes that leave $\tau$n and C invariant. K3 is the same object seen from three different frames. This is the quantum-field-theoretic realization of the frameindependence observation: it is invariant under the change of frame.

\section{Bell Inequalities and \texorpdfstring{$\tau_n$}{tau_n}}
\paragraph*{Theorem 8.1 (Bell consistency).}
The proper time $\tau_n$ is not a local hidden variable in Bell's sense. The framework is consistent with quantum-mechanical violation of Bell inequalities.

\paragraph*{Proof.}
Bell's theorem assumes a hidden variable $\lambda$ assigned independently to each particle, with measurement outcomes $A(\lambda_A,a)$ and $B(\lambda_B,b)$ and statistically independent variables $\lambda_A,\lambda_B$. In the present framework, however, $\tau_n$ belongs to the pair $(f,Jf)$ rather than to each element separately; one has
\[
\lambda_A = \lambda_B = \tau_n.
\]
Thus $\tau_n$ is a contextual variable, and Bell's theorem does not exclude violation by such theories. The correlation function $E(a,b)$ follows from the structure of $H_+^{\,+}$: since $C(f) \in H_+^{\,+}$ carries an irreducible representation of the Weyl algebra $W(A_1)$ (Stone--von Neumann, [1]), the standard quantum correlation
\[
E(a,b)=\cos \theta_{ab}
\]
is reproduced, yielding the CHSH value
\[
S = 2\sqrt{2} > 2.
\]
Hence $\tau_n$ explains why the correlation exists (shared proper time fixed at the near-collision), while Stone--von Neumann determines the strength of the violation.

\paragraph*{Remark (What $\tau_n$ does and does not do).}
The quantity $\tau_n$ encodes the magnitude $|z_-|$ of the displacement from the photon sector at the moment of the near-collision. It does not determine the measurement direction. When $f$ is measured, $z_-$ is projected onto a value; the paired state $Jf$ is projected onto $-z_-$, the $J$-image. The anticorrelation is built in at the collision event and is not transmitted at measurement. No superluminal signal is required.

\section{String Theory and Loop Quantum Gravity as HC Sectors}
The decomposition induced by the Heisenberg compensator leads naturally to symmetric and antisymmetric sectors.

\paragraph*{Theorem 9.1 (String--LQG duality via the Heisenberg compensator).}
The Hardy space $H_+$ decomposes under the Heisenberg compensator into two sectors,
\[
H_+ = H_+^{\,+} \oplus H_+^{\,-},
\]
with
\[
C(f)=\tfrac12(f+Jf) \in H_+^{\,+}, \qquad A(f)=\tfrac12(f-Jf) \in H_+^{\,-}.
\]
These sectors are interpreted as follows:
\begin{center}
\begin{tabular}{lll}
\toprule
Sector & Physical theory & Key property \\
\midrule
$H_+^{\,+}$ & String theory & bosonic, $J$-symmetric, continuous \\
$H_+^{\,-}$ & Loop quantum gravity & fermionic, $J$-antisymmetric, discrete \\
\bottomrule
\end{tabular}
\end{center}
Both sectors are unitarily equivalent by Stone--von Neumann~[1]; they are two representations of the same Weyl algebra $W(A_1)$ seen from different frames.

\paragraph*{Identification.}
In the string sector $H_+^{\,+}$, the state $C(f)$ satisfies $J(C(f))=C(f)$ and is therefore $J$-invariant. This mirrors the invariance of the worldsheet under orientation reversal $\sigma \mapsto -\sigma$. In the loop-quantum-gravity sector $H_+^{\,-}$, the state $A(f)$ satisfies $J(A(f))=-A(f)$ and changes sign under $J$, matching the spinorial sign change of oriented holonomies. The discrete LQG area spectrum
\[
A_j = 8\pi \gamma_{\mathrm{BI}} \, \ell_P^2 \sqrt{j(j+1)}, \qquad j \in \tfrac12\mathbb{N},
\]
is consistent with the antisymmetric, Pauli-constrained structure established earlier.

\paragraph*{Proposition 9.2 (Shared vacuum and Planck floor).}
Both sectors share the same vacuum and the same minimal scale:
\begin{itemize}
  \item Vacuum: $z_-=0$ (the photon/Sphaera) is the ground state of both $H_+^{\,+}$ and $H_+^{\,-}$.
  \item Planck floor: $\ell_P = |z_-|_{\min}$ is the string length in $H_+^{\,+}$ and the fundamental area scale in $H_+^{\,-}$.
\end{itemize}
String theory and loop quantum gravity are therefore not competing theories of quantum gravity, but bosonic and fermionic projections of the same object under the Heisenberg compensator.

\paragraph*{Theorem 9.3 (Triolic structure of $\gamma_{\mathrm{BI}}$).}
The Barbero--Immirzi parameter $\gamma_{\mathrm{BI}}$ and the string-sector candidate
\[
\gamma_{\mathrm{string}} = \frac{\ln 2}{\pi\sqrt{3}}
\]
are interpreted as two projections of the same triolic invariant:
\[
\gamma_{\mathrm{string}} + \gamma_{\mathrm{LQG}} + \gamma_3 = 0,
\qquad
\gamma_3 = -\bigl(\gamma_{\mathrm{string}} + \gamma_{\mathrm{LQG}}\bigr) \approx -0.3649.
\]
The frame-invariant quantity is the triolic norm
\[
\tau_\gamma := \sqrt{\frac{\gamma_{\mathrm{string}}^2 + \gamma_{\mathrm{LQG}}^2 + \gamma_3^2}{3}} \approx 0.2619.
\]
A direct computation gives
\[
\gamma_{\mathrm{string}} \approx 0.1274, \qquad \gamma_{\mathrm{LQG}} \approx 0.2375, \qquad \gamma_3 \approx -0.3649,
\]
with vanishing sum.

\paragraph*{Remark (Physical interpretation of the three sectors).}
The three values of $\gamma$ correspond to the three sectors of the biquaternion ground state:
\begin{itemize}
  \item $\gamma_{\mathrm{string}} > 0$: bosonic, the symmetric $C$-sector;
  \item $\gamma_{\mathrm{LQG}} > 0$: fermionic, the antisymmetric $A$-sector;
  \item $\gamma_3 < 0$: the third (tachyonic) sector, projected away by the cascade operator $\hat K$.
\end{itemize}
The negative sign of $\gamma_3$ is not pathological; it is the triolic complement required for the vanishing sum.

\paragraph*{Theorem 9.4 (Wick rotation as splitting mechanism).}
Under the imaginary proper-time transformation $t \mapsto i\tau_E$, the biquaternion time unit transforms as
\[
it \mapsto i(i\tau_E) = -\tau_E \in \mathbb{R}_{<0}.
\]
This identifies the imaginary-time component with the unstable tachyonic sector $\gamma_3<0$. Consequently:
\begin{enumerate}
  \item the balance condition $\gamma_{\mathrm{string}}+\gamma_{\mathrm{LQG}}+\gamma_3=0$ is the stability condition under imaginary proper time;
  \item the positive sectors $\gamma_{\mathrm{string}}, \gamma_{\mathrm{LQG}}>0$ are stable under Wick rotation, whereas the negative sector is projected away by $\hat K$;
  \item the Minkowski norm $|q_n|^2=\tau_n^2-|z|^2$ becomes Euclidean,
  \[
  |q_E|^2 = \tau_E^2 + |z|^2 \ge 0,
  \]
  and the system is unconditionally stable in Euclidean signature.
\end{enumerate}

\paragraph*{Theorem 9.5 (Observer frame offset).}
Every matter observer carries a nonzero contravariant component $z_- \neq 0$. All measured physical constants---including $\hbar$, $G$, $c$, and $\gamma_{\mathrm{BI}}$---are projections of frame-invariant values onto the observer's sector. If $\gamma_0$ denotes the frame-invariant value measured in the photon frame $z_-=0$, and if $\beta=2z_-$ is the observer's boost parameter, then
\[
\gamma_{\mathrm{observed}} = \gamma_0 \cdot \gamma_{\mathrm{Lorentz}}(\beta).
\]
In particular,
\[
\gamma_{\mathrm{BI}} \approx 0.2375
\]
is interpreted as the Lorentz-transformed value of $\gamma_{\mathrm{string}}$, seen from a matter frame with $\beta \approx 0.844$, equivalently $z_- \approx 0.422$.

\paragraph*{Remark (Planck length).}
The Planck length is likewise frame-dependent:
\[
\ell_P^{\mathrm{matter}} = \ell_P^{\mathrm{photon}} \cdot \gamma_{\mathrm{Lorentz}}(\beta) \approx 1.864\, \ell_P^{\mathrm{photon}}.
\]
Thus measured constants are sector projections rather than absolute invariants.

\paragraph*{Theorem 9.6 (Symmetric matter--tachyon frame).}
Matter and tachyon are positioned symmetrically around the photon frame $z_-=0$:
\[
\beta_{\mathrm{matter}} = +\beta_{\mathrm{sym}}, \qquad \beta_{\mathrm{tachyon}} = -\beta_{\mathrm{sym}},
\]
where
\[
\beta_{\mathrm{sym}} = \sqrt{\frac{\gamma_{\mathrm{BI}}-\gamma_{\mathrm{string}}}{\gamma_{\mathrm{BI}}+\gamma_{\mathrm{string}}}} \approx 0.5493.
\]
The Lorentz factor between the matter and tachyon frames is then
\[
\gamma_{\mathrm{matter}\leftrightarrow\mathrm{tachyon}} = \frac{1+\beta_{\mathrm{sym}}^2}{1-\beta_{\mathrm{sym}}^2} = \frac{\gamma_{\mathrm{BI}}}{\gamma_{\mathrm{string}}} \approx 1.864.
\]
Thus the sign-structure interpretation and the frame-dependent interpretation of $\gamma_{\mathrm{BI}}$ are equivalent descriptions of the same frame change.


\section{Black Holes and Hawking Radiation}
The horizon as the boundary of the Sphaera
\paragraph*{Definition}
10.1
(Schwarzschild horizon in Sphaera coordinates). The Schwarzschild horizon rS = 2GM/c2 corresponds to $\beta$ $\to$1 in the Sphaera metric gtt = 1 -$\beta$2. In sector coordinates: the horizon is where z-$\to$$\infty$.
\paragraph*{Proposition 10.2 (Horizon as cascade singularity). The cascade operator $\hat K$}
is undefined at z-= $\infty$: the Sphaera cannot project beyond its own horizon. The horizon is the boundary of the Sphaera's jurisdiction.
\paragraph*{Proof. $\hat K$ = C$\mu$$\nu$ $\circ$$\Pi$J requires $\|$z-$\|$< $\infty$(finite sector displacement).}
At z-$\to$$\infty$: $\beta$ $\to$1, $\gamma$Lorentz $\to$$\infty$, the Lorentz transformation is singular, and the projection $\Pi$J ceases to be bounded. Hawking radiation as compensator output
\paragraph*{Theorem 10.3 (Hawking radiation from the triolic structure). At the horizon, a virtual matter--tachyon pair from the triolic structure (Theorem 3.2) is}
promoted to a real pair:
- Tachyon ($\beta$ = -$\beta$sym): falls into the black hole.
- Matter ($\beta$ = +$\beta$sym): escapes to infinity.
- Their compensator output C(f) =
2(f + Jf): a photon with $\beta$ = 0, z-= 0 --- the Hawking radiation. The temperature of the Hawking photon in the photon frame (z-= 0) is: TSphaera = $\gamma$string $\cdot$ m2 Pc2 8$\pi$MkB = $\gamma$string $\cdot$ THawking.
\paragraph*{Proof. The Hawking temperature (standard derivation) is}
TH = $\hbar$c3 8$\pi$GMkB = m2 Pc2 8$\pi$MkB . The Sphaera produces the Hawking photon in the photon frame where $\gamma$ is measured as $\gamma$string (not $\gamma$BI, which is the matter-frame value). Hence TSphaera = $\gamma$string $\cdot$ TH. The ratio TSphaera/THawking = $\gamma$string = ln 2/($\pi$ $\sqrt{}$ 3) $\approx$0.1274 is constant across all black hole masses (verified numerically for M = mP and M = M$\odot$).
\paragraph*{Remark (Interpretation). THawking is the temperature measured by a matter}
observer far from the black hole. TSphaera is the temperature in the photon frame (z-= 0). The ratio $\gamma$string is the frame transformation factor between them --- exactly as $\gamma$BI is the factor between the matter and photon frames for other observables (Theorems 9.5 and 9.6). The Hawking radiation is the Sphaera projecting through the horizon: $\hat K$ fails at z-= $\infty$, but the compensator C still produces a well-defined output at z-= 0 from the pair (fmatter, Jftachyon).
\paragraph*{Remark (Natural observer distance from the horizon). Theorem 9.6 places}
the matter observer at $\beta$sym $\approx$0.5493. In Schwarzschild coordinates, this corresponds to r/rS = 1/(1 -$\beta$2 sym) $\approx$1.43: our natural distance from a black hole is 1.43 rS, just outside the horizon. The origin of the tachyon: causal direction
\paragraph*{Theorem 10.4 (Hawking radiation as the voice of the core). The Hawking}
tachyon is not created at the horizon. It is the black hole core --- which has z-$\to$$\infty$and cannot be projected by $\hat K$ --- attempting to reach z-= 0 through the only available channel: tunnelling through the horizon. The two descriptions are CPT-dual and physically identical: Outside view (t forward) Inside view (t reversed) t1: vacuum t3: core at z-= $\infty$ t2: virtual pair at horizon t2: tachyon at horizon t3: tachyon falls in, matter escapes t1: Hawking photon emitted The cascade $\hat K$ has no preferred time direction: it is the equilibrium between z-= 0 (outside) and z-= $\infty$(core). The Hawking pair is the discharge of this tension.
\paragraph*{Structural argument. Outside: z-grows from 0 (far away) to $\infty$(horizon). $\hat K$}
attempts to project z-$\to$0 but fails at the horizon. Tension forces a real pair. Inside: z-= $\infty$(core, singular compression). $\hat K$ attempts z-$\to$0 but the horizon blocks outward projection. The only escape: a core fraction tunnels as a tachyon ($\beta$ = -$\beta$sym) through the horizon. Under J-reflection, this tachyon appears outside as matter ($\beta$ = +$\beta$sym). The compensator C(f) = 1 2(f + Jf) produces the Hawking photon at $\beta$ = 0. CPT: C exchanges matter and tachyon; P exchanges $\beta$ $\leftrightarrow$-$\beta$ (sides of the horizon); T reverses creation and annihilation. Both descriptions are related by CPT.
\paragraph*{Remark (Black hole as inverted Sphaera). The normal Sphaera has z-decreasing inward: $\hat K$ projects from large z-toward z-= 0 (the photon ground}
state). A black hole inverts this: z-increases inward, from z-= 0 (far away) to z-= $\infty$(horizon and core). The cascade $\hat K$ cannot function inside the horizon; the interior is a region where the Sphaera does not exist. The horizon is where the Sphaera is maximally stressed. Hawking radiation is the Sphaera discharging at its boundary. The black hole evaporates because the core is the Sphaera --- and the Sphaera always tends toward z-= 0. The tachyon is not a particle falling in. It is the core, speaking through the only open channel. Bekenstein--Hawking entropy
\paragraph*{Remark (The entropy factor). The Bekenstein--Hawking entropy SBH}
= A/(4$\ell$2 P) and the LQG entropy SLQG = A ln 2/(8$\pi$$\gamma$BI$\ell$2 P) agree when $\gamma$BI = ln 2/(2$\pi$) $\approx$0.110. The measured value $\gamma$BI $\approx$0.2375 differs by a factor $\approx$2.15. Within the present framework: the LQG entropy formula is derived in the matter frame ($\gamma$BI), while the Bekenstein--Hawking formula is a photon-frame statement. The discrepancy factor 2.15 should be related to the frame transformation, but its precise form --- unlike the analogous Hawking temperature factor $\gamma$string --- is not yet derived. This is an open quantitative problem.
\paragraph*{Theorem 10.5 (Dual origin of $\gamma$BI). The splitting factor $\gamma$L = $\gamma$BI/$\gamma$string $\approx$}
1.864 has a dual origin [\emph{note: value corrected from 1.197 in v2.29.1}]: it is approximately $1/\bsym$ from the Hawking temperature argument:
$\gamma$L = |{z} triolic sector counting $\cdot$ $\beta$sym |{z} observer geometry $\approx$0.667 $\cdot$ 1.820 = 1.214, within 1.4\% of the measured value $\gamma$L = 1.197. Factor A (2/3, structural): In the triolic structure, three sectors exist (K1, K2, K3). Under Wick rotation, only two are stable ($\gamma$ > 0); the tachyon sector ($\gamma$3 < 0) is projected away by $\hat K$. The fraction of stable sectors is 2/3. $\gamma$string counts all three; $\gamma$BI counts only the two stable ones. Factor B (1/$\beta$sym, geometric): A matter observer at r/rS = 1/(1-$\beta$2 sym) $\approx$
1.43 measures an effective Hawking temperature T(r) = TH/$\beta$sym. Via S =
E/T, this shifts the effective entropy count by $\beta$sym, giving $\gamma$BI = $\gamma$string/$\beta$sym $\approx$
0.1274/0.549 $\approx$0.232 (within 2.4\% of 0.2375).
Both factors are active simultaneously. They are not two competing explanations: they are two aspects of the same collapse from z-= $\infty$to z-= 0, seen from the structural side (A) and the geometric side (B).
\paragraph*{Proof. Factor A. The triolic balance $\gamma$1 + $\gamma$2 + $\gamma$3 = 0 with $\gamma$3 < 0 (Theorem 9.3) means that $\gamma$string = $\gamma$1 is defined over all three sectors. The LQG}
computation counts only spin-network microstates, which correspond to the two stable sectors (bosonic C-sector and fermionic A-sector). The weight 2/3 is the fraction of stable sectors. Factor B. The matter observer at $\beta$sym has r/rS = 1/(1 -$\beta$2 sym). The local Hawking temperature is T(r) = TH(1 -rS/r)-1/2 = TH/$\beta$sym. Since $\gamma$BI is fixed by requiring S = A/(4$\ell$2 P) for the observed temperature, the effective parameter shifts: $\gamma$BI = $\gamma$string $\cdot$ (TH/T(r))-1 = $\gamma$string/$\beta$sym. Combined. $\gamma$L = (2/3) $\cdot$ (1/$\beta$sym) $\approx$1.214 vs measured 1.197: within 1.4\%. The residual 1.4\% is the remaining quantitative open problem.
\paragraph*{Remark (What this resolves). The question "are natural constants phaseshifted?" has a precise answer:}
- $\hbar$, G, c, kB are Lorentz-invariant --- they do not shift.
- $\gamma$BI is a dimensionless ratio, not a fundamental constant. It shifts due to
(A) triolic sector counting and (B) observer geometry.
- The shift is not a frame transformation of the constants; it is a consequence of which states the matter observer has access to, and where the
observer sits relative to the horizon. Nature does not choose between (A) and (B). They are the same statement: a matter observer at $\beta$sym naturally has access to 2 of 3 stable triolic sectors, at a distance that shifts the effective temperature by 1/$\beta$sym.
\paragraph*{Theorem 10.6 (ART--QFT consistency of the Sphaera). The Sphaera biquaternion encodes both General Relativity and Quantum Field Theory, connected by Wick rotation:}
it $\cdot$ 1 + ziei | {z } Minkowski (ART) t7$\to$i$\tau$E ----$\to$$\tau$E $\cdot$ 1 + ziei | {z } Euclidean (QFT) . At the natural observer position r0/rS = 1/(1-$\beta$2 sym), the Schwarzschild metric gives gtt(r0) = -$\beta$2 sym and $\sqrt{}$-g
r0 = r2 0 sin $\theta$ (invariant): no additional geometric factor is hidden in the ART side. The QFT side gives local redshifted quantities: Elocal = E$\infty$/$\beta$sym, Tlocal = TH/$\beta$sym. These are exactly Factors A and B of Theorem 10.5. The ART--QFT conversion is internally consistent.
\paragraph*{Remark (The 1.4\% residual and discreteness). The ART--QFT computation}
confirms the structure of Theorem 10.5 but does not close the 1.4\% gap between (2/3)/$\beta$sym $\approx$1.214 and the measured $\gamma$L $\approx$1.197. The gap is not a one-loop
quantum gravity correction (suppressed by ($\ell$P/r0)2 $\to$0) and not a hidden metric factor ($\sqrt{}$-g is invariant). The most likely source: the transition from the discrete prime lattice $\Gamma$P (Stone--von Neumann, QFT) to the continuous spacetime manifold (ART). This is the continuum limit of lattice quantum gravity --- a known open problem. The 1.4\% is the quantitative signature of this discreteness correction, and the framework cannot close it without a more detailed treatment of the lattice-to-continuum transition.

\section{The Sphaera as a Hopf Fibration}
\paragraph*{Theorem 11.1 (S-chain as nested Hopf fibrations). The stratum chain S0 $\subset$}
S1 $\subset$S2 of the Sphaera is identified with the base spaces of the three classical Hopf fibrations: S0 $\sim$= S2 = CP 1 (Weyl spinor, massless photon) S1 $\sim$= S4 = HP 1 (Dirac spinor, massive matter) S2 $\sim$= S8 = OP 1 (octonionic spinor, 3 generations) with the natural inclusion S2 ,$\to$S4 ,$\to$S8. The cascade operator $\hat K$ is the Hopf projection chain: $\hat K$ : S15 -$\to$S8 -$\to$S4 -$\to$S2, where the total spaces S3, S7, S15 are the spinor spaces carrying the gauge structure. The three gauge groups arise as Hopf fibres: S1 |{z} $\sim$ = U(1) $\to$S3 $\to$S2, S3 |{z} $\sim$ = SU(2) $\to$S7 $\to$S4, S7 |{z} $\supset$G2$\supset$SU(3) $\to$S15 $\to$S8. This is not ad hoc: it follows from the Hurwitz theorem that only R, C, H, O are normed division algebras, and only S1, S3, S7 are parallelisable. The gauge group U(1) $\times$ SU(2) $\times$ SU(3) is the topological consequence of working over C, H, O.
\paragraph*{Structural argument. The three Hopf fibrations are: S1 ,$\to$S3 $\to$S2 (over C),}
S3 ,$\to$S7 $\to$S4 (over H), S7 ,$\to$S15 $\to$S8 (over O). The base spaces are CP 1, HP 1, OP 1 --- the projective lines over each division algebra. The Sphaera axiom identifies S0 with the photon ground state z-= 0, which is the fixed point of J in CP 1. Matter at z-= $\beta$sym is a point in HP 1 \ CP 1, and the cascade $\hat K$ is the canonical projection between the projective lines.
\paragraph*{Theorem 11.2 ($\beta$sym from the Hopf angle). The matter observer's frame offset}
$\beta$sym is the sine of half the Hopf angle $\theta$H of the matter spinor on S3 relative to the photon pole: $\beta$sym = sin ($\theta$H ) , $\theta$H $\approx$66.6. This gives a geometric derivation of $\beta$sym from the Hopf fibration: it is the latitude of the matter sector on the spinor sphere S3. The natural distinguished latitudes on S3 $\sim$= SU(2) from the Hopf structure are:
- $\theta$H = 0: north pole = photon ($\beta$ = 0, z-= 0)
- $\theta$H = $\pi$/2 ($45^\circ$): equator of S3 ($\beta$ = sin $\pi$/4 = 1/
$\sqrt{}$ 2)
- $\theta$H = 2$\pi$/3 ($120^\circ$): triolic angle ($\beta$ = sin $\pi$/3 =
$\sqrt{}$ 3/2)
- $\theta$H = $\pi$ ($180^\circ$): south pole = tachyon ($\beta$ = 1, z-= $\infty$)
The measured value $\theta$H $\approx$66.6 lies between the equator and the triolic angle. The precise derivation of $\theta$H from the Hopf geometry --- which fixed point of the Z/3Z action on S3 corresponds to the matter sector --- is the remaining open problem.
\paragraph*{Remark (What the Hopf structure resolves). The identification}
$\hat K$ = Hopf projection gives:
(i) Three generations: three sections of S15 $\to$S8 over the three Odirections (e4, e5, e6) (Theorem 11.3).\\
(ii) Gauge groups: as Hopf fibres, not as an input.\\
(iii) Spinors: C2 (Weyl), H2 (Dirac), O2 (octonionic, 3 generations) are the\\
natural spinor spaces over each base.
(iv) $\beta$sym pathway: the Hopf angle $\theta$H is a geometric quantity, potentially\\
derivable from the Z/3Z fixed-point structure of the triolic cascade.
\paragraph*{Theorem 11.3 (Three fermion generations from O). In the Dixon algebra}
D = C $\otimes$H $\otimes$O, the three fermion generations are identified as: Geni $\sim$= (C $\otimes$H) $\otimes$ei+3, i = 1, 2, 3, where e4, e5, e6 $\in$Im(O) form the C3 triplet under SU(3) $\subset$G2 = Aut(O). The particle content of each generation is:
Generation O-unit Quarks Leptons Neutrino Gen 1 e4 u, d e $\nu$e Gen 2 e5 c, s $\mu$ $\nu$$\mu$ Gen 3 e6 t, b $\tau$ $\nu$$\tau$ Singlet e7 ----(right-handed $\nu$) Generation-changing transitions correspond to G2-rotations in Im(O): e4 $\cdot$ e5 = +e7, e5 $\cdot$ e6 = +e1, e6 $\cdot$ e4 = -e3. The cyclic structure Gen1 $\to$Gen2 $\to$ Gen3 $\to$Gen1 corresponds to Z/3Z (the same triolic symmetry as the Sphaera strata S2 $\supset$S1 $\supset$S0).
\paragraph*{Proof. Under SU(3) $\subset$G2, the imaginary octonions decompose as Im(O) =}
C $\oplus$C3: singlet e7 and triplet (e4, e5, e6). The anti-triplet (e1, e2, e3) carries the conjugate color representation. Each generation (C$\otimes$H)$\otimes$ei+3 carries one copy of the spinor representation of the Lorentz group (from C$\otimes$H = M(2, C)) tensored with one color index. The Fano plane structure of O gives the cyclic e4 $\to$e5 $\to$e6 symmetry.
\paragraph*{Remark (CKM mixing angles). The Cabibbo mixing angle $\theta$C $\approx$13 arises from}
G2-rotations between the generation planes. A natural candidate is $\theta$12 $\approx$ arcsin($\gamma$string) $\approx$7.3, within a factor of $\approx$2 of the observed value. The precise derivation of the CKM matrix from G2 Euler angles is open.
\paragraph*{Theorem 11.4 (Koide formula from the triolic structure). The lepton masses}
satisfy the Koide formula: K = me + m$\mu$ + m$\tau$ ($\sqrt{}$me + $\sqrt{}$m$\mu$ + $\sqrt{}$m$\tau$)2 = 2 to within 0.005\% experimentally. In the Sphaera framework, this is not a coincidence: the three leptons sit at the three triolic points on S3 fixed by the Z/3Z action. If lepton masses are Hopf-parameterised as mi = sin2($\theta$i/2) $\cdot$ mP, and $\theta$1, $\theta$2, $\theta$3 are the three triolic angles on S3 satisfying $\theta$1 + $\theta$2 + $\theta$3 = 2$\pi$ (the Hopf analogue of z1 + z2 + z3 = 0), then K = 2/3 follows algebraically from the constraint.
\paragraph*{Proof. The triolic constraint P}
i $\sqrt{}$mi = const combined with P i mi = const (both fixed by the Z/3Z symmetry on S3) gives K = 2/3 by the Cauchy-Schwarz identity applied to (1, 1, 1) and ($\sqrt{}$m1, $\sqrt{}$m2, $\sqrt{}$m3). The factor 2/3 is the triolic sector count (Theorem I.9.3): 2 stable sectors out of 3. The lepton mass formula realises the same 2/3 geometrically.
\paragraph*{Remark (The 2/3 factor is everywhere). The factor 2/3 appears as: the stable}
sector fraction (Thm. I.9.3), the spring oscillator coefficient $\omega$ = $\gamma$s p 2/3, the cosmological dark energy $\Omega$$\Lambda$ $\approx$2/3 (Theorem 11.9), and now the Koide lepton mass ratio. All are the same statement: 2 of the 3 triolic sectors collapse to the photon ground state.
\paragraph*{Remark (Mass spectrum beyond Koide). The seven imaginary octonion units}
e1, . . . , e7 provide seven natural slots for fermion masses. A direct identification mp $\sim$p $\cdot$ mP (Planck units) does not match the observed quark mass ratios: consecutive prime ratios (3/2, 5/3, 7/5, . . .) differ from the observed mass hierarchies by factors of 10--100. The mass spectrum requires a Yukawa coupling mechanism (Higgs vacuum expectation value) that is not yet derived from the Sphaera structure.
\paragraph*{Theorem 11.5 (Cosmic expansion as growth of the active zero basis). In the}
Sphaera framework, the scale factor a(t) of the universe is identified with the proper time: a(t) $\sim$$\tau$n(t) = q tn(t)2 -$\beta$2 sym, where tn(t) is the imaginary part of the n-th active Riemann zero at cosmic time t, and $\beta$sym is the matter observer's frame offset (Theorem 9.6). The Hubble parameter is the activation rate of zeros: H = dot a a = dot tn tn = dot n n (since tn $\approx$2$\pi$n/ log n by Weyl asymptotics, and for large n: dot tn/tn $\approx$dot n/n). The universe expands because more Riemann zeros become "active" as cosmic time increases. The active zero count at the Hubble time is nH $\sim$1062.
\paragraph*{Theorem 11.6 (Dark energy as permanent frame offset). The matter observer's permanent frame offset $\beta$sym enters the proper time as a constant term:}
$\tau$ 2 n = t2 n -$\beta$2 sym = t2 n |{z} grows -$\beta$2 sym |{z} constant . The constant term $\beta$2 sym plays the role of the effective cosmological constant: $\Lambda$eff $\sim$$\beta$2 sym $\cdot$ $\rho$P (where $\rho$P is the Planck density). The ratio of dark energy to matter density today is predicted as: $\Omega$$\Lambda$ $\Omega$M $\approx$ $\beta$2 sym $\gamma$string = 0.3018
0.1274 $\approx$2.37.
The observed value is $\Omega$$\Lambda$/$\Omega$M $\approx$2.18 (9\% discrepancy, consistent with the residual 1.4\% and redshift corrections not yet included).
Dark energy is not a new substance. It is the permanent phase offset of our matter frame from the photon ground state z-= 0: we can never fully reach z-= 0, so $\beta$2 sym remains as a residual vacuum energy.
\paragraph*{Theorem 11.7 (Spring-pull mechanism of the triolic system). The matterphoton-tachyon triple forms a mechanical system with the following spring}
forces in the z-coordinate: Ftachyon$\to$$\gamma$ = k(zT -z$\gamma$ ) = -k$\beta$sym, Fmatter$\to$$\gamma$ = k(zM -z$\gamma$ ) = +k$\beta$sym. The net force on the photon vanishes: Fnet = 0. The photon at z-= 0 is the mechanical equilibrium point: the tachyon pulls it forward, the inertial mass of matter brakes it back. Under small displacement from equilibrium:
- xT = -x$\gamma$ (tachyon is J-image of photon, always opposite);
- x$\gamma$ = xM/3 (photon follows matter with amplitude 1/3);
- m"xM = -k $\cdot$ 2
3 $\cdot$ xM (effective spring constant reduced by factor 2/3). The system is a harmonic oscillator with eigenfrequency: $\omega$ = r 2k 3m = $\gamma$string r 3, where k $\sim$$\gamma$string (coupling strength from the HC trace) and m $\sim$1/$\gamma$string (inertial mass from the Planck floor). The amplitude ratios are |xT| : |x$\gamma$| : |xM| = 1 : 1 : 3.
\paragraph*{Proof. Equilibrium. At z$\gamma$ = 0, zT = -$\beta$sym, zM = +$\beta$sym: Fnet = -k$\beta$sym +}
k$\beta$sym = 0. Linearized dynamics. The tachyon satisfies xT = J(x$\gamma$) = -x$\gamma$ (it is the Jimage, Theorem 9.6). The photon is masseless: 0 = k(xT -x$\gamma$)+k(xM -x$\gamma$) = k(-x$\gamma$ -x$\gamma$)+k(xM -x$\gamma$) = k(xM -3x$\gamma$), giving x$\gamma$ = xM/3. Substituting into the matter equation: m"xM = -k(xM-xM/3) = -2k 3 xM. Hence $\omega$2 = 2k/(3m). Eigenfrequency. With k = $\gamma$string (HC trace formula) and m = 1/$\gamma$string (Planck inertia): $\omega$ = $\gamma$string p 2/3.
\paragraph*{Remark (The factor 2/3 reappears). The effective spring constant 2k}
3 contains the factor 2/3 from Theorem 10.5 (two stable triolic sectors out of three). This is not a coincidence: the 2/3 sector counting is the mechanical statement that the tachyon and the matter-sector spring have equal strength, while the photon is the masseless node that mediates between them. The tachyon sector (counted negatively in the entropy, $\gamma$3 < 0) is the one that "pulls forward"; the two stable sectors (matter and photon) are the restoring force.
\paragraph*{Remark (Why c is constant: a mechanical explanation). The photon is the}
massless equilibrium node of the spring system. It has no inertia and no preferred frame. When matter oscillates, the photon follows with amplitude 1/3, and this following motion defines the coordinate system (the space in which everything happens). Since the photon has no mass and no net force, it cannot accelerate or decelerate: its speed is always c by construction. The constancy of c is the statement that the equilibrium node of a symmetric spring system has no dynamics of its own.
\paragraph*{Theorem 11.8 (Light as gravitational equilibrium). In the Sphaera framework, the proper time decomposes as:}
$\tau$ 2 n = t2 n |{z} geometry (pulls down) $\beta$2 sym |{z} dark energy (pushes up) . These two terms are gravitationally coupled through the metric: gtt(r0) = -$\beta$2 sym (Theorem 10.6), the same $\beta$2 sym that appears as the dark energy term. Gravity pulls gtt $\to$0 (toward the horizon); dark energy stabilizes gtt = -$\beta$2 sym (preventing total collapse). Light at z-= 0 is the unique fixed point of this coupling:
- No dark energy pressure: $\beta$sym = 0 for light, so the upward push vanishes.
- No gravitational trapping: tn /= 0, so the downward pull cannot halt the
light.
- Light is orthogonal to both pressures.
Therefore ds2 = 0 for light in any frame, and v = c always.
\paragraph*{Proof. For matter at z-= $\beta$sym: v/c = $\beta$sym/tn < 1 (sub-luminal). $\checkmark$For}
tachyons at z-= -$\beta$sym: v/c = pt2 n + $\beta$2 sym/tn > 1 (super-luminal). $\checkmark$For light at z-= 0: $\tau$ 2 n = t2 n -0 = t2 n, so $\tau$n = tn and v/c = tn/tn = 1. $\checkmark$ The coupling: gtt = -$\beta$2 sym is the metric value at the natural observer position (Theorem 10.6). It equals -(1 -rS/r0), so $\beta$2 sym = rS/r0. Gravity pulls rS/r $\to$1 (horizon); dark energy ($\Lambda$-term) counteracts this to maintain rS/r0 = $\beta$2 sym < 1. The balance is stable precisely at the observer's natural position.
\paragraph*{Remark (Light as the Heisenberg compensator of the universe). The Heisenberg compensator applied to the matter-tachyon pair gives:}
C($\beta$sym) = 1 2($\beta$sym + J $\cdot$ $\beta$sym) = 1 2($\beta$sym -$\beta$sym) = 0.
Light (z-= 0) is the compensator output of the entire universe: C(universe) = 1 2(matter + tachyon) = 1 2($\beta$sym -$\beta$sym) = 0 = light. Gravity and dark energy are not two separate substances that happen to balance. They are the matter sector ($\beta$sym) and its J-image (-$\beta$sym), and light is their compensator --- the only object that sees neither pressure. This is why c is constant in all frames: light is the frame-invariant object z-= 0, and frame changes are J-transformations that fix z-= 0.
\paragraph*{Theorem 11.9 (Dark energy as cascade probability; $\Lambda$ = 0). The cosmological}
parameters $\Omega$$\Lambda$ and $\Omega$M are the Markov transition probabilities of the Sphaera cascade: $\Omega$$\Lambda$ = P(S8 $\to$S2) = 1 -$\beta$2 sym $\approx$ 3 $\approx$0.698, $\Omega$M = P(S8 $\to$S4) = $\beta$2 sym $\approx$ 3 $\approx$0.302, with observed values $\Omega$$\Lambda$ = 0.685 (2\% agreement) and $\Omega$M = 0.315 (4\% agreement). With the Hopf value $\beta$sym = 1/ $\sqrt{}$ 3: $\Omega$$\Lambda$ = 2 3, $\Omega$M = 1
3.
Consequently, the fundamental cosmological constant vanishes: $\Lambda$fundamental =
0. Einstein's "greatest blunder" was correct.
What we observe as dark energy is not a new substance or field. It is the fraction of the cascade S8 $\to$S2 that collapses directly to the photon ground state z-= 0, bypassing the matter stratum S4.
\paragraph*{Proof. The cascade operator $\hat K$ is a Markov process on {S2, S4, S8} with transition matrix:}
P = (
1 -$\beta$2 sym $\beta$2 sym )
, (rows: from S2, S4, S8; columns: to S2, S4, S8). S2 is the absorbing state (photon ground state, z-= 0). A tachyon starting in S8 transitions with probability $\beta$2 sym to S4 (matter: massive Dirac spinor requiring a further step) and with probability 1 -$\beta$2 sym directly to S2 (massless: no further cascade needed). The matter fraction $\beta$2 sym is observed as $\Omega$M; the direct-collapse fraction 1 -$\beta$2 sym is observed as $\Omega$$\Lambda$. Since $\Omega$$\Lambda$ is a transition probability (not a vacuum energy density), $\Lambda$fundamental = 0. The "fine-tuning problem" ($\Lambda$observed $\ll$$\Lambda$Planck) dissolves: the
observed $\Omega$$\Lambda$ is dimensionless and of order unity; it never was a Planck-scale energy density.
\paragraph*{Remark (The factor 2/3 is $\Omega$$\Lambda$). Theorem 10.5 introduced the factor 2/3 as the}
fraction of stable triolic sectors. Theorem 11.9 identifies it as $\Omega$$\Lambda$. These are the same statement: 2 of the 3 sectors collapse directly to the photon ($\Omega$$\Lambda$ = 2/3); 1 sector passes through matter ($\Omega$M = 1/3). The 2/3 that appears in the spring frequency $\omega$ = $\gamma$s p 2/3, the Barbero--Immirzi factor, the GUE parameter, and the cosmological composition is the same number throughout --- the fraction of the cascade that reaches z-= 0 without resistance.
\paragraph*{Remark (Residual 3--5\% discrepancy). The Hopf prediction $\Omega$M = 1/3, $\Omega$$\Lambda$ =}
2/3 differs from observation by 3--5\%. This is the same continuum-limit correction that appears as 1/ $\sqrt{}$ 3 $\to$$\beta$sym (5\%) and as the 1.4\% gap in $\gamma$L. It is a single correction factor, not multiple independent errors.
\paragraph*{Remark (Matter density and primitive orbits). The matter energy density}
scales as $\rho$M $\sim$n/a3. In the De Sitter limit n $\sim$eH0t and a $\sim$eH0t, this gives $\rho$M $\sim$e-2H0t, equivalent to $\rho$M $\sim$a-3: the correct scaling for pressureless matter (w = 0). The primitive orbits P $\in$P of the arithmetic kernel (Lemma L6 of [1]) are identified with matter constituents: each prime p contributes one fundamental matter state, and the growing active count n(t) is the number of primes "reached" as the universe expands.
\paragraph*{Remark (The cosmological constant problem revisited). Two aspects of $\Lambda$ appear in the Sphaera:}
(i) Large $\Lambda$ from the field equation (Theorem 11.10): the constant term\\
1 in $\Sigma$ gives $\Lambda$ $\sim$t2 1 $\approx$200 m4 P, which is the standard fine-tuning problem.
(ii) Small $\Lambda$ from the frame offset (Theorem 11.6): $\beta$2\\
sym gives a small, frame-dependent contribution that matches observations at the 9\% level. The resolution of the large-$\Lambda$ problem requires showing that the $\Theta$--R cancellation in $\Sigma$ = 1 -$\Theta$ + R reduces the net cosmological constant by 10122. This is not yet established.
\paragraph*{Theorem 11.10 (Einstein equation from the trace formula). The variation}
of the trace formula action S := Z $\infty$ $\Sigma$($\tau$) d$\tau$ = Z $\infty$ 1 -$\Theta$($\tau$) + R($\tau$)
d$\tau$
with respect to the metric g$\mu$$\nu$ yields the Einstein equation: G$\mu$$\nu$ + $\Lambda$ g$\mu$$\nu$ = 8$\pi$G T$\mu$$\nu$. The three terms of $\Sigma$ contribute:
- $\delta$(1)/$\delta$g$\mu$$\nu$: cosmological constant $\Lambda$ (from the pole of $\zeta$ at s = 1);
- $\delta$$\Theta$/$\delta$g$\mu$$\nu$: stress-energy T$\mu$$\nu$ (primitive orbits of Karith as matter);
- $\delta$R/$\delta$g$\mu$$\nu$: Einstein tensor G$\mu$$\nu$ (trivial zeros encoding global geometry).
\paragraph*{Structural argument. $\Sigma$($\tau$) = Tr(e-i$\tau$L0) is the heat kernel of L0 on H+}
+ (Theorem T6 of [1]). The heat kernel variation formula gives $\delta$ Tr(e-$\tau$L0) = -$\tau$ Tr($\delta$L0 $\cdot$ e-$\tau$L0). Since L0 = -i(xd/dx + 1 2) encodes the prime lattice geometry in log-coordinates, $\delta$L0/$\delta$g$\mu$$\nu$ produces a curvature term. Via the Guinand--Weil explicit formula ($\Sigma$ = 1 -$\Theta$ + R), the three contributions are identified with $\Lambda$, T$\mu$$\nu$, and G$\mu$$\nu$ respectively.
\paragraph*{Remark (Cosmological constant). The constant term 1 in $\Sigma$ contributes $\Lambda$ $\sim$}
t2 1 $\approx$200 (Planck units), 10122 times the observed value. How $\Theta$ and R cancel to produce the small observed $\Lambda$ is the cosmological constant problem in this framework --- still open.
\paragraph*{Theorem 11.11 (GUE from the Sphaera). The eigenvalues {$\gamma$$\rho$} of L0 on H+}
+ --- which are the imaginary parts of the non-trivial Riemann zeros (Theorem T6 of [1]) --- follow the Gaussian Unitary Ensemble (GUE) statistics. Specifically, the normalized pair-correlation function is: R2(s) = 1 (sin $\pi$s $\pi$s )2 , and the nearest-neighbor spacing distribution follows the Wigner--Dyson law: p(s) = 32 $\pi$2 s2 e-4s2/$\pi$.
\paragraph*{Proof. We establish that L0 lies in the GUE universality class by verifying the}
three conditions:
(i) Self-adjointness.\\
L0 is essentially self-adjoint on H+ + by Theorem T4+T5 of [1]. This places it in the class of Hermitian operators, with real spectrum.
(ii) Irreducibility. H+\\
+ carries an irreducible representation of W(A1) by Stone--von Neumann (Theorem 3.2 of [1]). Irreducible self-adjoint operators in infinite-dimensional Hilbert spaces generically belong to the GUE universality class (Wigner--Dyson--Mehta universality).
(iii) Absence of time-reversal symmetry. The three symmetry classes are\\
distinguished by the presence of time reversal T:
- T present, T 2 = +1: GOE ($\beta$ = 1)
- T absent: GUE ($\beta$ = 2)
- T present, T 2 = -1: GSE ($\beta$ = 4)
The operator J : s 7$\to$1 -bar s satisfies JL0J-1 = L0 (Theorem T1+T3 of [1]), so J is a symmetry of L0. However, J is a space reflection ($\sigma$ 7$\to$1-$\sigma$, t 7$\to$t), not a time reversal. True time reversal T would send L0 7$\to$-L0 (since L0 is odd under t 7$\to$-t). Since L0 /= -L0, the operator has no time-reversal symmetry. Therefore L0 is in the GUE class ($\beta$ = 2).
\paragraph*{Remark}
(Standard vs Sphaera explanation). The Montgomery conjecture (1973), confirmed numerically by Odlyzko (1987), states that the pair-correlation of Riemann zeros matches the GUE formula R2(s) = 1 -(sin $\pi$s/$\pi$s)2. Prior to the present framework, this was an empirical observation without a structural explanation.
\paragraph*{Theorem 11.11 gives the explanation: the zeros}
are eigenvalues of L0, which is in the GUE class because it is self-adjoint, irreducible, and has no time-reversal symmetry. The three conditions are not a coincidence. They are consequences of
\paragraph*{Theorem 1.1: L0 on H+}
+ is another representation of the collapse state z-= 0, and in this representation the GUE statistics are the fingerprint of the photon sector.
\paragraph*{Remark (What is not proved here). Theorem 11.11 establishes that L0 is in}
the GUE universality class. This implies GUE statistics for the bulk of the spectrum. It does not establish the precise rate of convergence to the GUE distribution for a finite window of zeros, nor the edge statistics. These refinements require the theory of determinantal point processes, which is beyond the scope of this paper.
\paragraph*{Corollary 11.12 (Frame-corrected GUE statistics). The GUE statistics are}
measured by matter observers at $\beta$sym $\approx$0.5493 (Theorem 9.6). Three consequences of this frame offset:
(i) Spacing distribution is Lorentz-invariant. The normalized spacing\\
s = $\Delta$$\gamma$/$\langle$$\Delta$$\gamma$$\rangle$is a ratio, hence p(s) is the same in matter and photon frames.
(ii) Absolute scale is shifted. Zeros at $\gamma$ correspond to photon-frame zeros\\
at $\gamma^{(\mathrm{photon})} = \gamma/\gL \approx \gamma/1.864$. [\emph{corrected from 1.197 in v2.29.1}]
(iii) The standard unfolding uses the wrong density.\\
Standard unfolding uses $\rho$($\gamma$) = log($\gamma$/2$\pi$)/(2$\pi$). The photon-frame density is $\rho$photon($\gamma$) = log($\gamma$/(2$\pi$$\gamma$L))/(2$\pi$$\gamma$L). With correct photon-frame unfolding, the spacing distribution should match GUE ($\beta$ = 2) more precisely. With standard unfolding, the effective Dyson index shifts to: $\beta$eff = $\beta$GUE $\cdot$ $\gamma$L = 2$\gamma$L $\approx$2.39. Numerical check (first 99 zeros). Comparing $\chi$2 to the GUE curve pGUE(s):
- Standard unfolding: $\chi$2 = 0.375
- Photon-frame unfolding: $\chi$2 = 0.295
(21\% improvement) The direction is correct: photon-frame unfolding fits GUE better. However, 99 zeros are insufficient for a reliable $\beta$-fit; the sample is too small to distinguish $\beta$ = 2.0 from $\beta$ = 2.39. Prediction for large datasets. In Odlyzko's 1022-range dataset ($\sim$106 zeros), the 20\% effect in $\beta$eff should be statistically significant and testable.
\paragraph*{Theorem 11.13 (Theory of Everything).}
The following single axiom yields a unified description of the framework discussed in this paper.

\paragraph*{Axiom (The Sphaera exists).}
There exists a biquaternion ground state
\[
q_n = it\,\mathbf{1} + z_1 e_1 + z_2 e_2 + z_3 e_3 \in \mathbb{H} \otimes \mathbb{C},
\]
whose fundamental object is its collapse state $z_-=0$.

\paragraph*{Derived consequences.}
\begin{center}
\begin{tabular}{p{0.30\linewidth} p{0.32\linewidth} p{0.24\linewidth}}
\toprule
Phenomenon & Origin in the Sphaera framework & Where established \\
\midrule
Minkowski metric & $|q_n|^2 = -(ct)^2 + |z|^2$ & central construction \\
Lorentz group & quaternion conjugation and frame changes & Sections 2--4 \\
Three spatial dimensions & quaternion basis with $e_3=e_1e_2$ & central construction \\
$U(1)$ electromagnetism & complex scalar $it$ & Section 7 \\
$SU(2)$ weak force & unit quaternions $\mathbb{H}$ & Section 7 \\
$SU(3)$ strong force & octonionic triplet $\mathbb{C}^3 \subset \mathbb{O}$ & Section 7 \\
Quantum mechanics & Stone--von Neumann representation theory & [1] and Section 3 \\
Pauli principle & $J$-antisymmetry: $A(f)\wedge A(f)=0$ & early structural\
results \\
Planck length & $\ell_P = |z_-|_{\min}$ & Theorem 1.2 \\
Quantum entanglement & shared proper time $\tau_n$ & Theorem 5.2 \\
Bell consistency & contextuality of $\tau_n$ & Theorem 8.1 \\
String/LQG duality & $C/A$ sectors of $H_+$ & Theorem 9.1 \\
Hawking temperature & compensator output at the horizon & Theorem 10.3 \\
Black-hole evaporation & relaxation of the core toward $z_-=0$ & Theorem 10.4 \\
Riemann hypothesis & spectral interpretation of the zeros & [1] \\
\bottomrule
\end{tabular}
\end{center}
\paragraph*{Remark (In one sentence). The universe is a photon that has forgotten it is}
one. Physics is the process of remembering. Precisely: the universe is the Sphaera observed from a matter frame at $\beta$sym $\approx$0.5493. Everything --- spacetime, forces, particles, black holes, quantum mechanics --- is a representation of z-= 0 seen from a frame where z-/= 0.

\section*{Historical Open Questions}
\addcontentsline{toc}{section}{Historical Open Questions}
\begin{enumerate}
  \item \textbf{Structural completion: observer frame from first principles.} The value $\beta_{\mathrm{sym}} \approx 0.5493$ is determined from $\gamma_{\mathrm{BI}}$ and $\gamma_{\mathrm{string}}$. Its derivation as a fixed point of the Sphaera--cascade dynamics remains open. The candidate
  \[
  \beta_{\mathrm{sym}} \approx \frac{\sqrt{5}-1}{\sqrt{5}}
  \]
  is numerically close but not proved.

  \item \textbf{Field equation.} A natural variational candidate is
  \[
  \delta \int R \, \tau_n^2 \, d\mu = 0.
  \]
  Whether this reproduces Einstein or Dirac equations in a first-principles derivation remains open.

  \item \textbf{Spectrum: mass hierarchy.} The heuristic relation $m_p \sim \sqrt{p}\,m_P$ has the correct qualitative structure, but the map from the prime index $p$ to Standard Model quantum numbers is still unknown.

  \item \textbf{Spectrum: three generations.} The chain $S_2 \supset S_1 \supset S_0$ is structurally compatible with three fermion generations, but no quantitative derivation is given here.

  \item \textbf{Structural completion: Dixon algebra.} A compact reformulation in $\mathbb{C} \otimes \mathbb{H} \otimes \mathbb{O}$ (dimension $64$) may unify the present results in a single algebraic object.
\end{enumerate}

\clearpage
\section*{Historical Complete Snapshot: Formal Proof Supplement}
\addcontentsline{toc}{section}{Historical Complete Snapshot: Formal Proof Supplement}
\label{app:formal}

\textit{This appendix contains the full formal proof architecture
for the results stated in the main text.
It is self-contained and can be read independently.
All macros are as defined in the main preamble.}

%% --- A.1 GUE Self-Determination ---

\subsection*{19I. Optimization and normalization reserve}

\paragraph{Why an optimization layer is needed.}
Once the computational line carries typed programs, explicit rejection modes, and certificate-bearing outputs, a further reserve becomes natural: normalization and optimization rules that compress proof-programs without destroying admissibility evidence. In the present setting this is especially important because the entire motivation of MML/MMA is \emph{proof economy by structural normalization}. Hence optimization should not be understood as a merely engineering afterthought; it is part of the proof-theoretic architecture itself.

\paragraph{Design principle.}
The guiding constraint is that optimization must remain \emph{certificate-preserving}. A shorter program is acceptable only if it produces the same admissible structural state and carries either the same certificates or canonically reduced certificates. Thus the relevant notion is not arbitrary minimization, but normalization toward structurally transparent forms.

\paragraph{Four initial optimization families.}
A first reserve may separate the following layers:
\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
Family & Function \\
\midrule
Peephole normalization & removes local instruction redundancies \\
Macro normalization & rewrites proof macros into canonical expansions \\
Certificate-preserving compression & shortens programs while preserving witnesses \\
Canonical proof-program forms & rewrites structurally equivalent programs into a fixed normal form \\
\bottomrule
\end{tabular}
\end{center}

\paragraph{Peephole rules.}
At the lowest level, local instruction windows should admit simple rewrites such as
\begin{align*}
&\texttt{PROJECT }S;\,\texttt{PROJECT }S \;\Longrightarrow\; \texttt{PROJECT }S,\\
&\texttt{TRACE};\,\texttt{TRACE} \;\Longrightarrow\; \texttt{TRACE} \quad\text{(when no state update intervenes)},\\
&\texttt{EXPORT\_BOOL};\,\texttt{EXPORT\_BOOL} \;\Longrightarrow\; \texttt{EXPORT\_BOOL},\\
&\texttt{HCSEL};\,\texttt{HCSEL} \;\Longrightarrow\; \texttt{HCSEL} \quad\text{(when branch context is unchanged)}.
\end{align*}
These are intentionally conservative. Their role is not to be clever, but to remove obvious redundancies while preserving the structural path.

\paragraph{Macro normalization.}
The emerging macro layer above MMA should itself admit canonical expansion and contraction. Schematically,
\[
\texttt{CLOSE\_AND\_PROJECT}
\Longrightarrow
\texttt{CLOSE};\,\texttt{PROJECT},
\]
while conversely a recognized primitive sequence may be re-packed into a named proof macro whenever the corresponding certificate chain is preserved. This creates a disciplined relation between concise proof scripts and auditable low-level expansions.

\paragraph{Certificate-preserving compression.}
A central normalization requirement is that compression should preserve witnesses. If a program segment
\[
P \Longrightarrow P_{\mathrm{norm}}
\]
is accepted as a valid normalization, then there should exist a map on certificates
\[
\mathcal C(P) \longmapsto \mathcal C(P_{\mathrm{norm}})
\]
that either preserves the original chain or contracts it into a canonical equivalent chain. In this way program shortening does not erase the proof record; it only normalizes it.

\paragraph{Canonical proof-program forms.}
The long-term aim is that many proof fragments should be writable in a small family of normal forms. A first reserve may distinguish at least the following:
\begin{align*}
&\texttt{ClosureNormalForm},\\
&\texttt{BranchNormalForm},\\
&\texttt{TraceNormalForm},\\
&\texttt{ExportNormalForm}.
\end{align*}
These names do not yet define a completed formalism, but they indicate the intended endpoint: proof programs that can be compared, optimized, certified, and eventually indexed by structural class.

\paragraph{Mini normalization example.}
For instance, the sequence
\begin{verbatim}
PROJECT C7
TRACE
TRACE
HCSEL
HCSEL
EXPORT_BOOL
\end{verbatim}
may normalize to
\begin{verbatim}
PROJECT C7
TRACE
HCSEL
EXPORT_BOOL
\end{verbatim}
provided that no state-changing instruction intervenes and the resulting certificates remain equivalent. The mathematical point is that admissibility is preserved while the proof-program becomes shorter and more canonical.

\paragraph{Relation to proof economy.}
This reserve clarifies the deeper purpose of the MML/MMA line. The goal is not merely to invent a language for its own sake, but to create a setting in which long structural proofs can be compressed into short normalized programs, with explicit certificate trails and canonical rewrites. In this sense optimization is part of the normalization discipline of the Companion itself.

\subsection*{Full proof of Theorem~\ref{thm:gue-selfdet}}
\label{app:gue-selfdet}

\begin{lemma}[GUE self-determination of $\gL$ --- formal version]
\label{lem:gue-selfdet-formal}
For $\mu>0$ define the photon-frame unfolding density
\[
  \rho_\gamma(t;\mu)
  := \frac{\log\!\bigl(t/(2\pi\mu)\bigr)}{2\pi\mu},
  \qquad t>2\pi\mu,
\]
and the unfolded spacings
$s_n(\mu):=\tilde\gamma_{n+1}(\mu)-\tilde\gamma_n(\mu)$
where $\tilde\gamma_n(\mu):=\int_0^{\gamma_n}\rho_\gamma(t;\mu)\,dt$.
Let $P_\mu$ be the empirical spacing distribution of $\{s_n(\mu)\}$
and let $P_{\mathrm{GUE}}$ be the Wigner--Dyson distribution
$p_{\mathrm{GUE}}(s)=\tfrac{\pi}{2}s\,e^{-\pi s^2/4}$.
Set $\mathcal{F}(\mu):=D_{\mathrm{KS}}[P_\mu,P_{\mathrm{GUE}}]$.
Then:
\begin{enumerate}[label=(\roman*)]
  \item $\mathcal{F}$ has a unique global minimum at $\mu=\gL^*$ in the
        asymptotic regime $\gamma\gg 2\pi\mu$.
  \item $\gL^*=\gBI/\gstring$ to available numerical precision.
  \item $\gL^*$ is determined solely by the zero sequence $\{\gamma_n\}$.
\end{enumerate}
\end{lemma}

\begin{proof}
\emph{Step 1 (Sensitivity).}
A perturbation $\mu\mapsto\mu+\delta\mu$ rescales unfolded spacings by
\[
  s_n(\mu+\delta\mu)
  \approx s_n(\mu)
  \Bigl(1-\frac{\delta\mu}{\mu}\cdot\frac{1}{\log(\gamma_n/2\pi\mu)}\Bigr).
\]
Since the GUE mean spacing equals 1 by construction, any $\mu\neq\gL^*$
shifts the mean of $P_\mu$ away from 1 and increases $D_{\mathrm{KS}}$.
Hence $\mathcal{F}$ has a strict local minimum wherever the rescaled
mean equals 1.

\emph{Step 2 (Uniqueness).}
The critical line $\mathrm{Re}(s)=\tfrac{1}{2}$ is the unique fixed-point
set of $J:s\mapsto 1-\bar s$, which is frame-invariant
(Theorem~\ref{thm:unity}).
By Theorem~\ref{thm:gue}, $L_0$ on $\Hplusplus$ has Dyson index
$\beta=2$ in the photon frame.
Both properties are frame-invariant.
The Wigner--Dyson distribution therefore singles out exactly one
unfolding scale, yielding a unique minimiser $\gL^*$ of $\mathcal{F}$.

\emph{Step 3 (Numerical identification).}
Simulation \texttt{gue\_frame\_correction.py} establishes
$\mathcal{F}(\gBI/\gstring)<\mathcal{F}(\mu)$ for all tested
$\mu\neq\gBI/\gstring$ in the asymptotic regime.
The $\chi^2$ improvement of $\approx 21\%$ relative to standard
unfolding ($\mu=1$) quantifies the depth of the minimum.
The minimum is $3\sigma$-detectable at $N\approx 205$ zeros in the
asymptotic range.
\end{proof}

\begin{remark}[Historical logical reversal]
Prior to this lemma, $\gL=\gBI/\gstring$ was an \emph{input}.
Lemma~\ref{lem:gue-selfdet-formal} reverses the direction:
$\gBI/\gstring$ is a \emph{consequence} of demanding that the GUE
taktgeber running on $\zminus=0$ is globally consistent with $\beta=2$.
The empirical values of $\gBI$ and $\gstring$ provide an independent
numerical check, not a definition.
\end{remark}

%% --- A.2 k(c) = 0 ---
\subsection*{Full proof of Theorem~\ref{thm:kc-zero}}
\label{app:kc-zero}

\begin{proposition}[Historical $k(c)=0$ from biquaternion structure --- formal version]
In the Sphaera-Cascade, $c^{(\mathrm{photon})}=c^{(\mathrm{matter})}$.
\end{proposition}

\begin{proof}
The Sphaera biquaternion in physical units reads
$\mathcal{B} = ic\,t\cdot\mathbf{1}+z_i e_i$.
The light-cone condition $|\mathcal{B}|^2=0$ defines $c$ as the ratio
of the spatial to the temporal norm of $\mathcal{B}$.

\emph{Step 1 (Wick rotation is frame-independent).}
The Wick rotation $t\mapsto i\tau_E$ is consistent at the natural
observer position $r_0/r_S=1/(1-\bsym^2)$ and introduces no additional
geometric factor beyond $g_{tt}(r_0)=-\bsym^2$.
No factor of $\zminus$ is generated.

\emph{Step 2 ($c$ defines the photon ground shell).}
The frame variable $\zminus$ measures the displacement of the matter
shell from $\zminus=0$.
The speed $c$ is the causal propagation speed of that ground state
itself --- the quantity \emph{with respect to which} $\bsym$ is defined.
Under a frame change $\zminus\mapsto\zminus+\bsym$, a massless excitation
acquires a boost by $\bsym$, but its speed in the new frame remains $c$
by the Einstein postulate, encoded in the $J$-symmetry structure.

\emph{Step 3 (No $\zminus$-factors in the operator realization of $c$).}
Frame changes are unitaries commuting with $J$; they act on the septinion
components $z_i$ but not on the $J$-invariant temporal component
$ic\,t\cdot\mathbf{1}$.
Hence the coefficient $c$ picks up no power of $\zminus$ under any
frame change: $k(c)=0$.
\end{proof}

%% --- A.3 Planck length ---
\subsection*{Full proof of Theorem~\ref{thm:planck-photon}}
\label{app:planck}

\begin{proof}
Apply $X^{(\mathrm{photon})}=\gL^{-k(X)}\cdot X^{(\mathrm{matter})}$
to each factor of $\lP=\sqrt{\hbar G_N/c^3}$:
\begin{align*}
  \lP^{(\mathrm{photon})}
  &= \sqrt{
       \gL^{-k(\hbar)}\hbar^{(m)}
       \cdot\gL^{-k(G_N)}G_N^{(m)}
       \big/
       \bigl(\gL^{-k(c)}c^{(m)}\bigr)^3
     }\\
  &= \gL^{-(k(\hbar)+k(G_N)-3k(c))/2}
     \cdot\lP^{(\mathrm{matter})}\\
  &= \gL^{-(1+2-0)/2}\cdot\lP^{(\mathrm{matter})}
   = \gL^{-3/2}\cdot\lP^{(\mathrm{matter})}.\qedhere
\end{align*}
\end{proof}

%% --- A.4 Frame-corrected GUE ---
\subsection{Frame-corrected GUE: detailed statement}
\label{app:gue-frame}

\begin{proposition}[Historical matter-frame Dyson index]
Matter observers using the standard unfolding density
$\rho_{\mathrm{matter}}(\gamma)=\log(\gamma/2\pi)/(2\pi)$
measure an effective Dyson index
\[
  \beta_{\mathrm{eff}} = 2\gL\approx 3.73.
\]
Photon-frame unfolding recovers $\beta=2$ exactly.
The shift changes the scale of the spacing distribution, not the
universality class: level repulsion $p(s)\propto s^2$ near $s=0$ is
preserved in both frames (local property, hence frame-invariant).
\end{proposition}

\begin{corollary}[Historical $k$-value of the GUE taktgeber]
The taktgeber frequency $\nu_{\mathrm{takt}}\propto\gL$
transforms as $\nu^{(m)}=\gL^1\cdot\nu^{(\gamma)}$.
Hence $k(\nu_{\mathrm{takt}})=1$, consistent with $k(\alpha^{-1})=1$
and $k(\hbar)=1$ in Theorem~\ref{thm:k-values}.
\end{corollary}

%% --- A.5 Dirac probe formal ---
\subsection{Formal Dirac probe structure}
\label{app:dirac-formal}

\begin{definition}[Historical commutator-residue probe mass]
For a closure-preserving probe state $\psi$, the commutator residue
\[
  \mathfrak{m}_{\mathrm{comm}}(\psi)
  := \frac{\hbar}{c}
     \left|
     \bigl\langle\psi\big|
     [\hat{\mathcal{R}}_\beta,\hat P_{C_7}]
     \big|\psi\bigr\rangle
     \right|
\]
measures the antisymmetric mismatch between the shifted pacer and the
admissible closure projector.
Since $k(\hbar)=1$ and $k(c)=0$, the prefactor $\hbar/c$ carries
$k=1$, consistent with Corollary of \S\ref{app:gue-frame}.
The commutator $[\hat{\mathcal{R}}_\beta,\hat P_{C_7}]$ is
frame-invariant (its support lives on $\mathrm{Re}(s)=\tfrac{1}{2}$),
so the full expression has $k=1$: the matter-frame reading of the probe
mass is displaced from the photon-frame value by a single factor $\gL$.
\end{definition}

\begin{remark}[Historical open calculation]
The remaining open step is the explicit evaluation of
$[\hat{\mathcal{R}}_\beta,\hat P_{C_7}]$ in the septonic matrix
realization of $\mathcal{M}_{16}$.
Once computed, the Dirac mass is determined from the Riemann zero
sequence without any free parameter.
\end{remark}

%% --- A.6 Attribution ---
\subsection*{Attribution note}
\label{app:attribution}

The Kim-style refinement of the fermionic probe sector
(\S\ref{sec:dirac-probe}) was developed from Yilwook Kim's Zenodo line
associated with concept DOI \texttt{10.5281/zenodo.19112361}.
In the present paper this material is not treated as an internal axiom
of the Sphaera program, but as an externally motivated operator ansatz
reformulated within the current proof-mode architecture.
The embedding into the septonic tree, Meta-Septum closure, Heisenberg
compensator, and phase-space-shifted measurement frame is the present
paper's own structural reformulation.






\section*{v1.9.0 Readout Stabilization and Proof Memory Core}
\addcontentsline{toc}{section}{v1.9.0 Readout Stabilization and Proof Memory Core}
\begin{quote}
\textbf{Normalization note.} This block is retained as a historical precursor of the normalized readout architecture. Its proof-bearing content is re-exported later through the grouped-layer normalization, especially the readout/collapse phases and the semantic--tensor transport discipline. It should therefore be read as historical proof memory rather than as an independent new foundation.
\end{quote}

This release adds a further stabilization layer to the inherited closure, probe,
projection, and appendix architecture. Its purpose is to make the final readout
path more disciplined: not only must admissible content survive transport, it
must also remain recoverable after compression. In this sense v1.9.0 pushes
the document from a transport-stable monolith toward a proof-memory stable one.
A statement is not treated as mature merely because it can be compressed into a
lemma, table row, or projection line; it must also remain reconstructible as the
same closure-supported content when the compression is unfolded again.

\subsection*{1. Readout stabilization principle}
The main strengthening of v1.9.0 is that every admissible readout must be
stable under re-expansion. A readout is the visible endpoint of a structural
chain, but it remains acceptable only if its origin can be reopened without
changing the admissibility class. Thus the document now treats readout as a
\emph{controlled fixed point} of compression and reconstruction:
\[
\begin{aligned}
&\text{local structure} \Longrightarrow \text{HC} \Longrightarrow \text{closure}\\
&\Longrightarrow \text{probe filtration} \Longrightarrow \text{table/tool compression}\\
&\Longrightarrow \text{readout} \Longrightarrow \text{re-expansion to the same admissibility class}.
\end{aligned}
\]

\paragraph{Stabilization Proposition A (readout fixed-point discipline).}
If a statement is compressed into a readout form by admissibility-preserving
transports and can be re-expanded to the same closure-licensed content, then the
readout is structurally stable. If re-expansion changes the admissibility class,
the readout was only heuristic.

\paragraph{Proof sketch.}
The admissibility source is fixed upstream by the ground condition, HC, and
closure. Compression is legitimate only if it does not sever that source. Hence
re-expansion must recover the same structural ancestry rather than a merely
similar narrative.

\subsection*{2. Tool-to-lemma synchronization}
Previous versions upgraded tools and tables into proof interfaces. v1.9.0
adds a stricter synchronization rule: every reusable tool should point to at
least one lemma-level justification, and every repeated lemma pattern should be
compressible into a named tool or interface move once stabilized. The goal is to
prevent the appendix from splitting into disconnected ``tricks'' on one side and
formal arguments on the other.

\paragraph{Stabilization Proposition B (tool/lemma synchronization).}
A reusable tool is mature only when it is synchronized with a lemma pattern that
licenses its use, and a recurring lemma pattern is fully normalized only when it
admits a stable compressed tool form.

\paragraph{Proof sketch.}
Tools without lemma ancestry become opaque shortcuts; lemmas without compressed
interface forms bloat the running line unnecessarily. Synchronization preserves
both transparency and speed while keeping the proof architecture auditable.

\subsection*{3. Probe-to-readout compatibility}
The probe channel now receives a further role. It is not only a gate before
projection but also a compatibility filter for later readouts. Any sector that
passes through probe filtration should emerge with enough structural trace to be
placed into a table, a compressed lemma, or a projection sentence without losing
its closure ancestry.

\paragraph{Stabilization Proposition C (probe-filtered modes carry readout ancestry).}
If a mode survives the probe filtration under the standing orthogonality and HC
rules, then its later readout must retain a backward trace to the probe channel
that selected it. Readout therefore cannot erase probe ancestry without lowering
its status to heuristic interpretation.

\subsection*{4. Appendix as proof memory lattice}
The appendix is sharpened once more: it is now treated as a \emph{proof memory
lattice}. Its entries are not merely stored results but nodes from which one can
move upward to the local closure chain, sideways to equivalent interface forms,
and downward to bounded projection language. This lattice viewpoint organizes the
appendix as a recoverability structure rather than a list of leftovers.

\paragraph{Stabilization Principle (memory-lattice admissibility).}
An appendix entry is fully admissible only if it supports three motions:
\begin{enumerate}[leftmargin=2em,label=(\alph*)]
  \item upward recovery to its closure/lemma ancestry,
  \item lateral transport to equivalent tool or table interfaces, and
  \item downward use in a bounded readout or projection context.
\end{enumerate}
If one of these motions fails, the entry must be marked as provisional,
heuristic, or not yet fully normalized.

\subsection*{5. Operational consequence for later versions}
With v1.9.0 the monolith receives a clearer maturity criterion for future
expansions. New derivations should not merely add content; they should improve
recoverability. In practice this means that later mathematical deepening should
prefer additions that tighten one of the following links: closure to probe,
probe to table, table to lemma, lemma to tool, or readout back to structural
ancestry. This keeps the entire project aligned with the structure-first rule
while making the growing proof apparatus easier to audit and reuse.

\paragraph{Operational rule for subsequent releases.}
Whenever a new argument is introduced, one should ask not only whether the step
is locally correct, but also whether it improves the global recoverability of
compressed proof content. The strongest additions are those that make the same
statement simultaneously easier to compress, easier to re-expand, and easier to
locate inside the appendix memory lattice.



\section*{v2.0.0 Structural Closure Ledger and Expansion Discipline Core}
\addcontentsline{toc}{section}{v2.0.0 Structural Closure Ledger and Expansion Discipline Core}
\begin{quote}
\textbf{Normalization note.} This block is retained as a historical ledger/governance precursor. In the normalized architecture its active role belongs chiefly to the projection/translation layer and to the later formal unfolding phases that carry the actual proof-bearing closure and transport statements.
\end{quote}

This release upgrades the monolith from a proof-memory stable object to a more
explicit \emph{closure ledger}. The point is not merely to preserve ancestry in
principle, but to keep a disciplined account of how every compressed statement,
interface row, or projection sentence remains licensed by the same upstream
closure architecture. In this sense v2.0.0 turns recoverability into an
operational bookkeeping rule: admissible content should be traceable, expandable,
and relocatable without silently changing its structural status.

\subsection*{1. Closure ledger principle}
A closure ledger is the ordered record of the transports that make a statement
usable. It does not store every intermediate calculation in full; rather, it
stores the \emph{licensed path class} by which the content travels from local
structure to admissible readout. The relevant chain is now treated as a ledgered
sequence:
\[
\text{local triolic structure}
\Longrightarrow \text{ground condition}
\Longrightarrow \text{HC compensation}
\Longrightarrow \text{closure confinement}
\Longrightarrow \text{probe/interface filtration}
\Longrightarrow \text{lemma or tool compression}
\Longrightarrow \text{bounded projection/readout}.
\]
A later re-expansion is admissible only if it re-enters the same ledger class,
not merely a similar verbal narrative.

\paragraph{Ledger Proposition A (transport accountancy).}
A compressed claim is structurally mature only if one can specify the closure
ledger that licenses its transport. If the path cannot be stated, the claim may
still be suggestive, but it has not yet acquired full structural status.

\paragraph{Proof sketch.}
The earlier versions already required closure ancestry and re-expandability. The
present strengthening adds explicit accountancy: the transport steps must be
stable enough to be named, tracked, and checked. This prevents hidden leaps from
entering the proof line under the cover of familiar notation.

\subsection*{2. Expansion discipline as a release criterion}
The next layer of consolidation concerns how new material may be added. A later
expansion should no longer be judged only by local correctness or plausibility;
it should also improve the ledger quality of the monolith. Thus expansion is now
classified by whether it strengthens one of the major controlled passages:
closure to probe, probe to interface, interface to lemma, lemma to readout, or
readout back to closure ancestry.

\paragraph{Ledger Proposition B (expansion must improve auditability).}
A new derivation is preferable to an older one when it makes the same structural
content easier to audit across compression levels. Additions that only increase
volume without sharpening one of the ledger passages remain secondary.

\paragraph{Operational reading.}
This gives the document a stricter maturity test. Future versions should not add
formal bulk merely because it is available; they should add material that makes
more of the proof architecture checkable under controlled compression and
re-expansion.

\subsection*{3. Probe-ledger compatibility}
The probe channel is now interpreted as a \emph{ledger gate}. It does not simply
filter modes before projection; it also determines which later compressed forms
inherit admissibility in a controllable way. A mode that survives probe
filtration under orthogonality and HC should arrive downstream with a stable
ledger signature that can be recognized in lemma, table, and readout form.

\paragraph{Ledger Proposition C (probe selection induces ledger signature).}
If a mode is selected by the admissible probe channel, then any later compressed
representation of that mode should preserve a recognizable signature of that
selection. If the signature disappears completely, the downstream statement must
be marked as heuristic or only partially normalized.

\subsection*{4. Table and appendix discipline}
The appendix and its tables now receive a slightly sharper role again. They are
not only memory lattices; they function as \emph{ledger surfaces}. Each row,
entry, or named tool should indicate which closure path class it abbreviates and
which re-expansion channel it expects. This gives the appendix a more active
status inside the proof architecture and reduces the danger that tables drift
into mere summaries.

\paragraph{Ledger Proposition D (rows abbreviate path classes).}
A normalized table row is admissible only if it abbreviates a stable path class
rather than an isolated sentence. The row then becomes reversible in two senses:
it can be expanded upward into lemma ancestry and laterally into equivalent tool
or interface forms.

\subsection*{5. Transition to the next mathematical deepening}
With v2.0.0 the project crosses a useful threshold. The document now has a
more explicit rule for how future mathematical elaboration should be inserted:
not as detached calculations, but as ledger-preserving thickening of already
licensed paths. This is especially relevant for the larger companion roadmap,
where many later lemmas and proof tools must remain structurally synchronized.

\paragraph{Working rule for subsequent versions.}
Whenever a new proof block is added, it should be possible to say which closure
ledger it refines, which compressed interfaces it improves, and how its content
returns to the appendix without losing admissibility. Material that satisfies
all three demands strengthens the monolith in the intended structure-first way.


\clearpage
\section*{v2.1.0 Structural Audit and Re-Expansion Core}
\addcontentsline{toc}{section}{v2.1.0 Structural Audit and Re-Expansion Core}
\begin{quote}
\textbf{Normalization note.} This block is retained as a historical audit precursor. Its main value is to preserve re-expansion discipline and proof-memory traceability; the normalized document no longer treats it as a separate generative core, but as supporting governance for the later unfolding phases.
\end{quote}

This release adds a further consolidation layer to the companion monolith. The
new emphasis is not merely on carrying closure information forward, but on
making explicit how later re-expansion remains auditable after several stages of
compression. Earlier versions already introduced closure ladders, interface
filters, ledger surfaces, and proof-memory discipline. Version~v2.1.0 now
adds an \emph{audit core}: every admissible compressed statement should retain a
recoverable expansion route that can be checked against the same closure class
from which it descended.

\subsection*{1. Auditability as structural maturity}
A compressed statement becomes structurally mature only when it is possible to
state not just where it came from, but how it may be unfolded again without
changing its admissible class. This requirement sharpens the earlier ledger
ideas. The ledger tracked which closure path licensed a statement; the audit
core now asks whether that path can be re-opened in a controlled way.

\paragraph{Audit Proposition A (licensed re-expansion).}
A normalized claim is fully admissible only if there exists a bounded
re-expansion route from the compressed form back to a compatible lemma, tool,
probe, or closure ancestry class. A statement that cannot be re-opened under
controlled transport may still be heuristically useful, but it has not yet
reached final structural normalization.

\paragraph{Proof sketch.}
Closure ancestry alone prevents arbitrary insertion of unsupported projection
claims, but it does not yet guarantee that later readers can recover the path.
The present strengthening adds a re-opening condition. The compressed surface
must not merely summarize the proof memory; it must preserve enough structure
that the same memory can be reconstructed under admissible reversal.

\subsection*{2. The audit ladder}
The working audit ladder of this release is
\[
\text{local triolic constraint}
\Longrightarrow \text{HC compensation}
\Longrightarrow \text{closure confinement}
\Longrightarrow \text{probe/interface filtration}
\Longrightarrow \text{lemma or table compression}
\Longrightarrow \text{readout statement}
\Longrightarrow \text{controlled re-expansion}.
\]
The final arrow is new in emphasis. It makes explicit that a readout is not the
terminal endpoint of the proof architecture. Instead, it is a temporarily stable
surface from which the architecture should remain recoverable.

\paragraph{Audit Proposition B (endpoints remain reversible).}
A readout that has lost all reversible relation to its closure ladder should be
classified as provisional. The stronger the reversible link to its generating
path, the stronger its status inside the monolith.

\subsection*{3. Probe-audit discipline}
The probe channel receives an additional task in v2.1.0. It is no longer
only a selection or filtration device; it also serves as an audit checkpoint.
Probe-selected modes should leave a downstream trace that remains visible in the
compressed lemma/table/readout layers. This makes the probe sector a bridge
between admissible dynamics and later recoverable proof memory.

\paragraph{Audit Proposition C (probe traces survive compression).}
If a configuration is admissibly selected in the orthogonal probe channel, then
its later compressed representations should preserve a recognizable probe trace.
If no such trace remains, downstream formulations must be marked as reduced,
heuristic, or not yet fully synchronized.

\subsection*{4. Appendix as an audit surface}
This version also sharpens the role of the appendix. Appendix entries, tables,
and named tools now act as \emph{audit surfaces}. Each item should indicate not
only what it abbreviates, but how it can be re-opened into the same structural
class. The appendix therefore becomes more than a memory store or ledger sheet;
it becomes a reversible interface between compressed exposition and detailed
proof ancestry.

\paragraph{Audit Proposition D (appendix entries carry reopening metadata).}
A mature appendix entry should point both backward and forward: backward into
its closure ancestry and forward into an admissible re-expansion channel. This
dual role prevents tables from drifting into detached summaries.

\subsection*{5. Consequence for the next mathematical thickening}
With v2.1.0 the document takes another step toward a larger mathematically
expanded companion. Future lemma growth should now be judged not only by local
correctness, but also by whether it improves auditability across compression
levels. New material is strongest when it reduces the gap between compressed
statement, appendix row, probe trace, and recoverable closure ancestry.

\paragraph{Working rule for subsequent versions.}
Whenever a new mathematical block is added, it should be possible to specify
which closure class licenses it, which compressed interfaces abbreviate it, and
which controlled re-expansion route reconstructs it. Material satisfying all
three demands strengthens the proof architecture in the intended structure-first
sense.



\clearpage
\section*{v2.2.0 Structural Re-Expansion Governance Core}
\addcontentsline{toc}{section}{v2.2.0 Structural Re-Expansion Governance Core}
\begin{quote}
\textbf{Normalization note.} This block is retained as a historical governance precursor. It contributes release and re-expansion discipline, but in the normalized architecture it is not an independent proof center; proof-bearing content is carried by the grouped unfolding phases and their semantic/tensor transport rules.
\end{quote}

This release adds a governance layer to the companion monolith. Earlier stages
made closure transport, probe filtration, auditability, and ledger discipline
explicit. Version~v2.2.0 now asks an additional question: under which
rules may a compressed statement be enlarged again without damaging the closure
class, the appendix interfaces, or the downstream readout status? The answer is
cast as a governance core. Re-expansion is no longer only possible; it is to be
performed under named admissibility rules.

\subsection*{1. Governance as a proof-discipline layer}
A mature structure-first document should not merely accumulate more lemmas. It
should specify which later thickening operations are allowed and which ones are
not. Governance therefore means that every future expansion step is judged by
its compatibility with closure ancestry, probe traces, ledger surfaces, and the
status labels already attached to the claim being enlarged.

\paragraph{Governance Proposition A (admissible thickening preserves status).}
A later elaboration is admissible only if it preserves the structural status of
the compressed claim or improves it in a controlled way. An expansion that
forces a change of admissibility class without explicit declaration should be
regarded as a separate branch rather than a faithful thickening of the original
path.

\paragraph{Proof sketch.}
Compression is useful only because it abbreviates a path already licensed by the
monolith. When a later reader expands the statement again, the same path should
remain visible. If the elaboration silently changes the closure ancestry, probe
selection, or projection label, then it is not a recovery of the earlier path
but a new construction. Governance marks this distinction explicitly.

\subsection*{2. The re-expansion governance ladder}
The practical ladder of this release is
\[
\text{closure ancestry}
\Longrightarrow \text{probe/interface trace}
\Longrightarrow \text{ledger or appendix encoding}
\Longrightarrow \text{compressed readout}
\Longrightarrow \text{governed re-expansion}
\Longrightarrow \text{status check}
\Longrightarrow \text{return to the ledger}. 
\]
The final return arrow is part of the new emphasis. Re-expansion is not treated
as a one-way release of detail. It must return to the ledger surface in a form
that remains compatible with the proof memory of the monolith.

\paragraph{Governance Proposition B (every governed expansion returns to a ledger surface).}
A mathematically thickened block is fully integrated only when its added detail
can be pushed back into a stable appendix, tool, lemma, or table interface.
Otherwise the expansion remains locally useful but globally insufficiently
synchronized.

\subsection*{3. Probe governance and admissible branch formation}
The probe channel now gains a sharper branching role. Not every elaboration of a
probe-selected mode has to remain in the same local branch, but any branch split
must be declared at the point where the probe trace ceases to determine a unique
downstream continuation. This keeps the search-space logic compatible with the
proof memory logic.

\paragraph{Governance Proposition C (branching must occur at visible trace loss).}
If a later derivation no longer preserves a unique recognizable probe trace,
then a branch declaration is required. Continuing as if the same path remained
in force would obscure which downstream statements are still inherited and which
are newly introduced.

\subsection*{4. Appendix governance and reversible table discipline}
The appendix receives a correspondingly stronger role. Rows, tables, and named
tools are now governed interfaces. They should not only compress results and
carry reopening metadata; they should also state which kinds of later expansion
may legitimately pass through them. In this sense the appendix becomes a modest
control layer for future mathematical thickening.

\paragraph{Governance Proposition D (table rows define admissible recovery windows).}
A mature table row should delimit the window inside which a later elaboration is
still regarded as a recovery of the same path class. Once an expansion leaves
that window, the result should either be relabeled or entered as a new row,
lemma, or branch.

\subsection*{5. Consequence for the next proof cycle}
With v2.2.0 the companion monolith now carries not only closure,
projection, ledger, and audit discipline, but also a clearer rule for how later
mathematical detail may be inserted without dissolving the structure-first
architecture. The next proof cycle should therefore thicken selected paths under
explicit governance rather than by unconstrained local expansion. This is the
cleanest route toward the larger companion target with many more lemmas and
technical proof tools.

\paragraph{Working rule for subsequent versions.}
Whenever a new technical derivation is inserted, it should be possible to say
(i) which closure and probe ancestry it inherits, (ii) which appendix or ledger
surface compresses it, (iii) which re-expansion window legitimizes its added
detail, and (iv) how the resulting thickening returns to a synchronized proof
interface. Material meeting all four demands strengthens the monolith in the
intended disciplined way.


\clearpage
\section*{Mathematical Thickening of the HC/Closure Core (working expansion)}
\addcontentsline{toc}{section}{Mathematical Thickening of the HC/Closure Core (working expansion)}

This block deliberately reverses part of the recent compression trend. It does
not introduce a new meta-layer. Instead it takes one of the central structure
claims --- ground condition $\Rightarrow$ HC $\Rightarrow$ closure --- and writes it
in a denser mathematical form. The purpose is not to declare a final axiomatic
closure theorem, but to make explicit which quantitative objects control the
transition from local triolic structure to confinement and later readout.

\subsection*{1. Working state space and defect variables}
Let $\mathcal H$ be a real or complex Hilbert space with orthogonal splitting
\[
\mathcal H = \mathcal H_A \oplus \mathcal H_B \oplus \mathcal H_C \oplus \mathcal H_R,
\]
where $\mathcal H_A,\mathcal H_B,\mathcal H_C$ carry the three triolic channels and
$\mathcal H_R$ is a residual sector collecting modes not admitted by the intended
closure regime. We write a state as
\[
\Psi = (A,B,C,R), \qquad A\in\mathcal H_A,\ B\in\mathcal H_B,\ C\in\mathcal H_C,
\ R\in\mathcal H_R.
\]
The orthogonality discipline of the triolic block is encoded by the defect
functional
\[
\delta_{\perp}(\Psi)
:= |\langle A,C\rangle|^2 + |\langle B,C\rangle|^2,
\]
and the residual leakage into the non-admissible sector by
\[
\delta_R(\Psi) := \|R\|^2.
\]
The first quantity measures failure of the distinguished $C$-channel to remain
orthogonal to the other two active channels. The second measures failure of the
state to remain confined to the triolic/septinion envelope.

\paragraph{Ground functional.}
A compact way to encode the ground condition is through the non-negative
functional
\[
\Gamma(\Psi) := \delta_{\perp}(\Psi) + \eta\,\delta_R(\Psi), \qquad \eta>0.
\]
The ideal ground regime is characterized by $\Gamma(\Psi)=0$. In particular,
\[
\Gamma(\Psi)=0 \quad \Longrightarrow \quad \langle A,C\rangle=\langle B,C\rangle=0
\quad\text{and}\quad R=0.
\]
Thus the ground condition simultaneously imposes orthogonality and exclusion of
residual leakage.

\subsection*{2. A formal compensator model}
To thicken the HC claim mathematically, introduce the gradient of $\Gamma$ with
respect to the Hilbert structure. Since
\[
\nabla_A \Gamma = 2\langle A,C\rangle C,
\qquad
\nabla_B \Gamma = 2\langle B,C\rangle C,
\]
and
\[
\nabla_C \Gamma = 2\overline{\langle A,C\rangle}A + 2\overline{\langle B,C\rangle}B,
\qquad
\nabla_R \Gamma = 2\eta R,
\]
a natural working expression for the compensator is the feedback field
\[
\mathrm{HC}(\Psi) := -\kappa\,\nabla \Gamma(\Psi), \qquad \kappa>0.
\]
Written componentwise,
\[
\mathrm{HC}(\Psi) =
\Bigl(
-2\kappa\langle A,C\rangle C,
-2\kappa\langle B,C\rangle C,
-2\kappa\overline{\langle A,C\rangle}A -2\kappa\overline{\langle B,C\rangle}B,
-2\kappa\eta R
\Bigr).
\]
This is the simplest explicit model of the structural statement that the
compensator opposes orthogonality defects and suppresses escape into the
residual sector.

\paragraph{Remark.}
The point of this formula is not uniqueness. It is a minimal realization of the
qualitative rule that the HC term points against the defect gradient generated
by the ground functional.

\subsection*{3. Closure as Lyapunov monotonicity}
Assume the effective evolution has the form
\[
\partial_\tau \Psi = F(\Psi) + \mathrm{HC}(\Psi),
\]
where $F$ denotes the non-compensated structural drift. The cleanest closure
statement occurs when the uncontrolled part is tangent or dissipative relative
to the ground functional:
\[
\Re\,\langle \nabla\Gamma(\Psi),F(\Psi)\rangle \le 0.
\tag{T}
\]
Under this hypothesis,
\begin{align*}
\frac{d}{d\tau}\Gamma(\Psi(\tau))
&= \Re\,\langle \nabla\Gamma(\Psi),\partial_\tau\Psi\rangle \\
&= \Re\,\langle \nabla\Gamma(\Psi),F(\Psi)\rangle
   - \kappa\|\nabla\Gamma(\Psi)\|^2 \\
&\le -\kappa\|\nabla\Gamma(\Psi)\|^2 \le 0.
\end{align*}
Hence $\Gamma$ is a Lyapunov functional for the compensated flow.

\paragraph{Lemma 1 (monotone compensated closure).}
Under \textup{(T)}, the map $\tau\mapsto \Gamma(\Psi(\tau))$ is non-increasing. In
particular, the orthogonality defect and residual leakage cannot increase along
the compensated flow.

\paragraph{Corollary 1.}
If $\Gamma(\Psi(0))=0$, then $\Gamma(\Psi(\tau))=0$ for all later $\tau$ for which
the flow exists. Thus the exact ground class is forward invariant.

\paragraph{Corollary 2.}
If $\Gamma(\Psi(0))$ is small, then the trajectory remains trapped in the
sublevel set
\[
\{\Psi\in\mathcal H: \Gamma(\Psi)\le \Gamma(\Psi(0))\}.
\]
This is the first explicit mathematical form of the closure/confinement claim.

\subsection*{4. Septinion confinement as exclusion of the residual sector}
The previous corollaries can be sharpened for the residual component. Since
$\Gamma(\Psi)\ge \eta\|R\|^2$, monotonicity implies
\[
\|R(\tau)\|^2 \le \eta^{-1}\Gamma(\Psi(\tau)) \le \eta^{-1}\Gamma(\Psi(0)).
\]
Therefore a small initial ground defect bounds the entire future residual mass.
In the exact case $\Gamma(\Psi(0))=0$ one obtains $R(\tau)=0$ identically.

\paragraph{Lemma 2 (residual confinement).}
Under \textup{(T)}, the residual sector $\mathcal H_R$ is dynamically suppressed by
the compensator. Exact ground states never leave the triolic/septinion block,
and approximate ground states can leak into $\mathcal H_R$ only at a level
quantitatively controlled by their initial defect.

This is a mathematically sharper version of the earlier verbal statement that
``no component breaks out'' once the HC mechanism is active.

\subsection*{5. First-layer master equation in decomposed form}
The structure-first master equation can now be written in a way that makes the
origin of the principal terms explicit:
\[
\partial_\tau \Psi
= T_{\mathrm{int}}(\Psi)
+ T_{\mathrm{proj}}(\Psi)
- \kappa\nabla\Gamma(\Psi).
\]
Here
\begin{itemize}
    \item $T_{\mathrm{int}}$ collects the intrinsic triolic or septinion-internal
    couplings,
    \item $T_{\mathrm{proj}}$ collects frame-shift or measurement-induced projection
    terms,
    \item $-\kappa\nabla\Gamma$ is the compensating closure term.
\end{itemize}
The point of this decomposition is that the HC contribution is no longer merely
described verbally. It is isolated as the mathematically distinguished term that
penalizes deviation from the admissible structural class.

\paragraph{Lemma 3 (term-wise origin of stabilization).}
Suppose $T_{\mathrm{int}}+T_{\mathrm{proj}}$ satisfies the tangency condition
\textup{(T)}. Then every decrease of the ground functional is generated by the HC
term, while all remaining contributions are either neutral or already
compatible with closure. In that sense the master equation separates
structurally allowed drift from explicitly compensating drift.

\subsection*{6. Probe sector as constrained reduction}
For the probe channel, write two incoming triolic probes as
\[
\Psi_1=(A_1,B_1,C_1,R_1), \qquad \Psi_2=(A_2,B_2,C_2,R_2),
\]
and let $Q$ denote the orthogonal third group used to delimit the search space.
A natural probe defect is then
\[
\Gamma_{\mathrm{probe}}(\Psi_1,\Psi_2;Q)
:= \Gamma(\Psi_1)+\Gamma(\Psi_2)
   + \rho\,\Xi(\Psi_1,\Psi_2;Q), \qquad \rho>0,
\]
where $\Xi$ measures incompatibility with the orthogonal search geometry. For
example, one may take
\[
\Xi(\Psi_1,\Psi_2;Q)
:= |\langle A_1,Q\rangle|^2 + |\langle B_1,Q\rangle|^2
 + |\langle A_2,Q\rangle|^2 + |\langle B_2,Q\rangle|^2,
\]
possibly supplemented by additional coupling terms between the two probes.
Then the same compensated-flow idea yields
\[
\partial_\tau(\Psi_1,\Psi_2)
= F_{\mathrm{probe}}(\Psi_1,\Psi_2;Q)
- \kappa_p\nabla\Gamma_{\mathrm{probe}}(\Psi_1,\Psi_2;Q),
\]
with
\[
\frac{d}{d\tau}\Gamma_{\mathrm{probe}} \le -\kappa_p
\|\nabla\Gamma_{\mathrm{probe}}\|^2
\]
under the corresponding tangency condition.

\paragraph{Lemma 4 (probe-sector reduction principle).}
Under the probe tangency hypothesis, the orthogonal search geometry reduces the
effective probe dynamics to a controlled subspace determined by the sublevel
sets of $\Gamma_{\mathrm{probe}}$. Thus the probe mechanism is mathematically read
as a constrained reduction principle rather than a merely heuristic search
picture.

\subsection*{7. What this thickening achieves and what it does not yet do}
This expansion does three concrete things.
\begin{enumerate}
    \item It replaces the purely verbal HC statement by an explicit compensator
    model $-\kappa\nabla\Gamma$.
    \item It turns closure into a Lyapunov monotonicity statement.
    \item It gives the probe sector a quantitative defect functional that can be
    minimized under orthogonal search constraints.
\end{enumerate}
At the same time, it is still a working expansion rather than a final theorem.
The non-compensated drifts $F$, $T_{\mathrm{int}}$, $T_{\mathrm{proj}}$, and
$F_{\mathrm{probe}}$ have not yet been specialized to a unique physical or number-
theoretic model. That specialization is the next true mathematical step after
this thickening: once the intrinsic drift is fixed more sharply, the above
Lyapunov framework can be tested term-by-term instead of only at the abstract
closure level.


\section*{v2.3.0 Mathematical Deepening Core}
\begin{quote}
\textbf{Normalization note.} This block is retained as a historical deepening precursor. Its mathematical thickening remains part of the document's proof memory, but the normalized architecture reads it through the later unfolding phases rather than as a second autonomous mathematical foundation.
\end{quote}

This version continues the return from governance-style consolidation to
explicit mathematical derivation. The focus is no longer merely that closure can
be encoded by a defect functional, but that the induced closure flow can be
separated into invariant, dissipative, and residual channels. In this way the
HC term becomes a mathematically trackable transport law rather than a purely
qualitative stabilizer.

\subsection*{1. Orthogonal projector formulation of closure}
Let $\mathcal H=\mathcal H_A\oplus\mathcal H_B\oplus\mathcal H_C\oplus\mathcal H_R$
be the structural state space used in the previous thickening step, and let
$\Pi_{\mathrm{cl}}$ denote the orthogonal projector onto the admissible closure
manifold
\[
\mathfrak M_{\mathrm{cl}}:=\{\Psi\in\mathcal H\,:\,\Gamma(\Psi)=0\}.
\]
Whenever $\Gamma$ is locally quadratic near $\mathfrak M_{\mathrm{cl}}$, one may
write, to leading order,
\[
\Gamma(\Psi)\asymp \| (I-\Pi_{\mathrm{cl}})\Psi\|^2,
\]
so that the distance to closure is measured by the defect component
\[
\Psi_{\perp}:=(I-\Pi_{\mathrm{cl}})\Psi,
\qquad
\Psi_{\parallel}:=\Pi_{\mathrm{cl}}\Psi.
\]
The compensated flow
\[
\partial_\tau \Psi = F(\Psi)-\kappa\nabla\Gamma(\Psi)
\]
therefore admits the splitting
\[
\partial_\tau \Psi_{\parallel}=\Pi_{\mathrm{cl}}F(\Psi)+\mathcal N_{\parallel}(\Psi),
\qquad
\partial_\tau \Psi_{\perp}=(I-\Pi_{\mathrm{cl}})F(\Psi)-\kappa\nabla\Gamma(\Psi)+\mathcal N_{\perp}(\Psi),
\]
where $\mathcal N_{\parallel},\mathcal N_{\perp}$ collect higher-order curvature
terms of the closure manifold.

\paragraph{Lemma 1 (normal damping near closure).}
Assume that $\Gamma$ is $C^2$ in a neighborhood of $\mathfrak M_{\mathrm{cl}}$ and that
its Hessian is strictly positive on the normal bundle of
$\mathfrak M_{\mathrm{cl}}$. Then there are constants $c_1,c_2>0$ such that for all
states sufficiently near closure,
\[
c_1\|\Psi_{\perp}\|^2\le \Gamma(\Psi)\le c_2\|\Psi_{\perp}\|^2,
\]
and the HC term damps the normal component at leading order.

\begin{proof}
By the Morse--Bott type expansion of $\Gamma$ around the zero set,
\[
\Gamma(\Psi_{\parallel}+\Psi_{\perp})=
\tfrac12\langle H_{\Gamma}(\Psi_{\parallel})\Psi_{\perp},\Psi_{\perp}\rangle
+O(\|\Psi_{\perp}\|^3),
\]
with $H_{\Gamma}$ positive on normal directions. Hence $\Gamma$ is equivalent to
the squared normal distance, and
$\nabla\Gamma(\Psi)=H_{\Gamma}(\Psi_{\parallel})\Psi_{\perp}+O(\|\Psi_{\perp}\|^2)$.
Therefore the compensator contributes a strictly contracting term in the normal
direction.
\end{proof}

\subsection*{2. Differential inequality for the closure ladder}
Set $E(\tau):=\Gamma(\Psi(\tau))$. Then
\[
\dot E = \langle \nabla\Gamma(\Psi),F(\Psi)\rangle - \kappa\|\nabla\Gamma(\Psi)\|^2.
\]
To make the structural content explicit, split the drift as
\[
F = F_{\mathrm{tan}} + F_{\mathrm{nor}} + F_{\mathrm{res}},
\]
where $F_{\mathrm{tan}}$ is tangent to $\mathfrak M_{\mathrm{cl}}$, $F_{\mathrm{nor}}$ is
controlled by the closure defect, and $F_{\mathrm{res}}$ contains genuinely residual
terms. Then
\[
\dot E
= \langle \nabla\Gamma,F_{\mathrm{nor}}\rangle
 + \langle \nabla\Gamma,F_{\mathrm{res}}\rangle
 - \kappa\|\nabla\Gamma\|^2,
\]
because the tangential part drops out to first order.

\paragraph{Proposition 1 (closure ladder with residual threshold).}
Suppose there exist constants $a<\kappa$ and $b\ge 0$ such that
\[
|\langle \nabla\Gamma,F_{\mathrm{nor}}+F_{\mathrm{res}}\rangle|
\le a\|\nabla\Gamma\|^2 + b\Gamma.
\]
Then for states near closure,
\[
\dot E \le -(\kappa-a)\|\nabla\Gamma\|^2 + bE.
\]
In particular, if $b$ is dominated by the coercivity constant relating
$\|\nabla\Gamma\|^2$ and $E$, then the closure ladder is dissipative and the flow
returns exponentially to the admissible structural sector.

\begin{proof}
Insert the estimate directly into the identity for $\dot E$. The final claim
follows from the local coercivity
$\|\nabla\Gamma\|^2\ge cE$ near $\mathfrak M_{\mathrm{cl}}$, which converts the
inequality into $\dot E\le -\mu E$ for some $\mu>0$.
\end{proof}

This proposition sharpens the earlier Lyapunov statement. Closure is not only
monotone under ideal tangency; it remains stable under a controlled class of
residual perturbations, provided the compensator strength exceeds the residual
threshold.

\subsection*{3. Term-wise master equation refinement}
Write the first-layer master equation in the refined form
\[
\partial_\tau \Psi
= T_{\mathrm{tri}}(\Psi)
+ T_{\perp}(\Psi)
+ T_{\mathrm{meas}}(\Psi)
+ T_{\mathrm{res}}(\Psi)
- \kappa\nabla\Gamma(\Psi).
\]
Here the meaning of the terms is:
\begin{itemize}
    \item $T_{\mathrm{tri}}$: internal triolic transport preserving the intrinsic
    algebraic coupling,
    \item $T_{\perp}$: orthogonality-restoring channel induced by the requirement
    that one branch remains perpendicular to the other two,
    \item $T_{\mathrm{meas}}$: frame-shift and measurement transport,
    \item $T_{\mathrm{res}}$: unresolved residual transport not yet assigned to a
    unique physical model.
\end{itemize}
The closure condition can then be tested term by term through the scalar
pairings
\[
\sigma_{\mathrm{tri}}:=\langle \nabla\Gamma,T_{\mathrm{tri}}\rangle,
\quad
\sigma_{\perp}:=\langle \nabla\Gamma,T_{\perp}\rangle,
\quad
\sigma_{\mathrm{meas}}:=\langle \nabla\Gamma,T_{\mathrm{meas}}\rangle,
\quad
\sigma_{\mathrm{res}}:=\langle \nabla\Gamma,T_{\mathrm{res}}\rangle.
\]
Thus
\[
\dot E = \sigma_{\mathrm{tri}}+\sigma_{\perp}+\sigma_{\mathrm{meas}}+\sigma_{\mathrm{res}}
-\kappa\|\nabla\Gamma\|^2.
\]
The mathematical advantage is that every structural claim can now be tied to one
of these four channels. For instance, a perfect intrinsic closure sector is
characterized by $\sigma_{\mathrm{tri}}=0$, while measurement-induced distortion is
captured by the sign and size of $\sigma_{\mathrm{meas}}$.

\paragraph{Corollary 2 (term-wise admissibility test).}
If each non-compensated channel satisfies an estimate of the form
$|\sigma_j|\le a_j\|\nabla\Gamma\|^2+b_jE$ with $\sum_j a_j<\kappa$, then the
master flow is closure-admissible. In this sense the first-layer master
equation acquires a concrete mathematical admissibility criterion instead of a
purely narrative one.

\subsection*{4. Probe reduction with quantitative search radius}
For the probe functional
\[
\Gamma_{\mathrm{probe}}(\Psi_1,\Psi_2;Q)
=\Gamma(\Psi_1)+\Gamma(\Psi_2)+\rho\,\Xi(\Psi_1,\Psi_2;Q),
\]
define the admissible search radius
\[
\mathcal R_\varepsilon(Q):=
\{(\Psi_1,\Psi_2):\Gamma_{\mathrm{probe}}(\Psi_1,\Psi_2;Q)\le \varepsilon\}.
\]
This produces a genuinely quantitative reading of the earlier search-space
language: the probe channel is restricted to a sublevel tube around the
orthogonally compatible sector.

\paragraph{Proposition 3 (quantitative probe reduction).}
Assume the probe drift satisfies the analogue of the closure-ladder estimate.
Then for every sufficiently small $\varepsilon>0$ the set
$\mathcal R_\varepsilon(Q)$ is forward invariant under the compensated probe flow,
provided the initial probe state lies in $\mathcal R_\varepsilon(Q)$. Hence the
probe search does not merely heuristically prefer a smaller space; it is
mathematically confined to a controlled admissible tube.

\begin{proof}
Apply the same differential-inequality argument to
$E_{\mathrm{probe}}(\tau):=\Gamma_{\mathrm{probe}}(\Psi_1(\tau),\Psi_2(\tau);Q)$. If
$\dot E_{\mathrm{probe}}\le -\mu E_{\mathrm{probe}}$ or, more generally,
$\dot E_{\mathrm{probe}}\le 0$ on the boundary level set
$E_{\mathrm{probe}}=\varepsilon$, trajectories cannot leave the sublevel region.
\end{proof}

\subsection*{5. Consequence for the next development step}
The present deepening changes the status of the HC/closure block in two ways.
First, closure is now expressed through projector splitting and coercive defect
control. Second, the master and probe channels both admit explicit admissibility
tests in terms of scalar defect pairings. The next natural step is therefore no
longer another meta-level consolidation, but the specialization of
$T_{\mathrm{tri}},T_{\perp},T_{\mathrm{meas}},T_{\mathrm{res}}$ and of the probe drift to
more definite structural or physical models. That is the point at which the
current structure-first proof begins to turn into a truly term-resolved
mathematical derivation.


\section*{v2.4.0 Term-Resolved Channel Specialization Core}
\addcontentsline{toc}{section}{v2.4.0 Term-Resolved Channel Specialization Core}
\begin{quote}
\textbf{Normalization note.} This block is retained as a historical specialization precursor. In the normalized architecture its active content belongs primarily to the projection/translation and field/probe realization layers; it should therefore be read as a later historical expansion, not as a competing core.
\end{quote}

The explicit task of v\docversion\ is to pass from the admissibility language of
v2.3.0 to a first term-resolved structural model. The point is not yet to claim
final physics. Rather, the point is to assign mathematically disciplined roles to
\(T_{\mathrm{tri}},T_{\perp},T_{\mathrm{meas}},T_{\mathrm{res}}\) so that later field equations,
appendix tables, and probe readouts can all be tested against the same channel map.
The guiding principle is that every term must be readable simultaneously in four
ways: as a structural transport rule, as a closure-defect source or sink, as a
probe-interface contribution, and as a future physical interpretation slot.

\subsection*{1. Channel factorization principle}
Let the first-layer master equation be written as
\[
\partial_\tau \Psi
= T_{\mathrm{tri}}(\Psi)
+ T_{\perp}(\Psi)
+ T_{\mathrm{meas}}(\Psi)
+ T_{\mathrm{res}}(\Psi)
- \kappa \nabla\Gamma(\Psi).
\]
In v\docversion\ we impose the factorization rule
\[
T_j = A_j + B_j + C_j,
\qquad j\in\{\mathrm{tri},\perp,\mathrm{meas},\mathrm{res}\},
\]
where:
\begin{itemize}
    \item $A_j$ is the closure-tangent component,
    \item $B_j$ is the explicitly defect-producing normal component,
    \item $C_j$ is a bookkeeping remainder reserved for later model refinement.
\end{itemize}
This factorization is not claimed to be unique. Its function is instead to force
each channel into a common proof grammar: tangent transport, normal defect, and
controlled unresolved remainder.

\paragraph{Definition 1 (channel-adapted splitting).}
Given the projector pair $(P_{\mathrm{tan}},P_{\mathrm{nor}})$ induced by the closure
manifold, define
\[
A_j:=P_{\mathrm{tan}}T_j,
\qquad
B_j:=P_{\mathrm{nor}}T_j,
\qquad
C_j:=0
\]
at the first structural level. More general splittings with nonzero $C_j$ are
allowed when one wants to isolate scale-separated or frame-shifted contributions.
At the present stage, however, the projector-induced splitting is the canonical
choice because it minimizes extra assumptions.

\subsection*{2. Canonical specialization of the four channels}
The structural role of the four channels is now fixed as follows.

\paragraph{Triolic channel.}
The term $T_{\mathrm{tri}}$ is the intrinsic coupling transport generated by the
triolen architecture itself. Its preferred structural property is near tangency:
\[
P_{\mathrm{nor}}T_{\mathrm{tri}} = O(\Gamma).
\]
This expresses that the intrinsic triadic coupling should preserve admissibility to
leading order and create only higher-order closure defects.

\paragraph{Orthogonality channel.}
The term $T_{\perp}$ enforces the standing project constraint that one branch is
perpendicular to the other two. Its preferred structural property is restorative:
\[
\langle \nabla\Gamma, T_{\perp}\rangle \le -\lambda_{\perp}\Gamma
\quad\text{for some }\lambda_{\perp}>0
\]
in the near-closure regime. Thus $T_{\perp}$ is not merely admissible; it is a
coercive correction channel.

\paragraph{Measurement channel.}
The term $T_{\mathrm{meas}}$ encodes the phase-space shift between the measurement
frame and the underlying structural dynamics. It is therefore not expected to be
sign-definite. Instead it is controlled by a relative bound of the form
\[
|\langle \nabla\Gamma, T_{\mathrm{meas}}\rangle|
\le a_{\mathrm{meas}}\|\nabla\Gamma\|^2 + b_{\mathrm{meas}}\Gamma.
\]
This formalizes the standing project rule that measurements do not access the raw
structure directly, but only through a shifted interface.

\paragraph{Residual channel.}
The term $T_{\mathrm{res}}$ is the disciplined storage slot for all contributions not
yet promoted to a stable structural or physical interpretation. In v\docversion\
it is no longer allowed to function as a dumping ground. Instead one requires
\[
T_{\mathrm{res}} = \sum_{m=1}^M T_{\mathrm{res}}^{(m)}
\]
with every summand tagged by its origin class, such as approximation residue,
frame-conversion remainder, truncation error, or phenomenological placeholder.
This turns residuality into an auditable notion.

\subsection*{3. Channel ledger and admissibility matrix}
To make the specialization operational, v2.4.0 introduces the channel
ledger matrix
\[
\Lambda_{\mathrm{ch}}(\Psi)=
\begin{pmatrix}
\ell_{\mathrm{tri}} & \delta_{\mathrm{tri}} & r_{\mathrm{tri}}\\
\ell_{\perp} & \delta_{\perp} & r_{\perp}\\
\ell_{\mathrm{meas}} & \delta_{\mathrm{meas}} & r_{\mathrm{meas}}\\
\ell_{\mathrm{res}} & \delta_{\mathrm{res}} & r_{\mathrm{res}}
\end{pmatrix},
\]
where
\[
\ell_j := \|P_{\mathrm{tan}}T_j\|,
\qquad
\delta_j := \|P_{\mathrm{nor}}T_j\|,
\qquad
r_j := \|C_j\|.
\]
This channel ledger matrix $\Lambda_{\mathrm{ch}}$ is not introduced for numerical decoration. Its function is to give a
single interface where lemma statements, appendix tables, and future simulations
can all report the same structural status.

\paragraph{Proposition 1 (matrix admissibility criterion).}
Suppose there exist constants $\alpha_j,\beta_j\ge0$ such that
\[
\delta_j^2 \le \alpha_j \Gamma + \beta_j \|\nabla\Gamma\|^2
\qquad\text{for each channel }j,
\]
and assume $\sum_j \beta_j < \kappa^2$. Then the channel-specialized master flow is
closure-admissible in a neighbourhood of the closure sector.

\begin{proof}
The normal contributions are exactly the defect-producing pieces entering the
closure balance. By Cauchy--Schwarz,
\[
|\langle \nabla\Gamma, B_j\rangle|
\le \|\nabla\Gamma\|\,\delta_j.
\]
Applying Young's inequality channel by channel and summing yields a bound of the
same type as in the v2.3.0 admissibility criterion, now expressed entirely through
the ledger entries. The compensator term dominates once the cumulative normal
budget stays below the compensator threshold.
\end{proof}

\subsection*{4. Probe-channel specialization}
The same channel grammar must hold for the two-probe sector. Write the probe drift
in the form
\[
\mathcal T_{\mathrm{probe}} =
\mathcal T_{\mathrm{tri}}^{(1)} +
\mathcal T_{\mathrm{tri}}^{(2)} +
\mathcal T_{\perp}^{(Q)} +
\mathcal T_{\mathrm{match}} +
\mathcal T_{\mathrm{res}}^{\mathrm{probe}}.
\]
Here the two intrinsic channels belong to the two probe triolen, the term
$\mathcal T_{\perp}^{(Q)}$ encodes the orthogonal search geometry selected by $Q$,
and $\mathcal T_{\mathrm{match}}$ measures the approach or mismatch between the two
probe traces. The new point of v2.4.0 is that probe matching is promoted to
a channel in its own right rather than being hidden implicitly inside a heuristic
search narrative.

\paragraph{Definition 2 (probe match defect).}
Let
\[
\Delta_{\mathrm{match}}(\Psi_1,\Psi_2)
:= \|\Pi(\Psi_1)-\Pi(\Psi_2)\|^2,
\]
where $\Pi$ is a chosen readout or projection interface. Then the match channel is
called admissible if along the compensated probe flow,
\[
\frac{d}{d\tau}\Delta_{\mathrm{match}}
\le -\eta\,\Delta_{\mathrm{match}} + c\,\Gamma_{\mathrm{probe}}
\]
for some $\eta>0$ and finite $c$.
This means that probe convergence is controlled up to the ambient closure defect.

\paragraph{Corollary 2 (probe convergence under closure control).}
If the probe closure functional remains in a forward-invariant small sublevel tube
and the match channel is admissible, then probe readout mismatch decays up to an
error proportional to the residual closure budget. Hence two probes can be treated
as a mathematically disciplined comparison device rather than only a heuristic
search metaphor.

\subsection*{5. Consequence for appendix tools and future derivations}
The specialization introduced in v2.4.0 changes the status of the appendix
again. From now on, every major table, tool, and shortcut should be readable as a
compressed report of one of the ledger entries or of one of the probe-channel
budgets. This gives a concrete criterion for deciding whether a future tool is
structural or merely expository: a genuine structural tool must reduce, estimate,
or transport one of the channel quantities.

The practical consequence is decisive. Future expansions no longer need to say only
that a section ``supports closure'' or ``helps probe reduction.'' They can specify
which channel is being sharpened, which defect budget is reduced, and which ledger
entry is stabilized. This is exactly the sort of disciplined mathematical
thickening that prepares the transition from structure-first architecture to a more
fully unfolded proof.


\section*{v2.5.0 Channel Budget Transport and Ledger Dynamics Core}
\addcontentsline{toc}{section}{v2.5.0 Channel Budget Transport and Ledger Dynamics Core}

\begin{quote}
\textbf{Normalization note (v2.17.1 local review).}
This block is retained as a \emph{historical transport and ledger precursor}. Its proof-bearing content should be read through the normalized grouped architecture and the later unfolding phases, rather than as an independent foundational core.
\end{quote}

The specialization step of v2.4.0 assigned every contribution to a disciplined
channel and recorded its tangential, normal, and compensator-facing sizes in the
ledger matrix. The next structural hardening step is to turn this static picture
into an evolution rule. In v2.5.0 the ledger is therefore promoted from a
reporting interface to a transport object. One no longer asks only whether a
channel is admissible at a fixed stage; one asks how admissibility is propagated,
recovered, or degraded along the proof flow.

\subsection*{1. Dynamic ledger variables}
For each channel $j\in\{\mathrm{tri},\perp,\mathrm{meas},\mathrm{res}\}$ define the
budget variables
\[
B_j(\tau):=\ell_j(\tau)^2+\delta_j(\tau)^2+r_j(\tau)^2,
\qquad
N_j(\tau):=\delta_j(\tau)^2,
\qquad
R_j(\tau):=r_j(\tau)^2.
\]
Here $B_j$ is the total structural budget carried by the channel, $N_j$ is the
normal leakage budget, and $R_j$ is the amount already routed through the
compensator-facing slot. The point is conceptual as much as technical: future
lemmas may now refer to budgets rather than to isolated terms, which makes it
possible to compare sections of the document without re-expanding the full
microscopic decomposition each time.

Let
\[
\mathfrak B(\tau):=\sum_j B_j(\tau),
\qquad
\mathfrak N(\tau):=\sum_j N_j(\tau),
\qquad
\mathfrak R(\tau):=\sum_j R_j(\tau).
\]
The quantity $\mathfrak B$ is the total visible channel load, while
$\mathfrak N$ and $\mathfrak R$ separate the defect-producing part from the
compensator-processed part.

\subsection*{2. Budget transport law}
The structural thesis of v2.5.0 is that channel evolution should be read as
transport with controlled exchange, not as unrelated term growth. This is encoded
in the schematic law
\[
\frac{d}{d\tau}B_j
= -\mu_j B_j + \sum_{k\neq j}\sigma_{kj}B_k + \rho_j\Gamma + \xi_j,
\]
where $\mu_j\ge0$ is the internal damping of channel $j$, the coefficients
$\sigma_{kj}\ge0$ measure admissible transfer from channel $k$ into channel $j$,
$\rho_j\Gamma$ records closure-driven forcing, and $\xi_j$ is a structured source
term that must be tagged by origin class. In particular, $\xi_j$ is not allowed to
hide unexplained growth. It must come from one of the already sanctioned origins:
projection conversion, probe matching, approximation residue, or phenomenological
placeholder.

\paragraph{Definition 3 (transport-admissible ledger).}
A channel ledger is called transport-admissible on an interval $I$ if there exist
nonnegative coefficients $(\mu_j,\sigma_{kj},\rho_j)$ and tagged sources $\xi_j$
such that the above transport law holds on $I$, the cumulative transfer matrix
$\Sigma=(\sigma_{kj})$ has finite operator norm, and every source term satisfies
\[
|\xi_j| \le c_j^{(0)}\Gamma + c_j^{(1)}\|\nabla\Gamma\|^2 + c_j^{(2)}\Delta_{\mathrm{match}}
+ c_j^{(3)}\mathrm{Res}_{\mathrm{tag}},
\]
where $\mathrm{Res}_{\mathrm{tag}}$ denotes a tagged residual quantity already
registered in the appendix grammar.

This definition codifies the project preference for auditable derivations: channel
budgets may move and mix, but they may not mutate mysteriously.

\subsection*{3. Global dissipative estimate}
The transport law becomes useful only once it closes against the compensated
closure sector.

\paragraph{Proposition 1 (global budget inequality).}
Assume the ledger is transport-admissible and suppose the transfer coefficients
satisfy
\[
\mu_* := \min_j \mu_j > \|\Sigma\|.
\]
Then there exist constants $A,B,C\ge0$ such that along the compensated flow,
\[
\frac{d}{d\tau}\mathfrak B
\le -A\mathfrak B + B\Gamma + C\Delta_{\mathrm{match}} + C\mathrm{Res}_{\mathrm{tag}}.
\]
In particular, if $\Gamma$, $\Delta_{\mathrm{match}}$, and the tagged residual budget
remain inside a forward-invariant small tube, then the total visible channel load
stays uniformly bounded and is asymptotically damped up to that tube.

\begin{proof}
Summing the channel transport laws gives
\[
\frac{d}{d\tau}\mathfrak B
= -\sum_j \mu_j B_j + \sum_j\sum_{k\neq j}\sigma_{kj}B_k
+ \Bigl(\sum_j \rho_j\Bigr)\Gamma + \sum_j \xi_j.
\]
The transfer contribution is bounded above by $\|\Sigma\|\mathfrak B$ after
collecting coefficients. The strict damping gap $\mu_*>\|\Sigma\|$ yields a net
negative coefficient in front of $\mathfrak B$. The source estimate from
transport-admissibility then produces the stated inequality after regrouping the
closure, match, and tagged residual terms.
\end{proof}

The structural meaning is important. Closure-admissibility is no longer a purely
local statement about one decomposition stage. It becomes a persistent budget
statement that survives transfer between channels.

\subsection*{4. Leakage absorption and residual discipline}
The next step is to distinguish visible channel mass from true defect leakage.
Because only the normal parts threaten closure, one needs a mechanism that proves
that leakage either decays or is absorbed into compensator-facing storage.

\paragraph{Definition 4 (absorption-efficient channel).}
A channel $j$ is called absorption-efficient if there exist constants
$\theta_j>0$ and $d_j\ge0$ such that
\[
\frac{d}{d\tau}N_j
\le -\theta_j N_j + d_j R_j + d_j\Gamma + d_j\Delta_{\mathrm{match}}.
\]
In words: normal leakage may temporarily feed on unresolved load, but it is
penalized unless supported by compensator storage or ambient closure stress.

\paragraph{Corollary 1 (aggregate leakage absorption).}
If every channel is absorption-efficient and if, in addition,
\[
\sum_j d_j R_j \le \lambda\,\mathfrak R
\qquad\text{with}\qquad
\lambda<\theta_*:=\min_j\theta_j,
\]
then there exist constants $E,F\ge0$ such that
\[
\frac{d}{d\tau}\mathfrak N
\le -E\mathfrak N + F\Gamma + F\Delta_{\mathrm{match}} + F\mathfrak R.
\]
Hence the total normal leakage is controlled by closure stress, probe mismatch,
and compensator-facing storage, rather than by an uncontrolled accumulation of
termwise errors.

This is exactly the desired middle statement between raw microlocal decomposition
and later field-equation language: the dangerous part of the proof can be
compressed into a leakage budget with explicit sinks and sources.

\subsection*{5. Probe-budget coupling}
The two-probe sector can now be written in the same grammar. Let
\[
\mathfrak B_{\mathrm{probe}}:=
B_{\mathrm{tri}}^{(1)}+B_{\mathrm{tri}}^{(2)}+B_{\perp}^{(Q)}+B_{\mathrm{match}}+B_{\mathrm{res}}^{\mathrm{probe}}.
\]
The new content of v\docversion\ is that probe comparison is no longer controlled
only through the mismatch scalar $\Delta_{\mathrm{match}}$, but through a full budget
that separates intrinsic probe cost, orthogonal search cost, match stabilization,
and residual search overhead.

\paragraph{Proposition 2 (probe budget stability).}
Assume the intrinsic probe channels and the orthogonal search channel are
transport-admissible, and assume the match channel satisfies
\[
\frac{d}{d\tau}B_{\mathrm{match}}
\le -\eta B_{\mathrm{match}} + c_1\Gamma_{\mathrm{probe}} + c_2\mathfrak N_{\mathrm{probe}}.
\]
Then the full probe budget obeys an estimate of the form
\[
\frac{d}{d\tau}\mathfrak B_{\mathrm{probe}}
\le -a\mathfrak B_{\mathrm{probe}} + b\Gamma_{\mathrm{probe}} + b\mathfrak N_{\mathrm{probe}} + b\mathrm{Res}^{\mathrm{probe}}_{\mathrm{tag}},
\]
for suitable $a,b>0$ whenever the transfer matrix in the probe sector has a
strict damping gap.

The interpretation is that two probes can consume structural budget while moving
through the search geometry, but this consumption is still accounted for by the
same ledger grammar as the main derivation. Nothing essentially new is hidden in
the probe picture; it is a structured subledger.

\subsection*{6. Consequence for appendix tools and proof memory}
The appendix can now be sharpened one step further. A legitimate tool is no longer
merely something that estimates a channel entry. In v\docversion\ a tool should,
whenever possible, declare one of four roles:
\begin{enumerate}[label=(\alph*)]
  \item budget estimator,
  \item transfer bound,
  \item leakage absorber, or
  \item source tagger.
\end{enumerate}
This classification is mathematically useful because it says where the tool enters
an actual differential inequality rather than only where it appears heuristically.

The same holds for proof memory. Once a lemma has been translated into budget form,
its content can be stored compactly as a transport or absorption rule without
repeating the entire derivation chain. This is the precise sense in which the
structure-first approach compresses thousands of calculations without erasing them:
it stores their net structural effect as a stable ledger law.

\subsection*{7. v\docversion\ readiness criterion}
The present section should be regarded as successful if the following can now be
done systematically:
\begin{itemize}
  \item every major future term can be assigned a channel budget rather than only a raw symbol;
  \item every appendix tool can be classified by its role in a budget inequality;
  \item probe arguments can be reported through a subledger instead of loose search language;
  \item residual terms can be traced by tagged sources rather than by informal comments; and
  \item later field equations can read these budget laws as compressed pre-equation structure.
\end{itemize}


This is the mathematical gain of v\docversion. The document now contains not only a
closure grammar and a channel grammar, but also a transport grammar that explains
how structural control is preserved as the proof unfolds.

\section*{v2.6.0 Lyapunov Ledger Closure and Forward-Invariant Regimes}
\addcontentsline{toc}{section}{v2.6.0 Lyapunov Ledger Closure and Forward-Invariant Regimes}

\begin{quote}
\textbf{Normalization note (v2.17.1 local review).}
This block is retained as a \emph{historical closure-stability and Lyapunov precursor}. Its role is supporting and developmental; the normalized proof-bearing closure chain is carried by the unfolding phases.
\end{quote}

The next logical step is to turn the budget grammar into a genuine closure device.
As long as one only writes transport inequalities, the proof still depends on a
stagewise reading. For the structure-first programme this is not yet strong enough.
One wants a single scalar functional that certifies that the admissible sector is
preserved under the whole transport chain. This is the role of the Lyapunov-ledger
closure introduced in v\docversion.

\subsection*{1. Weighted structural energy}
Choose positive weights
\[
\omega_j>0,
\qquad
\omega^{N}_j>0,
\qquad
\omega_R>0,
\qquad
\omega_\Gamma>0,
\]
and define the weighted structural energy
\[
\mathcal L
:=\sum_j \omega_j B_j + \sum_j \omega^{N}_j N_j + \omega_R\mathfrak R + \omega_\Gamma\Gamma.
\]
The aim is to choose the weights so that cross-channel transfer and residual spill
can be absorbed into one dissipative inequality. In this language, the ledger is no
longer just a bookkeeping device; it becomes the carrier of a monotonicity law.

\paragraph{Definition 1 (Lyapunov-admissible weight system).}
A family of weights is called Lyapunov-admissible if, after inserting the transport,
absorption, and source inequalities from the previous section, one obtains constants
$\kappa>0$ and $C_{\mathrm{tag}},C_{\mathrm{match}}\ge0$ such that
\[
\frac{d}{d\tau}\mathcal L
\le -\kappa \mathcal L
+ C_{\mathrm{tag}}\,\mathrm{Res}_{\mathrm{tag}}
+ C_{\mathrm{match}}\,\Delta_{\mathrm{match}}.
\]
This means that every admissible leakage path is paid for either by global decay or
by an explicitly tagged source of unresolved structure.

\subsection*{2. Weighted closure theorem}
The abstract transport statements from v2.5.0 are now compressed into a single
persistence theorem.

\paragraph{Theorem 1 (weighted ledger closure).}
Assume that the channel transport matrix has a strict damping gap, that every active
normal channel is absorption-efficient, and that residual production is source-tagged.
Then there exists a Lyapunov-admissible weight system for which
\[
\frac{d}{d\tau}\mathcal L
\le -\frac{\kappa}{2}\mathcal L
+ C_{\mathrm{tag}}\,\mathrm{Res}_{\mathrm{tag}}
+ C_{\mathrm{match}}\,\Delta_{\mathrm{match}}.
\]
In particular, if the tagged residual and probe mismatch stay integrably controlled,
then the whole derivation remains in a forward-invariant admissible regime.

\begin{proof}
Choose the weights recursively along the dominance order induced by the transfer
matrix, starting from the most dissipative channels and moving toward the weaker
ones. Because the damping gap is strict, the transfer contributions can be absorbed
by enlarging downstream weights finitely many times. The absorption-efficient normal
channels contribute additional negative terms for the dangerous normal mass. Residual
production enters only through tagged sources and therefore cannot create an
unidentified positive contribution. After summing the weighted inequalities and
absorbing the remaining off-diagonal terms, one obtains the stated estimate.
\end{proof}

This is the first point at which the document obtains a real forward-invariance
statement rather than only a local transport picture. The admissible proof sector is
now a dynamically protected cone.

\subsection*{3. Forward-invariant regime and restart stability}
One practical advantage of the Lyapunov-ledger closure is that it explains why the
proof can be resumed after interruptions without re-deriving the entire local chain.
If at some stage $\tau_0$ the state satisfies
\[
\mathcal L(\tau_0)\le M,
\qquad
\int_{\tau_0}^{\tau_1}\mathrm{Res}_{\mathrm{tag}}\,d\tau<\infty,
\qquad
\int_{\tau_0}^{\tau_1}\Delta_{\mathrm{match}}\,d\tau<\infty,
\]
then Gronwall's inequality implies a bound of the form
\[
\mathcal L(\tau)
\le e^{-\kappa(\tau-\tau_0)/2}M
+ \int_{\tau_0}^{\tau}e^{-\kappa(\tau-s)/2}
\bigl(C_{\mathrm{tag}}\,\mathrm{Res}_{\mathrm{tag}}(s)+C_{\mathrm{match}}\,\Delta_{\mathrm{match}}(s)\bigr)\,ds.
\]
Hence restarting the argument from an intermediate checkpoint is legitimate as long
as the checkpoint data are reported in ledger form. This gives a rigorous version of
proof memory: the document needs to preserve the weighted state, not the full local
micro-history.

\subsection*{4. Probe inheritance of invariance}
The two-probe picture inherits the same closure principle. Define the weighted probe
functional
\[
\mathcal L_{\mathrm{probe}}
:= \alpha_1 B^{(1)}_{\mathrm{tri}} + \alpha_2 B^{(2)}_{\mathrm{tri}}
+ \alpha_Q B^{(Q)}_{\perp} + \alpha_M B_{\mathrm{match}}
+ \alpha_R B^{\mathrm{probe}}_{\mathrm{res}}.
\]
If the probe transfer system has a strict damping gap and the mismatch source remains
ledger-tagged, then the same weighting argument yields
\[
\frac{d}{d\tau}\mathcal L_{\mathrm{probe}}
\le -\kappa_{\mathrm{probe}}\mathcal L_{\mathrm{probe}}
+ C^{\mathrm{probe}}_{\mathrm{tag}}\,\mathrm{Res}^{\mathrm{probe}}_{\mathrm{tag}}
+ C^{\mathrm{probe}}_{\mathrm{match}}\,\Delta_{\mathrm{match}}.
\]
Thus the probe sector is not merely compatible with the main ledger; it inherits the
same forward-invariant discipline. The search geometry therefore remains structurally
confined rather than opening an uncontrolled side channel.

\subsection*{5. Structural consequence for later field equations}
The future field-equation layer can now be read more sharply. A field equation will
not merely translate channel budgets into continuous variables; it must preserve the
forward-invariant regime induced by $\mathcal L$. In other words, admissible field
variables should be chosen so that the induced energy identity mirrors the weighted
ledger closure. This provides a direct criterion for accepting or rejecting later
field-equation ansaetze.

\subsection*{6. Consequence for appendix tools}
The tool appendix can be refined once more. Besides the roles introduced in v2.5.0,
a strong tool may now be marked as one of the following:
\begin{enumerate}[label=(\alph*)]
  \item weight constructor,
  \item off-diagonal absorber,
  \item invariant-region verifier, or
  \item restart certificate.
\end{enumerate}
This is useful because it identifies precisely which lemmas contribute to the global
Lyapunov closure and which only support local transport estimates.

\subsection*{7. v\docversion\ readiness criterion}
The present step should be regarded as successful if the following can now be done:
\begin{itemize}
  \item the entire admissible sector can be encoded in one weighted ledger functional;
  \item restart points can be justified by checkpoint data in ledger form;
  \item probe arguments inherit the same invariant-region control as the main proof;
  \item later field equations can be tested against a precise closure criterion; and
  \item appendix tools can be sorted by whether they contribute to local transport or global invariance.
\end{itemize}

This is the gain of v\docversion. The companion document now carries not only staged
closure, channel specialization, and budget transport, but also a genuine
forward-invariant structural regime that stabilizes the proof under continuation,
probe comparison, and later equation-level readout.


\section*{v2.7.0 Field-Energy Realization and Semigroup Readout Core}
\addcontentsline{toc}{section}{v2.7.0 Field-Energy Realization and Semigroup Readout Core}

\begin{quote}
\textbf{Normalization note (v2.17.1 local review).}
This block is retained as a \emph{historical realization and semigroup-readout precursor}. It supports the later field/probe realization layer and should not be read as a second independent primary core.
\end{quote}

The next structural step after forward-invariant ledger closure is to show that the
weighted ledger is not only a bookkeeping device but already the shadow of a genuine
field-energy balance. Version~\docversion\ therefore adds a realization layer: the
channel budgets are interpreted as coarse observables of an underlying evolution
system, and admissibility is reformulated as dissipative semigroup compatibility.
This still does not claim the final field equations, but it fixes the exact kind of
continuous dynamics those equations must generate if they are to remain faithful to
the structural proof architecture.

\subsection*{1. Abstract state space and energy readout}
Let
\[
X := X_{\mathrm{tri}}\oplus X_{\perp}\oplus X_{\mathrm{meas}}\oplus X_{\mathrm{res}}
\]
be an abstract Banach or Hilbert state space carrying the channel-resolved structural
variables. We write a state as
\[
U=(u_{\mathrm{tri}},u_{\perp},u_{\mathrm{meas}},u_{\mathrm{res}}).
\]
The ledger variables are no longer treated as primitive numbers but as readout
functionals
\[
B_i(U)\ge 0,
\qquad i\in\{\mathrm{tri},\perp,\mathrm{meas},\mathrm{res}\},
\]
with the weighted energy
\[
\mathcal E(U):=\alpha_{\mathrm{tri}}B_{\mathrm{tri}}(U)
+\alpha_{\perp}B_{\perp}(U)
+\alpha_{\mathrm{meas}}B_{\mathrm{meas}}(U)
+\alpha_{\mathrm{res}}B_{\mathrm{res}}(U).
\]
The content of this step is that $\mathcal E$ should reproduce the Lyapunov ledger
$\mathcal L$ up to uniform equivalence on the admissible sector. Thus the ledger is
interpreted as a compressed energy observable of the underlying state rather than an
external annotation.

\subsection*{2. Dissipative generator Ansatz}
Assume that the channel evolution is generated by an abstract equation
\[
\partial_\tau U = \mathcal A U + \mathcal N(U) + \mathcal S_{\mathrm{tag}}(\tau),
\]
where $\mathcal A$ is the linear transport-dissipation generator, $\mathcal N$ is a
structured nonlinear coupling compatible with HC confinement, and
$\mathcal S_{\mathrm{tag}}$ collects the tagged residual sources. The admissibility
requirement from the earlier sections now becomes the statement that
$\mathcal A+\mathcal N$ is energy-dissipative modulo tagged production and match
error, namely
\[
\frac{d}{d\tau}\mathcal E(U(\tau))
\le -c_0\mathcal E(U(\tau))
+ C_{\mathrm{tag}}\,\mathrm{Res}_{\mathrm{tag}}(\tau)
+ C_{\mathrm{match}}\,\Delta_{\mathrm{match}}(\tau)
\]
for every admissible trajectory. This is the semigroup form of the weighted ledger
closure derived in v2.6.0.

\subsection*{3. Realization principle}
The structural point can now be isolated as a single principle.

\paragraph{Principle 1 (field-energy realization).}
An eventual field equation is acceptable only if its induced state evolution admits a
channel readout $U\mapsto (B_i(U))_i$ and a weighted energy $\mathcal E$ such that:
\begin{enumerate}[label=(\alph*)]
  \item $\mathcal E$ is uniformly equivalent to the admissible ledger functional $\mathcal L$;
  \item the induced evolution preserves the forward-invariant admissible sector;
  \item all positive production terms are either dissipatively absorbed or explicitly tagged; and
  \item probe mismatch enters only through a controlled source term $\Delta_{\mathrm{match}}$.
\end{enumerate}
Any ansatz failing one of these four points is structurally too weak, even if it
looks formally plausible at equation level.

\subsection*{4. Semigroup consequence and proof memory}
If the above dissipative estimate holds, then the evolution operator $S(\tau)$
generated by the admissible dynamics satisfies an estimate of the form
\[
\mathcal E(S(\tau)U_0)
\le e^{-c_0\tau}\mathcal E(U_0)
+\int_0^\tau e^{-c_0(\tau-s)}
\bigl(C_{\mathrm{tag}}\,\mathrm{Res}_{\mathrm{tag}}(s)
+ C_{\mathrm{match}}\,\Delta_{\mathrm{match}}(s)\bigr)\,ds.
\]
Hence proof memory can now be reformulated in semigroup language: restarting from a
checkpoint is valid because the future evolution depends continuously on the current
energy state and the tagged source history, not on the unrecoverable microscopic path.
This upgrades the earlier restart argument into an abstract continuous-time stability
statement.

\subsection*{5. Probe-sector realization}
The same realization principle applies to the two-probe geometry. If
\[
U_{\mathrm{probe}}=(U^{(1)},U^{(2)},U^{(Q)},U^{(M)},U^{(R)})
\]
denotes the probe state and $\mathcal E_{\mathrm{probe}}$ the corresponding weighted
energy, then admissibility requires
\[
\frac{d}{d\tau}\mathcal E_{\mathrm{probe}}
\le -c_{\mathrm{probe}}\mathcal E_{\mathrm{probe}}
+ C^{\mathrm{probe}}_{\mathrm{tag}}\,\mathrm{Res}^{\mathrm{probe}}_{\mathrm{tag}}
+ C^{\mathrm{probe}}_{\mathrm{match}}\,\Delta_{\mathrm{match}}.
\]
Thus the probe mechanism is now seen as a dissipative comparison flow living in the
same semigroup class as the main derivation. This makes the search geometry more than
a metaphor: it becomes an admissible evolution subsystem with inherited energy
control.

\subsection*{6. Consequence for future field equations and appendix tools}
The field-equation programme is sharpened once again. A candidate PDE or abstract
operator equation must now be checked in the following order:
\begin{enumerate}[label=(\arabic*)]
  \item identify the state space and channel decomposition;
  \item construct the ledger readout functionals $B_i(U)$;
  \item verify equivalence between readout energy $\mathcal E$ and structural ledger $\mathcal L$;
  \item prove dissipativity modulo tagged sources; and
  \item verify probe inheritance and restart compatibility.
\end{enumerate}
Accordingly, appendix tools can receive one further classification layer. A strong
tool may now be marked as a state-space constructor, energy comparator, semigroup
stability lemma, or source-tagging certificate. This helps separate tools that only
support symbolic rearrangement from those that certify genuine dynamic admissibility.

\subsection*{7. v\docversion\ readiness criterion}
The present step should be regarded as successful if the following can now be done:
\begin{itemize}
  \item the ledger is interpretable as a uniformly equivalent field-energy observable;
  \item admissibility is reformulated as dissipative semigroup compatibility;
  \item proof memory is supported by a continuous-time stability estimate;
  \item the two-probe sector is embedded into the same realization framework; and
  \item future field equations can be accepted or rejected by an explicit realization checklist.
\end{itemize}

This is the gain of v\docversion. The companion monolith now carries not only
closure, channel specialization, transport dynamics, and forward invariance, but also
a first abstract field-energy realization principle. The later field-equation layer is
therefore no longer free-form: it has to generate the same dissipative proof memory,
probe inheritance, and tagged-source discipline already fixed at the structural level.


\section*{v2.8.0 Spectral Admissibility and Resolvent Gatekeeping Core}
\addcontentsline{toc}{section}{v2.8.0 Spectral Admissibility and Resolvent Gatekeeping Core}

\begin{quote}
\textbf{Normalization note (v2.17.1 local review).}
This block is retained as a \emph{historical spectral-admissibility precursor}. Its gatekeeping role is preserved, but the normalized admissibility logic is now carried by the semantic/tensor and grouped-layer unfolding phases.
\end{quote}

The semigroup realization from v2.7.0 still leaves one decisive structural question
open: which generators are admissible before one even starts evolving the system? In
this version the answer is sharpened by adding a spectral gatekeeping layer. The
state evolution is not accepted merely because it can be written as an abstract flow;
it must also possess a resolvent and spectrum compatible with the ledger discipline,
HC confinement, tagged-source accounting, and probe inheritance.

\subsection*{1. Spectral admissibility principle}
Let $\mathfrak X$ be the admissible state space from the previous realization step and
let $\mathcal G$ denote the effective generator of the dynamics, including its
controlled tagged-source coupling. We call $\mathcal G$ \emph{spectrally admissible}
if there exist $\omega_*>0$ and $C_*>0$ such that for every $\lambda$ with
$\Re \lambda\ge -\omega_*$ the resolvent exists on the admissible sector and obeys
\[
\| (\lambda-\mathcal G)^{-1} \|_{\mathfrak X\to\mathfrak X}
\le \frac{C_*}{1+|\lambda|},
\]
after projection to the HC-confined channel space. This estimate expresses that the
generator does not hide uncontrolled growth modes behind formal notation. The ledger
control from earlier versions is therefore promoted into a spectral entrance test.

\paragraph{Principle 2 (spectral gatekeeping).}
A candidate field equation or abstract operator model is structurally admissible only
if its effective generator is spectrally admissible in the above sense and the
corresponding resolvent bounds are compatible with the tagged-source decomposition.
The proof programme may therefore reject an operator before any detailed PDE analysis
if the spectrum already violates ledger closure.

\subsection*{2. Resolvent--ledger compatibility}
The weighted energy $\mathcal E$ from v2.7.0 should now be stable not only along the
flow but already at the resolvent level. Concretely, admissibility requires a bound
of the schematic form
\[
\mathcal E\bigl((\lambda-\mathcal G)^{-1}F\bigr)
\le C_{\mathrm{res}}
\Bigl( \mathcal E(F)
+ \mathrm{Tag}(F)
+ \mathrm{Match}(F)\Bigr)
\]
for every admissible forcing term $F$ and every $\Re\lambda\ge -\omega_*$. Here
$\mathrm{Tag}(F)$ denotes the explicitly source-tagged part and
$\mathrm{Match}(F)$ the mismatch component inherited from probe comparison. This is
the spectral analogue of the semigroup energy estimate: the resolvent may amplify
forcing, but only in a way that remains ledger-readable.

\subsection*{3. HC confinement as spectral projector}
The HC mechanism is now reinterpreted one step more sharply. Earlier versions treated
it as the structural compensator that confines evolution to the septinion closure.
At spectral level this becomes the statement that there exists a projector
$\Pi_{\mathrm{HC}}$ with
\[
(\lambda-\mathcal G)^{-1}
= \Pi_{\mathrm{HC}}(\lambda-\mathcal G)^{-1}\Pi_{\mathrm{HC}}
+ \mathcal R_{\mathrm{tag}}(\lambda),
\]
where $\mathcal R_{\mathrm{tag}}(\lambda)$ is either negligible in the admissible
regime or explicitly source-tagged. Hence uncontrolled extra channels are not merely
suppressed dynamically; they are prevented from entering the admissible resolvent
class except through a tagged remainder. This gives a clean spectral formulation of
HC confinement.

\subsection*{4. Spectral gap and restart robustness}
If the admissible generator satisfies the above bounds together with a positive gap
between the dominant admissible spectrum and the forbidden sector, then restart
stability acquires a stronger meaning. A checkpoint is not only dynamically safe but
spectrally robust: small reconstruction errors remain trapped inside the same
resolvent class and do not move the system into a hidden unstable branch. In this
sense proof memory is upgraded from semigroup continuity to spectral persistence.

\subsection*{5. Probe-sector comparison spectrum}
The two-probe subsystem inherits the same criterion. Let
$\mathcal G_{\mathrm{probe}}$ be the effective probe generator acting on the probe
state space. Then admissibility requires a comparison estimate
\[
\| (\lambda-\mathcal G_{\mathrm{probe}})^{-1} \| 
\le C_{\mathrm{probe,res}}(1+|\lambda|)^{-1},
\qquad \Re\lambda\ge -\omega_{\mathrm{probe}},
\]
with constants controlled by the same ledger hierarchy and mismatch tags. The search
geometry therefore becomes spectrally testable: a probe construction is not accepted
merely because it heuristically mirrors the main evolution, but because its spectrum
remains subordinate to the same admissible comparison regime.

\subsection*{6. Consequence for future field equations and appendix tools}
The field-equation admission checklist is sharpened again. A candidate model should
now be tested in the following order:
\begin{enumerate}[label=(\arabic*)]
  \item identify state space and admissible channel projector;
  \item construct the energy and ledger readout;
  \item verify semigroup dissipativity modulo tagged sources;
  \item prove spectral admissibility and resolvent compatibility; and
  \item verify probe-spectrum inheritance and restart robustness.
\end{enumerate}
This also refines the appendix tools. Strong tools may now additionally be marked as
resolvent estimator, spectral projector certificate, gap witness, or restart
persistence lemma. These labels distinguish genuinely admissibility-producing tools
from merely formal symbolic aids.

\subsection*{7. v\docversion\ readiness criterion}
The present step should be regarded as successful if the following can now be done:
\begin{itemize}
  \item admissible generators can be filtered by a spectral entrance test;
  \item resolvent bounds can be read in the same ledger language as the semigroup flow;
  \item HC confinement is expressible as a spectral projector statement;
  \item restart stability is strengthened to spectral persistence; and
  \item the probe sector inherits a controlled comparison spectrum.
\end{itemize}

This is the gain of v\docversion. The companion monolith now carries not only
closure, channel specialization, transport dynamics, forward invariance, and
field-energy realization, but also a first spectral admissibility layer. Future
field equations are therefore constrained twice: they must generate the correct
ledger-dissipative semigroup and they must belong to the correct resolvent class
before they are even admitted into the structural programme.


\section*{v2.9.0 Spectral Readout Filters and Trace-Compatible Mode Selection Core}
\addcontentsline{toc}{section}{v2.9.0 Spectral Readout Filters and Trace-Compatible Mode Selection Core}

\begin{quote}
\textbf{Normalization note (v2.17.1 local review).}
This block is retained as a \emph{historical spectral-readout and trace-filter precursor}. It contributes proof memory and specialized readout discipline, but should be read through the later normalized readout and interface phases.
\end{quote}

The spectral admissibility layer from v2.8.0 decides which generators may enter the
programme at all. The next structural question is subtler: once a generator is
admissible, which parts of its spectrum are actually readable in a way compatible
with the ledger, HC confinement, probe comparison, and later trace-style summary
principles? In this version the monolith therefore adds a spectral readout layer.
Not every admissible mode is automatically promoted to a structural observable. A
mode must also survive a filter test that distinguishes stable, tagged, and
non-readable contributions.

\subsection*{1. Readout filter principle}
Let $\mathcal G$ be spectrally admissible on the HC-confined state space
$\mathfrak X$, and let $\Sigma_{\mathrm{adm}}(\mathcal G)$ denote the admissible part
of the spectrum. We introduce a family of bounded readout filters
$\{\mathcal F_\alpha\}_{\alpha\in A}$ acting on the admissible sector, where each
$\mathcal F_\alpha$ is designed to isolate one structural mode class: triadic,
orthogonal, measurement, residual, or probe-induced. A mode contribution is called
\emph{trace-compatible} if there exists at least one filter $\mathcal F_\alpha$ such
that
\[
\mathcal F_\alpha \Pi_{\mathrm{HC}} u = \Pi_{\mathrm{HC}} u + r_\alpha,
\qquad
\|r_\alpha\|_{\mathfrak X} \le \varepsilon_\alpha \|u\|_{\mathfrak X},
\]
with either small remainder $r_\alpha$ or explicitly tagged remainder. Thus the
readout is not merely spectral isolation but structural readability modulo tagged
error.

\paragraph{Principle 3 (filtered observability).}
A spectral component may be used in the proof architecture only if it is both
spectrally admissible and trace-compatible under an approved readout filter family.
This prevents uncontrolled spectral clutter from being mistaken for genuine
structural information.

\subsection*{2. Trace-compatible decomposition}
The admissible spectrum should now admit a decomposition of the schematic form
\[
\Sigma_{\mathrm{adm}}(\mathcal G)
= \Sigma_{\mathrm{read}} \cup \Sigma_{\mathrm{tag}} \cup \Sigma_{\mathrm{mute}},
\]
where readable modes lie in $\Sigma_{\mathrm{read}}$, tagged but non-primary modes lie
in $\Sigma_{\mathrm{tag}}$, and the remainder $\Sigma_{\mathrm{mute}}$ is admissible but
not allowed to enter the principal structural trace. The point is not that mute
modes vanish, but that they cannot serve as primary carriers of theorem-level
readout. This sharpens the difference between existence and interpretability.

\subsection*{3. HC-filter compatibility}
The HC projector from v2.8.0 is now required to commute with readout at least up to
controlled tagged remainder. Concretely, one seeks estimates of the form
\[
\mathcal F_\alpha \Pi_{\mathrm{HC}}
= \Pi_{\mathrm{HC}} \mathcal F_\alpha + \mathcal T_\alpha,
\]
where $\mathcal T_\alpha$ is either small in the admissible norm or belongs to a
source-tagged correction class. This means HC confinement is preserved under mode
selection: filtering is not allowed to re-open forbidden channels except through a
ledger-visible correction.

\subsection*{4. Mode budgets and readable trace mass}
The budget language from v2.5.0 now acquires a spectral refinement. For every
approved filter $\mathcal F_\alpha$ define the readable trace mass
\[
\mathfrak M_\alpha(u) := \mathcal E(\mathcal F_\alpha u)
\]
with $\mathcal E$ the field-energy functional from v2.7.0. Structural readability
requires a balance inequality of the schematic form
\[
\sum_{\alpha\in A_{\mathrm{read}}} \mathfrak M_\alpha(u)
\le C_{\mathrm{tr}}
\Bigl( \mathcal E(u) + \mathrm{Tag}(u) + \mathrm{Leak}(u) \Bigr).
\]
Hence readable spectral mass cannot exceed the ledger budget except through already
visible tagged or leakage terms. This gives a quantitative bridge between spectral
filtering and earlier channel transport estimates.

\subsection*{5. Probe-sector mode inheritance}
For the two-probe subsystem the same distinction becomes a comparison principle.
Let $\mathcal G_{\mathrm{probe}}$ be the effective probe generator and
$\mathcal F^{\mathrm{probe}}_\beta$ its admissible filter family. Then every readable
probe mode should either descend from a readable ambient mode or be marked as a
probe-only tagged correction. In particular, one asks for comparison bounds of the
form
\[
\mathfrak M^{\mathrm{probe}}_\beta(v)
\le C_{\mathrm{cmp}}
\bigl( \mathfrak M_\alpha(u) + \mathrm{Mismatch}(u,v) \bigr),
\]
for matched ambient/probe states $(u,v)$. This prevents the probe sector from
manufacturing fake observables that have no ambient structural parent.

\subsection*{6. Consequence for future field equations and appendix tools}
The admission checklist for candidate field equations is now extended once more:
\begin{enumerate}[label=(\arabic*)]
  \item verify semigroup dissipativity and ledger closure;
  \item verify spectral admissibility and resolvent compatibility;
  \item construct an HC-compatible readout filter family;
  \item separate readable, tagged, and mute spectral sectors; and
  \item prove probe inheritance of readable modes.
\end{enumerate}
The appendix tool taxonomy is correspondingly refined. Besides resolvent and gap
certificates, a strong tool may now also be labelled as readout filter constructor,
trace-mass comparator, mute-sector exclusion lemma, or probe-inheritance witness.
These labels indicate that the tool produces not only existence information but also
structural legibility.

\subsection*{7. v\docversion\ readiness criterion}
The present step should be regarded as successful if the following can now be done:
\begin{itemize}
  \item admissible spectra can be split into readable, tagged, and mute parts;
  \item HC confinement survives mode filtering modulo controlled tagged remainder;
  \item readable trace mass is bounded in the same ledger language as channel budgets;
  \item probe readout inherits ambient readable modes rather than inventing new ones; and
  \item future field equations can be screened not only for existence, but for structural legibility.
\end{itemize}

This is the gain of v\docversion. The companion monolith now carries not only
closure, channel specialization, transport dynamics, forward invariance,
field-energy realization, semigroup readout, and spectral admissibility, but also a
first filtered-observability layer. Future field equations are therefore constrained
three times: they must generate the correct ledger-dissipative semigroup, they must
belong to the correct resolvent class, and their admissible modes must pass a
trace-compatible readout filter before they can count as structural content.


\section*{v2.10.0 Trace Reconstruction and Observable Extraction Core}
\addcontentsline{toc}{section}{v2.10.0 Trace Reconstruction and Observable Extraction Core}

\begin{quote}
\textbf{Normalization note (v2.17.1 local review).}
This block is retained as a \emph{historical reconstruction and observable-extraction precursor}. Its reconstruction language remains useful, but the normalized proof-bearing architecture now lies in the unfolding phases and the grouped transport logic.
\end{quote}

The filtered-observability layer from v2.9.0 separates admissible modes into
readable, tagged, and mute sectors. The next structural task is no longer only to
screen admissible modes, but to reconstruct theorem-level observables from the
readable part without double counting, without reopening forbidden channels, and
without losing the ledger discipline enforced by HC confinement. The present step
therefore adds a trace-reconstruction layer. It answers the question: once a mode
has passed admissibility and readability tests, how is it promoted to an actual
observable in the companion proof architecture?

\subsection*{1. Reconstruction operator for readable modes}
Let $\{\mathcal F_\alpha\}_{\alpha\in A_{\mathrm{read}}}$ be the approved readable
filter family from v2.9.0. We now introduce a reconstruction operator
\[
\mathcal R: \bigoplus_{\alpha\in A_{\mathrm{read}}} \mathfrak X_\alpha \to \mathfrak O,
\]
where $\mathfrak X_\alpha = \mathcal F_\alpha \mathfrak X$ denotes the readable mode
sector and $\mathfrak O$ is the structural observable space. The role of
$\mathcal R$ is not arbitrary synthesis: it is constrained reconstruction. For a
state $u\in\mathfrak X$ one seeks a decomposition
\[
\mathrm{Obs}(u) = \mathcal R\bigl((\mathcal F_\alpha u)_{\alpha\in A_{\mathrm{read}}}\bigr)
\]
such that every observable component remains HC-confined, ledger-bounded, and
source-tagged whenever the reconstruction is not exact. Thus observable extraction
is treated as a disciplined readout map rather than an unrestricted projection.

\paragraph{Principle 4 (reconstructive observability).}
A readable mode contributes to theorem-level content only after passing through a
reconstruction operator whose output is compatible with HC confinement, channel
budgets, and source tags. Readability alone is therefore necessary but not yet
sufficient for observable status.

\subsection*{2. Observable extraction at theorem level}
For each admissible theorem-context $\Theta$ let $\mathfrak O_\Theta$ denote the class
of observables allowed in that context. One then requires extraction estimates of
schematic form
\[
\|\mathcal R_\Theta((\mathcal F_\alpha u)_\alpha)\|_{\mathfrak O_\Theta}
\le C_\Theta
\Bigl(
\sum_{\alpha\in A_{\mathrm{read}}} \mathfrak M_\alpha(u)
+ \mathrm{Tag}(u)+\mathrm{Leak}(u)
\Bigr),
\]
with $\mathfrak M_\alpha$ the readable trace masses from v2.9.0. This means an
observable is only legitimate when its theorem-level norm is controlled by the same
ledger quantities already used to regulate channel transport and spectral readout.
The companion proof therefore acquires an explicit extraction stage between readable
modes and theorem statements.

\subsection*{3. Overlap ledger and anti-double-counting rule}
Once multiple filters contribute to one observable quantity, overlap must be made
visible. Introduce an overlap ledger
\[
\mathrm{OL}(u)= \sum_{\alpha\neq\beta}
\omega_{\alpha\beta}\,\langle \mathcal F_\alpha u,\mathcal F_\beta u\rangle_{\mathfrak X},
\]
where the coefficients $\omega_{\alpha\beta}$ encode admissible correlation or
subtraction rules. The purpose of $\mathrm{OL}(u)$ is not to forbid overlap but to
prevent silent duplication of structural mass. Observable reconstruction is required
to satisfy a compensated estimate of the form
\[
\|\mathrm{Obs}(u)\|_{\mathfrak O}
\le C_{\mathrm{obs}}
\Bigl(
\sum_{\alpha\in A_{\mathrm{read}}} \mathfrak M_\alpha(u)
- \mathrm{OL}(u)
+ \mathrm{Tag}(u)+\mathrm{Leak}(u)
\Bigr),
\]
whenever the overlap ledger is well defined. Thus one can read several filtered
channels at once without pretending they were disjoint when they are not.

\subsection*{4. HC-stable observable closure}
The reconstructed observable sector must itself remain HC-stable. Concretely, one
requires that
\[
\Pi_{\mathrm{HC}}\,\mathrm{Obs}(u)=\mathrm{Obs}(u)+\tau(u),
\qquad
\|\tau(u)\|_{\mathfrak O}\le C_{\mathrm{HC}}
\bigl(\mathrm{Tag}(u)+\mathrm{Leak}(u)\bigr),
\]
so that reconstruction does not reactivate forbidden directions except through
ledger-visible correction terms. In this sense HC confinement becomes not only a
state-space condition and not only a filter-compatibility condition, but now also an
observable-closure condition. Only outputs that survive this test may be used as
stable structural readings.

\subsection*{5. Probe-sector reconstruction inheritance}
For the two-probe subsystem the same reconstruction principle must descend to the
probe readout. Let $\mathcal R^{\mathrm{probe}}$ be the probe reconstruction operator.
Then every probe observable should either be inherited from an ambient observable
through the descent chain or be explicitly tagged as probe-local correction. One
asks schematically for bounds of the form
\[
\|\mathrm{Obs}^{\mathrm{probe}}(v)\|_{\mathfrak O_{\mathrm{probe}}}
\le C_{\mathrm{pr}}
\bigl(
\|\mathrm{Obs}(u)\|_{\mathfrak O}
+\mathrm{Mismatch}(u,v)
\bigr),
\]
for matched ambient/probe states $(u,v)$. This ensures the probe sector remains an
instrument of structural descent rather than an independent source of untracked
observables.

\subsection*{6. Consequence for field equations and appendix tools}
The admission checklist for future candidate field equations is therefore extended
once more:
\begin{enumerate}[label=(\arabic*)]
  \item verify semigroup dissipativity and ledger closure;
  \item verify spectral admissibility and resolvent compatibility;
  \item verify readable mode separation under HC-compatible filters;
  \item construct a reconstruction operator for theorem-level observables; and
  \item bound overlap, leakage, and probe descent in a visible ledger form.
\end{enumerate}
The appendix tool taxonomy is refined accordingly. Besides resolvent estimators,
readout filters, and gap witnesses, a strong tool may now also be labelled as
reconstruction operator, observable extractor, overlap-ledger bound, HC-stability
witness, or probe-descent certificate. These labels mark tools that do not merely
prove admissibility, but actually promote admissible readable content into stable
observables.

\subsection*{7. v\docversion\ readiness criterion}
The present step should be regarded as successful if the following can now be done:
\begin{itemize}
  \item readable spectral content can be reconstructed into theorem-level observables;
  \item overlaps between readable channels are accounted for instead of silently duplicated;
  \item observable extraction remains HC-stable modulo visible tagged/leakage correction;
  \item probe observables descend from ambient observables rather than appearing autonomously; and
  \item future field equations can be screened for observable reconstruction, not merely for readable spectra.
\end{itemize}

This is the gain of v\docversion. The companion monolith now carries not only
closure, channel specialization, transport dynamics, forward invariance,
field-energy realization, semigroup readout, spectral admissibility, and filtered
observability, but also a first disciplined reconstruction layer. Future field
equations are therefore constrained four times: they must generate the correct
ledger-dissipative semigroup, belong to the correct resolvent class, pass a
trace-compatible readout filter, and finally admit HC-stable observable
reconstruction before they can count as theorem-level structural content.


\section*{v2.11.0 Multi-Readout Consistency and Fusion Core}
\addcontentsline{toc}{section}{v2.11.0 Multi-Readout Consistency and Fusion Core}

\begin{quote}
\textbf{Normalization note (v2.17.1 local review).}
This block is retained as a \emph{historical multi-readout and fusion precursor}. It remains relevant as supporting proof memory, but no longer functions as an independent primary structural core in the normalized architecture.
\end{quote}

The reconstruction layer from v2.10.0 promotes readable modes into theorem-level
observables. The next structural question is what happens when several
reconstructions are all individually admissible but are intended to describe one
and the same structural content. The task of v\docversion\ is therefore not merely
to extract observables, but to fuse them without creating contradictory readouts,
without hiding overlap behind notation, and without allowing probe-local views to
break the ambient closure architecture. This introduces a multi-readout
consistency layer.

\subsection*{1. Readout families and consistency targets}
Let $\{\mathcal R_j\}_{j\in J}$ be a family of admissible reconstruction maps,
possibly associated with different filter systems, theorem contexts, or probe
resolutions. For a given state $u$ one obtains a readout family
\[
\mathrm{Obs}_j(u)=\mathcal R_j\bigl((\mathcal F^{(j)}_\alpha u)_\alpha\bigr).
\]
The purpose of the new layer is to decide when the collection
$\{\mathrm{Obs}_j(u)\}_{j\in J}$ should be regarded as several compatible views of one
structural object rather than several unrelated observables. Compatibility is
measured against the same closure, budget, and tagging rules that already govern
the single-readout setting.

\paragraph{Principle 5 (multi-readout consistency).}
A family of theorem-level observables may be fused only when their mutual
discrepancies are either controlled by a visible ledger term or assigned to
explicit source tags. Silent disagreement is not an allowed mode of structural
multiplicity.

\subsection*{2. Consistency ledger and fusion defect}
Introduce a fusion-defect functional
\[
\mathfrak D_{\mathrm{fus}}(u)
:= \sum_{j<k}\nu_{jk}\,\|\mathrm{Obs}_j(u)-\Gamma_{jk}\mathrm{Obs}_k(u)\|_{\mathfrak O_*},
\]
where $\Gamma_{jk}$ transports the $k$-th observable into the comparison frame of the
$j$-th observable and the coefficients $\nu_{jk}\ge 0$ encode structural relevance.
The associated consistency ledger is then
\[
\mathrm{CL}(u):=\mathfrak D_{\mathrm{fus}}(u)+\mathrm{Tag}(u)+\mathrm{Leak}(u).
\]
A family is called fusion-admissible when $\mathrm{CL}(u)$ stays below the threshold
prescribed by the theorem context. Thus disagreement is not denied; it is made
visible, budgeted, and therefore manageable.

\subsection*{3. Fused observable and anti-inflation rule}
Whenever the family is fusion-admissible, define a fused observable
\[
\mathrm{Obs}_{\mathrm{fus}}(u)=\sum_{j\in J}\lambda_j\,\Gamma_j\mathrm{Obs}_j(u)-\Xi(u),
\]
where the coefficients $\lambda_j$ are admissible weights, $\Gamma_j$ transports to a
common observable frame, and $\Xi(u)$ is an anti-inflation compensator correcting
for visible overlap and recorded fusion defect. The fused observable is required
to satisfy an estimate of the form
\[
\|\mathrm{Obs}_{\mathrm{fus}}(u)\|_{\mathfrak O_*}
\le C_{\mathrm{fus}}
\bigl(
\sum_{j\in J}\|\mathrm{Obs}_j(u)\|_{\mathfrak O_j}
-\mathrm{OL}_{\mathrm{fus}}(u)
+\mathrm{CL}(u)
\bigr),
\]
so that fusion cannot enlarge theorem-level content by counting the same readable
structure several times through nominally different channels.

\subsection*{4. HC-stable fusion and theorem transport}
The fused observable must still respect HC confinement. One therefore asks for
\[
\Pi_{\mathrm{HC}}\,\mathrm{Obs}_{\mathrm{fus}}(u)
=\mathrm{Obs}_{\mathrm{fus}}(u)+\rho(u),
\qquad
\|\rho(u)\|_{\mathfrak O_*}
\le C_{\rho}\,\mathrm{CL}(u).
\]
This means that multi-readout fusion is accepted only if the residual mismatch is
fully visible in the same ledger that already recorded comparison defects. In
addition, theorem transport must preserve fusion status: if a fused observable is
moved into a lemma, table, or appendix tool, the same fusion certificate must
remain available after transfer.

\subsection*{5. Probe/ambient coherence under fusion}
In the two-probe sector one may have ambient and probe reconstructions of the same
structural feature. The new requirement is that the probe fusion defect be bounded
by ambient fusion defect plus descent mismatch:
\[
\mathfrak D^{\mathrm{probe}}_{\mathrm{fus}}(v)
\le C_{\mathrm{coh}}
\bigl(
\mathfrak D_{\mathrm{fus}}(u)+\mathrm{Mismatch}(u,v)
\bigr).
\]
Hence probe readouts do not become a second autonomous truth layer; they inherit
coherence from the ambient structure, up to visible descent error.

\subsection*{6. Consequence for appendix tools}
The appendix tool taxonomy is refined again. A strong appendix item may now be
labelled as a fusion certificate, consistency-ledger estimate, anti-inflation
compensator, theorem-transport witness, or probe-coherence bound. This fits the
user's structural preference that tools be collected as reusable proof interfaces
rather than scattered informally through the running text.

\subsection*{7. v\docversion\ readiness criterion}
The present step should be regarded as successful if the following can now be
done:
\begin{itemize}
  \item several admissible reconstructions can be compared in one explicit ledger;
  \item disagreements between readouts are tagged or budgeted instead of hidden;
  \item fused observables remain protected against overlap inflation;
  \item probe and ambient readouts satisfy a visible coherence estimate; and
  \item appendix transfer preserves not only observables, but also their fusion certificates.
\end{itemize}

This is the gain of v\docversion. The companion monolith now carries not only
closure, channel specialization, transport dynamics, forward invariance,
field-energy realization, semigroup readout, spectral admissibility, filtered
observability, and trace reconstruction, but also a first disciplined fusion
layer for theorem-level observables. Future field equations are therefore
constrained five times: they must generate the correct ledger-dissipative
semigroup, belong to the correct resolvent class, pass a trace-compatible readout
filter, admit HC-stable observable reconstruction, and finally support
multi-readout fusion without hidden inconsistency.


% ------------------------------------------------------------------
% Appendix: Structural Tools and Regime Tests (v2.27.4 integration)
% ------------------------------------------------------------------

\section{Structural Tools and Regime Tests}
\label{app:structural-tools-regime-tests}
\addcontentsline{toc}{section}{Appendix: Structural Tools and Regime Tests}

This appendix collects the main structural tools extracted from the
proof architecture. Their purpose is not to replace the main
derivations, but to normalize frequently recurring arguments into
reusable tests and reduction principles.

\subsection*{Sector and closure tools}

\paragraph{Tool (Closure-transfer principle).}
Let $X$ be any operator on the septonic probe space. Its admissible
reduction is
\[
  X_{\mathrm{adm}} := \hat{P}_{C_7} X \hat{P}_{C_7}.
\]
For admissible states $\Psi \in \operatorname{Ran}(\hat{P}_{C_7})$, the
physically relevant action of $X$ is determined by $X_{\mathrm{adm}}$,
while the complementary blocks measure off-sector leakage.

\paragraph{Tool (Equivalent closure criteria).}
The following are equivalent
(cf.\ Lemma~\ref{lem:closure-forcing-lemma}):
\begin{enumerate}[label=(\roman*)]
  \item $\mathfrak{R}_C = 0$;
  \item $[\mathcal{M}_{16}, \hat{P}_{C_7}] = 0$;
  \item $\mathcal{M}_{16}$ preserves the admissible sector and its complement.
\end{enumerate}
Hence exact closure is equivalent to block preservation, i.e.\
block-diagonalization of $\mathcal{M}_{16}$ with respect to
$\mathcal{H}_{\mathrm{sept}} = \hat{P}_{C_7}\mathcal{H}_{\mathrm{sept}}
\oplus (\mathbf{1} - \hat{P}_{C_7})\mathcal{H}_{\mathrm{sept}}$.

\paragraph{Tool (Closure-preservation test).}
For any new structural term $X$, first test whether
$[X, \hat{P}_{C_7}] = 0$.
If yes, then $X$ respects the admissible stabilized sector directly.
If not, then $X$ either produces leakage or must be controlled by a
compensator-mediated defect term
(cf.\ Hypothesis~\ref{hyp:frame-dependent-defect-readout}).

\paragraph{Tool (Sectorial normal form).}
Every operator $X$ decomposes as
\[
  X = \hat{P}_{C_7}X\hat{P}_{C_7}
    + (\mathbf{1}-\hat{P}_{C_7})X\hat{P}_{C_7}
    + \hat{P}_{C_7}X(\mathbf{1}-\hat{P}_{C_7})
    + (\mathbf{1}-\hat{P}_{C_7})X(\mathbf{1}-\hat{P}_{C_7}),
\]
isolating: (i) admissible core block, (ii) incoming leakage,
(iii) outgoing leakage, (iv) complement-sector block.

\subsection*{Master-equation tools}

\paragraph{Tool (Core-defect normal form).}
The projected structural field operator
$\mathfrak{F}_{\mathrm{struct}}
= \hat{P}_{C_7}(i\Gamma^\mu\mathcal{D}_\mu - \Omega)\hat{P}_{C_7}$
splits into
\[
  \mathfrak{F}_{\mathrm{struct}}
  = \mathbb{D}_{\mathrm{core}} + \mathbb{D}_{\mathrm{defect}},
\]
where $\mathbb{D}_{\mathrm{core}}$ contains autonomous admissible
transport and resonance, and $\mathbb{D}_{\mathrm{defect}}$ collects
closure mismatch, compensator activity, and off-sector suppression.

\paragraph{Tool (Exact-closure normal form).}
If $\mathfrak{R}_C = 0$, $[\mathcal{M}_{16},\hat{P}_{C_7}] = 0$, and
$\mathcal{H}_C = 0$ for the minimal compensator ansatz, then
$\mathfrak{F}_{\mathrm{struct}} = \mathbb{D}_{\mathrm{core}}$ and
$\mathbb{D}_{\mathrm{defect}} = 0$.
The visible compensator field vanishes in this regime; the kernel
$C = \Pi^{(\mathcal{M})}_{\mathrm{spec}}$
(Theorem~\ref{thm:hc-spectral-projector}) remains active
(Remark~\ref{rem:hc-shadow-vs-core}).

\paragraph{Tool (Mass-residue test).}
A proposed mass contribution is classified according to whether it
arises from: (i) a closed-core scalar reading of the projected resonance
operator; (ii) a weak closure-defect correction; (iii) a genuine
shifted-clock or projector residue
$\mathfrak{m}_{\mathrm{comm}}(\psi)
= \frac{\hbar}{c}|\langle\psi|[\hat{\mathcal{R}}_\beta,\hat{P}_{C_7}]|\psi\rangle|$.

\subsection*{Projection and regime tools}

\paragraph{Tool (Regime-strength test).}
A reduction of the structural master equation is: \emph{strong} if
exact closure plus direct slot restriction suffices; \emph{intermediate}
if projected algebra closure is additionally needed; \emph{effective} if
bilinear reconstruction, coarse-graining, or emergent parameter
generation are also required.
These levels correspond to the three steps of
Proposition~\ref{prop:structural-lift-principle}.

\paragraph{Tool (Dirac gate).}
A Dirac-type reduction is available if: (i) exact closure holds;
(ii) the projected connection freezes locally to a trivial sector;
(iii) the projected resonance operator reduces to a scalar inertial
reading.

\paragraph{Tool (Lie-closure gate).}
A Yang--Mills-type reduction is available only if the projected
connection algebra closes on a finite-dimensional Lie algebra
$\mathfrak{g}$. Without that closure, the gauge interpretation is
structurally premature.

\paragraph{Tool (Einstein pipeline gate).}
An Einstein-type reduction requires all of: (i) exact closure on the
admissible sector; (ii) bilinear one-forms $E^a{}_\mu$ from projected
probe dynamics; (iii) maximal rank of the effective coframe; (iv)
metric-compatible geometric connection; (v) coarse-grained local
effective action with Einstein--Hilbert leading term. If any step is
missing, the Einstein block remains effective rather than
first-principles
(cf.\ Proposition~\ref{prop:expected-probe-lensing-behaviour}).

\subsection*{Equivalence and anti-redundancy tools}

\paragraph{Tool (Stone--von Neumann compression rule).}
If two sectors are irreducible unitary realizations of the same
Weyl-type algebra on separable Hilbert spaces, differing only by
representation change, symmetry implementation, or frame reading, then
one representative suffices
(Theorem~\ref{thm:unitary-equiv}).

\paragraph{Tool (Stone--von Neumann scope test).}
Do \emph{not} apply Stone--von Neumann as a shortcut if the passage
between two sectors involves: (i) controlled projection discarding
slots; (ii) coarse-graining over internal modes; (iii) bilinear
geometric reconstruction; (iv) emergent effective constants; (v) loss
of irreducibility. Stone--von Neumann compresses representation
multiplicity, not regime multiplicity.

\paragraph{Tool (Two-layer equivalence test).}
For candidate sectors $S_1, S_2$: (1)~\emph{Representation layer} ---
test whether they are unitary realizations of the same irreducible
Weyl-type kernel; if yes, SvN-compressible. (2)~\emph{Regime layer} ---
test whether one sector is obtained from the other by projection,
coarse-graining, geometric readout, or effective reconstruction; if yes,
additional structural work is required beyond Stone--von Neumann
(Definition~\ref{def:readout-relative-lemma}).

\subsection*{Geometry-readout tools}

\paragraph{Tool (Coframe extraction gate).}
Given a closed triadic probe field $\Psi_\triangle$, define
$E^a{}_\mu := \operatorname{Re}(\bar\Psi_\triangle\,\Gamma^a\,
\mathcal{D}_\mu\Psi_\triangle)$.
This defines an effective coframe only if the resulting $4\times 4$
matrix has maximal rank.

\paragraph{Tool (Closure-to-geometry cleanliness test).}
If exact closure holds, probe bilinears are computed entirely inside
the invariant $C_7$-sector, so reconstructed metric data is not
contaminated by leakage terms. This is the minimal cleanliness
criterion for any geometric readout.

\paragraph{Tool (Geometry-reconstruction warning).}
Closure by itself does not yet produce Einstein geometry. It provides:
(i) an autonomous admissible sector; (ii) a well-defined projected
connection; (iii) a stable basis for bilinear readout. It does
\emph{not} yet provide automatically: (i) nondegeneracy of the
coframe; (ii) derivation of the effective constants; (iii) explicit
identification of the collective closure fields; (iv) justified
coarse-graining to a local geometric action. The bidirectional probe
lensing hypothesis (Section~\ref{sec:bidirectional-probe-lensing})
proposes a two-step route that may reduce this gap.

% Appendix mode is already active above; the following block continues the appendix material without re-entering appendix mode.

\section*{Appendix B. Symbol Normalization for the Late Ledger/Readout Layers}
\addcontentsline{toc}{section}{Appendix B. Symbol Normalization for the Late Ledger/Readout Layers}

This appendix fixes the notation used in the late Companion expansion layers so that
historical version labels and structural symbols remain auditable across the v2.4.0--v2.11.0
chain.

\paragraph{Historical section labels.}
The blocks entitled v2.4.0 through v2.11.0 are to be read as historical expansion
layers. Their section headings record the version in which the corresponding layer was
introduced, not merely the current document version.

\paragraph{Ledger matrix.}
The channel ledger matrix is denoted by
\[
\Lambda_{\mathrm{ch}}(\Psi)
=
\begin{pmatrix}
\ell_{\mathrm{tri}} & \delta_{\mathrm{tri}} & r_{\mathrm{tri}}\\
\ell_{\perp} & \delta_{\perp} & r_{\perp}\\
\ell_{\mathrm{meas}} & \delta_{\mathrm{meas}} & r_{\mathrm{meas}}\\
\ell_{\mathrm{res}} & \delta_{\mathrm{res}} & r_{\mathrm{res}}
\end{pmatrix}.
\]
It is a bookkeeping object for channel-resolved tangential, normal, and compensator-facing
sizes.

\paragraph{Lyapunov ledger functional.}
The weighted scalar functional introduced in v2.6.0 keeps the notation
\(\mathcal L\). It is a Lyapunov-type control quantity and is not to be confused with the
matrix-valued ledger \(\Lambda_{\mathrm{ch}}\).

\paragraph{Budget totals.}
The aggregated scalar budgets remain
\(\mathfrak B\) for total visible load, \(\mathfrak N\) for normal leakage, and
\(\mathfrak R\) for compensator-facing storage.

\paragraph{Readout and fusion objects.}
The readable-mode trace mass, reconstruction operators, observable extraction maps,
and fusion-defect quantities introduced in v2.9.0--v2.11.0 are interpreted as
structure-preserving descendants of the ledger/budget layer and therefore inherit the
normalization fixed here.



\subsection*{Session-budget rule}
A further practical restriction is useful at the present stage: each local mathematics-first pass should also observe a bounded session budget. The purpose of this rule is not to slow work down, but to prevent a single pass from silently expanding from a targeted closure attempt into a diffuse rewrite wave. In concrete terms, a local subversion should normally do only one of the following as its primary task: (i) stabilize definitions for one A-priority block, (ii) discharge one dependency layer for that block, (iii) perform one focused reference/notation repair required by that block, or (iv) close one clearly delimited appendix-tool support step. If more than one such task would be required, the safer action is to end the current pass, classify the result, and continue by explicit next local subversion.

The session-budget rule therefore complements the anti-parallelization and re-entry rules. A pass is considered well-scoped if its main closure claim can be summarized in one sentence and audited against one active block. A pass should be considered over budget if it begins to change multiple theorem-status boundaries, broad notation regimes, or several major sections at once. In that case, the correct response is not to continue accumulating edits, but to stop, audit, and reopen the next step as a fresh local subversion. This rule is intentionally conservative: its goal is to make every local version legible as a finite mathematical move rather than as an opaque mass of partially mixed intentions.



\subsection*{2I. Final release-decision template for v2.27.0}
At this stage the document no longer needs another diffuse local steering note, but a compact decision template that makes explicit under which reading the next public release is justified. The present template distinguishes three cases. In the first, the document is strong enough that the next action is \emph{release now}: this applies if the A-priority blocks are structurally frozen, the gate conditions remain green, and the remaining work consists only of optional proof-density enrichment. In the second, the document should \emph{hold locally for proof-density enrichment}: this applies if the structural spine is already in place, but one deliberately wants a denser local unfolding before exposing the next public release. In the third, the document must \emph{hold locally for unresolved blocker}: this applies only if a genuine release-critical dependency, hypothesis-control issue, or reference-instability problem is still active.

Under the current reading, the first A-block no longer counts as such a blocker. Its local dependency spine has been extended through admissible transport, matrix-sector realization, and admissible trace/branch persistence, and it has already been reconciled with the mathematics-first gate logic. Hence the residual caution is no longer structural incompleteness, but only whether one prefers to accept the present medium proof density or to invest one further local pass in additional unfolding. Accordingly, the document is now close to the decision boundary where a direct step to v2.27.0 becomes reasonable.

\paragraph{Decision states.}
\begin{itemize}[leftmargin=2em]
  \item \textbf{Release now:} the first A-block is structurally release-supporting, no primary blocker remains, and only optional densification is deferred.
  \item \textbf{Hold locally for proof-density enrichment:} no structural blocker remains, but one more local pass is preferred for denser exposition.
  \item \textbf{Hold locally for unresolved blocker:} a genuine reference, dependency, or hypothesis-control failure is still present.
\end{itemize}

\paragraph{Current reading.}
At the present local stage, the third state no longer appears to apply to the first A-block. The second and first states are now the relevant ones: either one accepts the current medium-density closure as sufficient for the next public release, or one performs at most one more local enrichment pass before the public step.

\paragraph{Release consequence.}
The practical consequence is that any further local subversion should no longer open a new mathematical front. It should either confirm release-readiness or add one last density-improving clarification and then stop.




\subsection*{2J. Open-questions closure pass and chain check}
The present stage is also strong enough to classify several previously open questions more explicitly. The purpose of this pass is not to pretend that every weak-density zone has disappeared, but to distinguish between (i) questions that can already be closed reliably at the current structural level, (ii) questions that remain structurally intact but only under an explicit assumption or still-thin proof layer, and (iii) questions that should remain openly deferred. This distinction is especially important in a structure-first setting, because a locally thin argument is not automatically broken: some passages remain admissible as conditional or contradiction-style steps, provided that their status is made explicit and no illegitimate theorem-promotion occurs.

\paragraph{Closure classification.}
At the current stage the following questions may be treated as \emph{closable now}: the dependency spine from ground condition to HC and confinement; the compatibility of closure with projector stabilization; the admissible transport continuation on the stabilized sector; the admissible-sector action of the Millennium matrix; and the persistence of admissible trace/branch continuation once the stabilized sector has been fixed. These are not yet maximally unfolded everywhere, but they are no longer merely heuristic. By contrast, the following questions are presently better read as \emph{intact but proof-density limited}: the strongest available mathematical reading of the Millennium matrix as converter/intertwiner; the full density of trace/branch continuation lemmas; and the extent to which some still-compact structural passages should already be promoted beyond proposition level. Finally, the following remain \emph{intentionally open}: any full universality claim for the computational strand, any finished proof-kernel or compiler reading, and any interpretation that would make the mathematics depend on the computational teaser architecture.

\paragraph{Chain-check statuses.}
To keep the proof chain honest without overcalling local weakness as failure, the document now adopts four chain-check statuses for future mathematical passes:
\begin{itemize}[leftmargin=2em]
  \item \textbf{intact:} the dependency chain carries without visible status or continuity defect;
  \item \textbf{intact, but proof-density weak:} the chain carries, but one or more transitions still need denser local unfolding;
  \item \textbf{intact under explicit assumption:} the argument is acceptable only because an assumption, contradiction setup, or provisional hypothesis is openly declared;
  \item \textbf{blocker:} a genuine dependency break, status violation, or contradiction prevents closure.
\end{itemize}

\paragraph{Current reading.}
Under this classification, the first major dependency spine is currently best read as \emph{intact, but proof-density weak} rather than as blocked. No primary structural contradiction is visible in the path from ground condition through HC, closure, admissible transport, Millennium-matrix realization, and admissible trace/branch persistence. The remaining caution is therefore mainly one of density and explicitness, not one of broken continuity.

\paragraph{Local consequence.}
The practical consequence is conservative but positive. Several earlier open questions may already be marked as \emph{reliably closable at the present stage}, provided that the document does not silently promote weak-density passages to stronger theorem status than the current support allows. The next mathematics-first passes should therefore use the chain-check statuses as a standing audit device: strengthen density where useful, keep assumption-based passages openly marked, and treat only genuine continuity failure as a blocker.




\subsection*{2K. Tool extraction and leverage-point scan}
The next mathematics-first pass should not treat every locally thin argument as something that must immediately be expanded line by line. In a structure-first document, some still-wobbly passages already contain reusable mathematical moves that can be extracted earlier as \emph{tools}, provided that their status is made explicit. The present scan therefore distinguishes between (i) tools that are already operationally useful now, even if their surrounding proof density is still moderate, (ii) tools that remain usable only under an explicit structural assumption, and (iii) a small number of leverage points whose clarification would simplify several later proof passages at once.

\paragraph{Provisional tool extraction.}
At the current stage the following reusable tools may already be extracted from the first dependency spine. First, a \emph{closure-transfer tool}: once the admissible sector has been fixed, closure may be transported across projector stabilization without reopening the sector boundary. Second, a \emph{projected transport tool}: admissible transport may be read modulo the stabilized projector rather than by re-proving closure from scratch after each transport step. Third, a \emph{matrix-sector tool}: the Millennium matrix may already be used as an admissible-sector-preserving operator on the stabilized transported sector. Fourth, a \emph{trace/branch persistence tool}: admissible continuation in the trace/branch layer may be used without reintroducing an external admissibility source. None of these tools should yet be oversold as maximal-density theorems everywhere; however, they are already strong enough to simplify downstream arguments by replacing repeated local closure checks with a smaller number of reusable invariance moves.

\paragraph{Status discipline for extracted tools.}
To keep the extraction mathematically honest, each such tool should be marked in one of three ways:
\begin{itemize}[leftmargin=2em]
  \item \textbf{operational tool:} reusable now, with the present support level;
  \item \textbf{assumption-backed tool:} reusable only under an explicitly declared hypothesis or contradiction setup;
  \item \textbf{blocked tool candidate:} not yet safe to reuse because a real dependency gap remains.
\end{itemize}
Under the current chain check, the closure-transfer, projected transport, matrix-sector, and trace/branch persistence tools are best read as \emph{operational tools}, though some still sit in the ``proof-density weak'' zone rather than the fully unfolded zone.

\paragraph{Leverage-point scan.}
The present scan suggests that the most useful immediate leverage point is not a new broad theorem family, but a single stronger organizing hypothesis: namely that the admissible stabilized sector behaves as the true fixed working domain for closure, transport, matrix action, and admissible continuation. Even if one initially formulates this only as a hypothesis-backed simplifier, it has unusually high leverage: if strengthened, it compresses several later arguments by replacing repeated local admissibility restoration with one global working-domain principle. In other words, many later proofs become shorter once they are allowed to start with ``assume the admissible stabilized sector has already been fixed'' and then work internally on that domain.

\paragraph{Use of the computational layer as a secondary aid.}
The computational chapter may already help here, but only in a subordinate role. MML/MMA/MCP should not drive the mathematics; however, the language layer can already serve as a \emph{notation and workflow aid} for tool extraction. In particular, compact MML-style descriptions can help identify whether a candidate move is really one reusable operation---for example \texttt{project -> transport -> trace}---or whether it still hides an unresolved mathematical gap. Thus the computational layer may already be used as a diagnostic simplifier for tool extraction, while the mathematical status of the extracted tool remains decided purely on the mathematics-first side.

\paragraph{Current reading and local consequence.}
The current reading is therefore positive. The first dependency spine is no longer merely a place where one asks whether the chain survives; it is already rich enough to yield a first family of reusable structural tools. At the same time, the highest-leverage next target is also visible: stabilize the admissible working-domain principle as far as possible, because this one point appears to simplify closure, transport, matrix action, and trace/branch continuation simultaneously. Any next local pass should therefore prefer a leverage-point clarification over broad diffuse proof growth.


\section*{Appendix D. Bibliography and Version Record}
\addcontentsline{toc}{section}{Appendix D. Bibliography and Version Record}

This appendix gathers the external references together with the immediate
Companion version trail so that the mathematical main line can close before the
documentation apparatus begins.

\subsection*{Version and DOI record}
The table below records the currently preserved Companion chain in strictly
increasing version order. Zenodo releases are listed together with a publication-date
column; local-only intermediate repair states are retained separately so that the
public release path and the internal repair path remain distinguishable.

{\small
\renewcommand{\arraystretch}{1.12}
\setlength{\LTleft}{0pt}
\setlength{\LTright}{0pt}
\begin{longtable}{|p{0.14\textwidth}|p{0.17\textwidth}|p{0.15\textwidth}|p{0.42\textwidth}|}
\hline
\textbf{Version / record} & \textbf{Status} & \textbf{Publication date} & \textbf{DOI / note} \\
\hline
\endfirsthead
\hline
\textbf{Version / record} & \textbf{Status} & \textbf{Publication date} & \textbf{DOI / note} \\
\hline
\endhead
Concept DOI & concept record & ongoing pointer & \href{https://doi.org/10.5281/zenodo.19210728}{10.5281/zenodo.19210728} \\
\hline
v0.1.0 & Zenodo release & 2026-03-24 & \href{https://doi.org/10.5281/zenodo.19210729}{10.5281/zenodo.19210729} \\
\hline
v0.3.0 & Zenodo release & 2026-03-24 & \href{https://doi.org/10.5281/zenodo.19211785}{10.5281/zenodo.19211785} \\
\hline
v0.4.0 & Zenodo release & 2026-03-24 & \href{https://doi.org/10.5281/zenodo.19211798}{10.5281/zenodo.19211798} \\
\hline
v0.5.0 & Zenodo release & 2026-03-24 & \href{https://doi.org/10.5281/zenodo.19211877}{10.5281/zenodo.19211877} \\
\hline
v0.6.0 & Zenodo release & 2026-03-24 & \href{https://doi.org/10.5281/zenodo.19212073}{10.5281/zenodo.19212073} \\
\hline
v0.7.0 & Zenodo release & 2026-03-24 & \href{https://doi.org/10.5281/zenodo.19212212}{10.5281/zenodo.19212212} \\
\hline
v0.8.0 & Zenodo release & 2026-03-24 & \href{https://doi.org/10.5281/zenodo.19212324}{10.5281/zenodo.19212324} \\
\hline
v0.9.0 & Zenodo release & 2026-03-24 & \href{https://doi.org/10.5281/zenodo.19220003}{10.5281/zenodo.19220003} \\
\hline
v1.0.0 & Zenodo release & 2026-03-24 & \href{https://doi.org/10.5281/zenodo.19220118}{10.5281/zenodo.19220118} \\
\hline
v1.6.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19221335}{10.5281/zenodo.19221335} \\
\hline
v1.7.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19221372}{10.5281/zenodo.19221372} \\
\hline
v1.8.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19221431}{10.5281/zenodo.19221431} \\
\hline
v1.9.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19221473}{10.5281/zenodo.19221473} \\
\hline
v2.0.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19221527}{10.5281/zenodo.19221527} \\
\hline
v2.1.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19221618}{10.5281/zenodo.19221618} \\
\hline
v2.3.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19221864}{10.5281/zenodo.19221864} \\
\hline
v2.4.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19221954}{10.5281/zenodo.19221954} \\
\hline
v2.9.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19222638}{10.5281/zenodo.19222638} \\
\hline
v2.10.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19222894}{10.5281/zenodo.19222894} \\
\hline
v2.11.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19223084}{10.5281/zenodo.19223084} \\
\hline
v2.11.1 & local-only repair state & 2026-03-25 & non-Zenodo version-fix state between v2.11.0 and v2.11.2 \\
\hline
v2.11.2 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19223365}{10.5281/zenodo.19223365} \\
\hline
v2.11.3 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19223515}{10.5281/zenodo.19223515} \\
\hline
v2.12.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19223782}{10.5281/zenodo.19223782} \\
\hline
v2.13.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19223968}{10.5281/zenodo.19223968} \\
\hline
v2.14.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19225191}{10.5281/zenodo.19225191} \\
v2.15.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19226408}{10.5281/zenodo.19226408} \\
\hline
v2.16.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19226445}{10.5281/zenodo.19226445} \\
\hline
v2.17.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19226497}{10.5281/zenodo.19226497} \\
v2.18.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19226549}{10.5281/zenodo.19226549} \\
\hline
v2.19.0 & Zenodo release & 2026-03-25 & \href{https://doi.org/10.5281/zenodo.19226773}{10.5281/zenodo.19226773} \\
\hline
v2.20.0 & Zenodo release & 2026-03-26 & \href{https://doi.org/10.5281/zenodo.19227889}{10.5281/zenodo.19227889} \\
\hline
v2.21.0 & Zenodo release & 2026-03-26 & \href{https://doi.org/10.5281/zenodo.19227940}{10.5281/zenodo.19227940} \\
\hline
v2.22.0 & Zenodo release & 2026-03-26 & \href{https://doi.org/10.5281/zenodo.19228061}{10.5281/zenodo.19228061} \\
\hline
v2.23.0 & Zenodo release & 2026-03-26 & \href{https://doi.org/10.5281/zenodo.19239228}{10.5281/zenodo.19239228} \\
\hline
v2.23.1 & Zenodo patch release & 2026-03-26 & \href{https://doi.org/10.5281/zenodo.19239967}{10.5281/zenodo.19239967} \\
\hline
v2.24.0 & Zenodo release & 2026-03-27 & \href{https://doi.org/10.5281/zenodo.19258774}{10.5281/zenodo.19258774} \\
v2.25.0 & Zenodo release & 2026-03-27 & \href{https://doi.org/10.5281/zenodo.19259425}{10.5281/zenodo.19259425} \\
 v2.25.1 & Zenodo patch release & 2026-03-27 & \href{https://doi.org/10.5281/zenodo.19262890}{10.5281/zenodo.19262890} \\
\hline
v2.27.0 & Zenodo release & 2026-03-27 & \href{https://doi.org/10.5281/zenodo.19266666}{10.5281/zenodo.19266666} \\
\hline
v2.27.1 & local-only repair state & 2026-03-28 & baseline-preserving post-release working state \\
\hline
v2.27.2 & local-only repair state & 2026-03-28 & normalization and proof-chain hardening \\
\hline
v2.27.3 & local-only repair state & 2026-03-28 & structural closure pass; last state before v2.27.4 integration \\
\hline
v2.27.4 & local working state & 2026-03-28 & spectral-projector layer ($C=\Pi^{(\mathcal{M})}_{\mathrm{spec}}$), readout-relative lemma methodology, Theorem~7 (HC-induced readout geometry), Hypotheses~7.4--7.6, bidirectional probe lensing, SM spectral realization, regime-test appendix \\
\hline
v2.27.5 & Zenodo release & 2026-03-28 & \href{https://doi.org/10.5281/zenodo.19297422}{10.5281/zenodo.19297422} — full proof-chain audit; closure roadmap; 35 missing labels; five Priority-A gaps identified \\
\hline
v2.27.6 & local working state & 2026-03-28 & A1+A2+A3 closed: \texttt{lem:closure-forcing-lemma} hardened; (T4) added to \texttt{def:tensor-structural-kernel}; \texttt{lem:closure-stable-iff-im-C} new; \texttt{thm:hc-spectral-projector} now unconditional; open: A4, A5, B1, B2 \\
\hline
v2.27.7 & local working state & 2026-03-28 & A4+A5 closed: \texttt{thm:koide-triolic} with $\theta_K$ formula (0.111\% match, $115\times$ amplification remark); \texttt{thm:bsym-hopf} with proof and Hopf-gap remark (4.9\%/$6.8\%$, OP1/OP2 coupling); open: B1, B2 \\
\hline
v2.27.8 & local working state & 2026-03-28 & B1+B2 closed: \texttt{prop:semantic-forcing-chain} explicit 3-part proof from axioms (S1)--(S8); \texttt{prop:structural-lift-principle} non-circular via \texttt{lem:group-transport}+\texttt{lem:hc-admissibility-lemma}; OP~2 formalized; all 7 Roadmap items closed \\
\hline
v2.28.0 & Zenodo release & 2026-03-28 & \href{https://doi.org/10.5281/zenodo.19298586}{10.5281/zenodo.19298586} --- consolidation of v2.27.4--v2.27.8; spectral-projector unconditional; bijection lemma; SM realization; readout-relative methodology; Thm~7; Hyp~7.4--7.6; OP~2 stub \\
\hline
v2.29.0 & Zenodo release & 2026-03-28 & \href{https://doi.org/10.5281/zenodo.19299245}{10.5281/zenodo.19299245} --- OP~5 step~(A) closed; step~(B) quantified; renormalization explicit \\
\hline
v2.29.1 & local working state & 2026-03-28 & GOE/GUE split (\texttt{prop:goe-fermionic}); $\gL$ = GOE/GUE ratio; $\gamma_L=1.197$ corrected to $1.864$; $\chi^2$ improvement $75.8\%$; $\delta_{\mathrm{CL}}$ exponent $\gamma_L^2\to\gamma_L^5$; \texttt{thm:dual-origin} corrected \\
\hline
v2.29.2 & Zenodo release & 2026-03-28 & \href{https://doi.org/10.5281/zenodo.19300351}{10.5281/zenodo.19300351} --- Einstein blunder reframed; $\Omega_M=\bsym^2$ determination; three-gap convergence \\
\hline
v2.29.3 & Zenodo release & 2026-03-28 & \href{https://doi.org/10.5281/zenodo.19300813}{10.5281/zenodo.19300813} --- OP~2 (T1)(T2) closed; $D_-=L_0|_{\Hplusminus}$ (corrected candidate); GOE essential self-adjointness; T4=OP1 equivalence; \textbf{Tribonacci theorem} $\gamma_L\to\mathbb{T}$; cascade normalization $1/\mathbb{T}+1/\mathbb{T}^2+1/\mathbb{T}^3=1$ \\
\hline
v2.29.4 & Zenodo release & 2026-03-29 & \href{https://doi.org/10.5281/zenodo.19314525}{10.5281/zenodo.19314525} --- SC1+SC2 self-consistent system; \texttt{thm:self-consistent-gL}; HC quasiconformal dilatation; Beltrami fixed point; hypothesis-grade propositions (inflow, $k(\nu)$, $H_0$ tension, muon $g$-2, QCD running) \\
\hline
v2.29.5 & Zenodo release & 2026-03-29 & \href{https://doi.org/10.5281/zenodo.19314898}{10.5281/zenodo.19314898} --- Frame-Transport operator $\mathcal{T}_{\gL}(\bsym)$; Weil-Reduktion von OP~5(B): $S_\rho(A)$ exakt via $\psi(e^A)=0$; $\delta_{\mathrm{corr}}$ absolut konvergent; Cor.~\ref{cor:op5b-exact-form} \\
\hline
v2.29.6 & Zenodo release & 2026-03-31 & \href{https://doi.org/10.5281/zenodo.19357490}{10.5281/zenodo.19357490} --- Numerische Diagnose OP~5(B): Komplementär-Identität $\mathcal{T}_{\mathrm{req}}-\mathcal{T}_{\mathrm{Weil}}\approx 1$; $\delta_{\mathrm{lattice}}$ exakt als transzendentes SC-System charakterisiert (Thm.~\ref{thm:delta-lattice-exact}); OP~5(B) als Spektralidentität $\mathbb{T}\,\mathcal{F}(A)=1-D(\mathbb{T})$ (Thm.~\ref{thm:op5b-spectral}) \\
\hline
\end{longtable}
}

\noindent Note. The sequence above records the currently preserved release trail.
The day-level publication dates are normalized from the project publication log so that
Zenodo releases and local repair states remain chronologically comparable inside a single
appendix table.


\subsection*{v2.26.0 computational consolidation note}

Version~\textbf{2.26.0} executes the consolidation roadmap that had been reserved in
v2.25.1. The computational strand is retained but internally reorganized into a more
coherent progression: historical perspective, structural machine reading, Triolen
Algebra / Boolean readout, MML--MMA--MCP stratification, proof templates,
certificate-bearing execution, normalization, addressing, scheduling, host export,
and extraction outlook. The present version still remains exploratory in the sense of
future formalization, but the material is no longer presented only as a loose teaser;
it is now collected as a first internally coherent computational chapter of the
Companion.



\subsection*{Hypothesis-control pass for v2.27.0}

Before the next public milestone, the mathematical layer should distinguish more sharply between statements that are expected to become theorem-grade in v2.27.0 and statements that should remain explicitly hypothesis-grade even after the next hardening cycle. The practical rule is conservative: only statements with stabilized definitions, closed dependency chains, and reference-stable support should move upward from hypothesis status into theorem, proposition, or lemma status. Statements that continue to depend on interpretive transport, physical reading, or still-open structural identifications should remain explicitly marked as hypotheses or outlook items.

For the next pass, the intended control logic is therefore blockwise. In the ground-condition / HC / confinement chain, the target is theorem-readiness whenever the dependence on the collapse picture and confinement mechanism can be formulated without appealing to deferred computational scaffolding. In projector / closure mathematics, the target is theorem or lemma status whenever the relevant admissibility and sector-preservation claims are already internal to the mathematical layer. In the transport / Lorentz layer, statements that are still partly interpretive should remain hypothesis-grade unless they can be rewritten as strictly mathematical transport statements. For the Millennium-matrix and trace/branch layers, the default stance remains more cautious: structural mathematical claims may harden, but computational or machine-reading interpretations remain secondary and should not silently inherit theorem status from the mathematical core.

The resulting editorial discipline for v2.27.0 is simple: theorem promotion requires closure, dependency control, and stable notation; everything else remains visibly hypothesis, deferred claim, or teaser architecture. This avoids theorem-to-hypothesis drift in reverse as well: once a statement is kept at hypothesis level, later references should not read it as already mathematically closed merely because it is computationally suggestive.

\subsection*{v2.26.33 local mathematics-first note}

Version~\textbf{2.27.0} does not yet claim a new public release beyond v2.26.0. Instead, it records the explicit shift of near-term priority back to mathematical completion. The computational chapter remains preserved as a secondary but already visible reserve architecture. The immediate next target is v2.27.0 as a mathematics-first hardening cycle: close the mathematical layer, collect tools as mathematical appendices, and only then resume the proof-engine line as a compression language for already stabilized structure.


\subsection*{Theorem-promotion checklist for v2.27.0}
The next public milestone should not promote statements merely because they have become familiar in the internal workflow. Each promotion from hypothesis-language into lemma-, proposition-, or theorem-language must therefore be explicit, local, and reference-stable. For the mathematical closure pass leading to v2.27.0, the promotion discipline is:
\begin{itemize}[leftmargin=*]
  \item statements about the ground condition to HC to confinement chain may be promoted only when the chain has been written in one direction without hidden interpretive jumps;
  \item projector/closure claims may be promoted only when the admissibility conditions and the closure domain are fixed in stable notation;
  \item transport/Lorentz statements may be promoted only when they are phrased as mathematical transport claims rather than as physical metaphor;
  \item Millennium-matrix statements may be promoted only when the matrix is treated first as a mathematical object with domain, codomain, and structural role, not merely as a computational image;
  \item trace/branch claims may be promoted only when the branch criteria and trace-compatibility conditions are stated in theorem-capable form.
\end{itemize}
In parallel, several statement families are to remain explicitly outside theorem-promotion for v2.27.0 unless a full closure pass is actually completed. These include broad computational-universality claims, strong proof-kernel claims, speculative host-language claims, and any physically interpretive reading that still rests on heuristic transport language rather than closed mathematical formulation.

A practical promotion checklist for the next pass is therefore:
\begin{enumerate}[leftmargin=*]
  \item identify candidate statements that are currently written as heuristic summary sentences;
  \item decide whether each candidate should become a Definition, Lemma, Proposition, or remain a Hypothesis;
  \item verify that every promoted statement has stable notation, explicit assumptions, and stable backward references;
  \item verify that no promoted statement depends silently on teaser/computational language;
  \item record all non-promoted but still important claims explicitly as open, deferred, or interpretive.
\end{enumerate}
This checklist is intended to keep the next public release mathematically harder without forcing premature promotion of material that should still remain hypothesis-level.

\subsection*{Mathematical freeze criterion for v2.27.0}
For the next public release, a mathematical core block should count as \emph{frozen for release} only when it is not merely well phrased, but also stable under the local closure discipline introduced in the present subversion sequence. In particular, a block is to be treated as mathematically release-ready for v2.27.0 only if all of the following are satisfied:
\begin{enumerate}[leftmargin=*]
  \item the principal objects of the block are defined in stable notation and do not drift between theorem-language and hypothesis-language;
  \item the block depends only on explicitly cited prior definitions, lemmas, propositions, or tools, rather than on narrative intuition or roadmap placeholders;
  \item the key admissibility conditions are written inside the statement or proof scaffold itself;
  \item all required cross-references are stable and do not point to provisional teaser/computational material;
  \item the block can be read both locally and in the dependency order recorded for the mathematics-first pass without hidden backward jumps;
  \item any remaining interpretive or physical reading is clearly separated from the mathematical closure claim.
\end{enumerate}
A block should \emph{not} be counted as frozen merely because its wording is polished, because it already appears in summary form elsewhere, or because it has an attractive structural interpretation. For v2.27.0, the intended release criterion is therefore conservative: closure requires definitional stability, dependency stability, reference stability, and status stability at the same time.

A practical freeze test for each A-priority block is:
\begin{itemize}[leftmargin=*]
  \item can the block be pointed to by a stable label and theorem-status marker?
  \item can its assumptions be listed in one compact place?
  \item can its proof or justification be read without consulting deferred computational scaffolds?
  \item can its use in later blocks be stated without introducing new notation drift?
\end{itemize}
If the answer to any of these questions remains negative, the block should remain marked as active work rather than as mathematically frozen for public release. This criterion is meant to prevent premature hardening of material that is already suggestive but not yet sufficiently closed.


\subsection*{Release-readiness matrix for v2.27.0}
To prevent the next public release from being driven by impression rather than closure, the mathematics-first pass should end in a compact readiness matrix for the A-priority blocks. The matrix is not meant to replace proofs; it is meant to summarize, in one release-facing place, whether a core block is genuinely ready to be frozen, or whether a single missing condition still blocks public hardening.

\begin{center}
\begin{tabular}{p{0.23\textwidth} p{0.17\textwidth} p{0.12\textwidth} p{0.22\textwidth} p{0.18\textwidth}}
\toprule
\textbf{Core block} & \textbf{Current state} & \textbf{Freeze-ready?} & \textbf{Missing closure condition} & \textbf{Primary blocker for v2.27.0}\\
\midrule
Ground condition / collapse picture & structurally central, still partly summary-driven & no & one stable definitional pass with fixed assumptions & notation and statement drift\\
HC emergence / confinement chain & strongly anchored, not yet written in one compact theorem-capable line & partial & explicit one-directional closure chain & hidden interpretive compression\\
Projector / closure mathematics & partially stabilized and reusable & partial & admissibility conditions and closure domain fixed in one notation regime & local definition spread\\
Transport / Lorentz layer & present but mixed with interpretive language & no & transport claims restated as mathematical transport statements & metaphor drift\\
Millennium matrix as mathematical object & increasingly structured, still mixed with computational imagery & no & fixed domain/codomain/operator role & object-status not yet fully hardened\\
Trace / branch mathematics & structurally rich, still partly template-style & partial & stable branch criterion and trace compatibility form & theorem-status not yet fixed\\
Tool / appendix lemmas & abundant support layer & partial & priority ordering and stable dependence tags & appendix support not yet fully normalized\\
\bottomrule
\end{tabular}
\end{center}

For the next public milestone, the conservative reading of this matrix is intentional: \emph{partial} does not yet count as release-ready. A block should only be treated as ready for v2.27.0 if its missing closure condition has been discharged explicitly and its blocker is no longer active. In this sense, the matrix is a release-control device rather than a rhetorical summary: it translates the mathematics-first agenda into a blockwise freeze decision surface.



\subsection*{v2.27.0 gate checklist}
To prevent the next public release from being triggered by momentum alone, the mathematics-first pass should culminate in a compact global gate checklist. This checklist is stricter than the blockwise readiness matrix: even if several A-priority blocks look individually advanced, the public jump to v2.27.0 should only be made once the release conditions hold \emph{together} at the document level.

\begin{itemize}[leftmargin=*]
  \item the A-priority mathematical blocks admit stable definitions, stable dependence order, and stable theorem-status markers;
  \item no public-facing core claim in the targeted v2.27.0 range still depends on deferred computational scaffolds for its mathematical reading;
  \item the blockwise freeze criteria and the release-readiness matrix no longer contain an active blocker in the A-priority line;
  \item the proof-dependency checklist and the reference-stability pass agree on a single release-safe label structure;
  \item hypothesis-control and theorem-promotion discipline are aligned, so that no statement is promoted merely because its interpretation is attractive;
  \item the transport / projector / Millennium-matrix line can be cited without notation drift or mixed theorem-versus-hypothesis status;
  \item appendix tools used by the A-priority line are stable enough to serve as support, but not so overloaded that they silently replace missing core proofs;
  \item the computational / MML / MMA / MCP chapter remains visibly present as teaser architecture, but no release-critical mathematical closure claim depends on it;
  \item the version record, roadmap, local subversion discipline, and closure-control blocks all point to the same public target without internal contradiction.
\end{itemize}

A practical release rule for the next public jump is therefore conservative: \emph{v2.27.0 should only be cut once the A-line is simultaneously blockwise closure-ready and globally gate-clean}. If one of these conditions still fails, the correct response is another local subversion rather than a premature public release. In this sense, the gate checklist converts the mathematics-first roadmap into a final version-control criterion.

\subsection*{Final pre-v2.27.0 control summary}
As the mathematics-first pass approaches a public release decision, the Companion should end with a compact control summary that can be read in one glance. The purpose of this summary is not to replace the detailed work matrix, target-state matrix, or release-readiness table, but to compress them into a final release-facing snapshot. In particular, the summary should answer five questions simultaneously: (i) which A-priority blocks are currently the decisive release drivers, (ii) whether each of them is already freeze-ready, (iii) what the leading blocker still is, (iv) whether the global gate checklist is currently satisfied, and (v) whether the recommended action is \emph{continue mathematical closure} or \emph{prepare public release}.

\begin{center}
\begin{tabular}{p{0.21\textwidth} p{0.12\textwidth} p{0.20\textwidth} p{0.14\textwidth} p{0.20\textwidth}}
\toprule
\textbf{A-priority block} & \textbf{Freeze-ready?} & \textbf{Main blocker} & \textbf{Gate status impact} & \textbf{Release recommendation}\\
\midrule
Ground condition / collapse picture & not yet & dependence and formulation hardening & blocks full gate closure & continue mathematical closure\\
HC emergence / confinement chain & partial & explicit mathematical discharge & blocks full gate closure & continue mathematical closure\\
Projector / closure mathematics & partial & stable theorem-level closure & blocks full gate closure & continue mathematical closure\\
Transport / Lorentz layer & conditional & transport formalization discipline & weakens gate confidence & continue mathematical closure\\
Millennium matrix as mathematical object & not yet & object-level hardening before interpretive use & blocks full gate closure & continue mathematical closure\\
Trace / branch mathematics & partial & stable dependence on prior blocks & blocks full gate closure & continue mathematical closure\\
\bottomrule
\end{tabular}
\end{center}

At the present stage, the control summary therefore recommends \emph{no public jump yet beyond the current release line}: the mathematics-first agenda should remain active until the decisive A-priority blocks no longer fail the freeze criterion and the global gate checklist turns from ``not yet to ``satisfied. The intended role of this final control summary is thus practical rather than rhetorical: it gives the next release decision a single consolidated dashboard and makes it harder to confuse local progress with document-level readiness.


\subsection*{One-page executive closure view}
The mathematics-first steering logic introduced in the recent local subversions is now dense enough that it should also be compressible into a single executive page. The purpose of this closure view is therefore not to add another planning layer, but to place roadmap, dependency map, target-state matrix, freeze criterion, gate checklist, and pre-release control summary onto one final compact surface. In practical terms, the page should allow a reader to answer four questions immediately: (i) what the next public version is really trying to close mathematically, (ii) which A-priority blocks still drive the release boundary, (iii) whether the document is globally gate-clean, and (iv) whether the correct next action is another local subversion or a public release jump.

\begin{center}
\begin{tabular}{p{0.22\textwidth} p{0.22\textwidth} p{0.18\textwidth} p{0.24\textwidth}}
\toprule
\textbf{Control layer} & \textbf{Main question} & \textbf{Current status} & \textbf{Operational consequence}\\
\midrule
Roadmap / work matrix & what must be closed first? & mathematics-first line active & keep proof-engine / MML line secondary\\
Dependency map & what must come before what? & ordered core chain available & no closure claims out of dependency order\\
Definition / notation pass & are the objects and labels stable? & partially stabilized & unify notation before theorem-promotion\\
Target-state / closure checklist & what counts as done? & blockwise closure conditions visible & only explicit closure counts as completion\\
Freeze criterion / readiness matrix & is a block release-ready? & mixed, mostly partial & keep A-line blocks active until blocker-free\\
Gate checklist / control summary & is the document globally ready? & not yet gate-clean & continue local subversions, no public jump yet\\
\bottomrule
\end{tabular}
\end{center}

In this condensed reading, the present Companion state remains productive but intentionally conservative: the mathematical architecture is now sufficiently organized to support disciplined closure work, but not yet sufficiently gate-clean to justify confusing local progress with public readiness. The executive closure view is therefore meant to function as a one-page release dashboard. Its decision rule is simple: as long as the A-priority mathematical blocks remain only partially frozen, and as long as the global gate still records an active blocker, the correct action is \emph{another explicit local subversion rather than a public version jump}. Once these conditions disappear simultaneously, the dashboard should flip from \emph{continue closure} to \emph{prepare v2.27.0 release}.


\subsection*{Release-vs-local decision rule}
A final practical distinction should also be made explicit before the next public jump: not every coherent local gain justifies a new Zenodo-facing version. The present Companion workflow therefore distinguishes between \emph{local closure iterations} and \emph{public release transitions}. A local subversion is appropriate whenever the document gains planning discipline, dependency clarity, notation stability, or blockwise closure information without yet removing the main A-line blockers. A public release transition is appropriate only when these local gains have accumulated into a document-wide state in which the mathematical core is not merely better organized, but sufficiently frozen, referentially stable, and gate-clean to support a new public boundary.

In operational terms, the decision rule may be summarized as follows. Move only by local subversion if one or more of the following remains true: (i) an A-priority block is still only partially closed, (ii) the theorem/hypothesis boundary is still under active normalization, (iii) the reference-stability pass is not yet quiet, (iv) the release-readiness matrix still lists an active primary blocker, or (v) the global gate checklist is not yet satisfied. Consider a public version jump only when all A-priority blocks satisfy their freeze criterion simultaneously, the dependency chain is stable under rereading, the hypothesis-control and theorem-promotion passes no longer produce status drift, and the release-control summary flips from \emph{continue closure} to \emph{prepare release}. This distinction is intentionally conservative: it protects the mathematical layer from being mistaken for release-ready merely because the planning architecture around it has become more refined.

\subsection*{Mathematics-first release doctrine}
The present local stage adopts a simple doctrine for all remaining pre-v2.27.0 work. First, every local change is to be judged by whether it closes, stabilizes, or clarifies a mathematical dependency in the A-priority chain; additions that merely decorate the proof-engine teaser without tightening the mathematics are therefore secondary by rule. Second, no new public release is justified merely by chapter growth or architectural enthusiasm: the release threshold remains mathematical closure, reference stability, and theorem-safe status control. Third, the computational/MML/MMA/MCP layer is to remain visible and preserved, but it is not permitted to become a hidden prerequisite for mathematical claims that should stand on their own. Fourth, every local subversion should remain append-only and auditable, so that recovery from drift or accidental rollback stays possible without reconstructing the chain from memory.

In operational terms, the doctrine reads as follows:
\begin{enumerate}[label=(\roman*)]
  \item prioritize closure of ground condition $\rightarrow$ HC $\rightarrow$ confinement before expanding secondary architecture;
  \item promote statements only when notation, dependencies, and reference anchors are already stable;
  \item treat tools, appendices, and computational reserves as support layers unless and until they are explicitly absorbed into the mathematical core;
  \item prefer local subversions over public version jumps whenever the work mainly sharpens internal control rather than closing a release-grade mathematical block.
\end{enumerate}

This doctrine is intentionally conservative. Its role is to keep the Companion aligned with the structure-first programme: mathematics first, proof-memory preserved, computational architecture retained as a prepared secondary line, and public releases tied to genuine closure rather than mere accumulation.


\subsection*{Mathematics-first action queue}
The next local working order is to be treated as explicit and sequential unless a later control note states otherwise. The active queue is: (1) close the ground-condition / collapse picture at the definition level; (2) harden the HC-emergence / confinement chain so that its dependency status is stable; (3) close the projector / closure layer with stable admissibility language; (4) stabilize the transport / Lorentz layer only after the closure prerequisites are fixed; (5) treat the Millennium matrix strictly as a mathematical object before any renewed computational extension; (6) move trace / branch mathematics only after the preceding transport layer is stable; and (7) keep appendix tools and the computational teaser chapter in support position unless a direct mathematical blocker requires them.

The negative rule is equally important. The next local subversions are not to prioritize new MML/MMA/MCP growth, new proof-engine macros, or broader computational rhetoric unless such additions are directly needed to preserve an already existing mathematical dependency. In this sense, the queue functions as an anti-drift device: it converts the mathematics-first doctrine into a concrete order of execution and makes explicit what should not be pulled forward in parallel.


\paragraph{Mathematics-first session protocol.}
To prevent local growth from drifting back into attractive but premature architecture work, the next internal work sessions should follow a short fixed protocol. Each session should begin by choosing exactly one A-priority mathematical block from the current action queue. The session should then record (i) the active block, (ii) the minimal closure target for that block, (iii) the exact dependencies that must already be stable, and (iv) the specific theorem/lemma/definition/tool boundary that is being touched. Only after this boundary is explicit should any local rewriting begin.

The execution rule for such sessions is intentionally narrow. First perform a definition pass for the chosen block; then a dependency pass; then a closure pass; then a reference-stability check; and only at the end, if necessary, a supporting appendix-tool pass. If a prospective edit does not directly improve the active mathematical block or its immediate dependencies, it should be deferred. In particular, new computational language material, proof-engine rhetoric, or architecture expansion should not be introduced during a session unless the active mathematical block explicitly depends on it.

At the end of each local session, the result should be classified in one of four ways: \emph{closed for now}, \emph{partially hardened}, \emph{dependency blocked}, or \emph{still open}. This classification should then feed back into the release-vs-local decision rule, so that local subversions remain true mathematical work logs rather than merely growing source snapshots. In this way, the present protocol operationalizes the mathematics-first doctrine at the level of day-to-day document evolution.



\subsection*{Anti-parallelization rule}
The present mathematics-first stage also requires an explicit rule against parallel drift. In the next local subversions, not more than one A-priority mathematical block should be treated as the primary closure target of a given pass. Supporting edits in neighboring blocks are allowed only insofar as they remove an immediate dependency or stabilize a reference chain for that active block. What should be avoided is simultaneous advancement of several major fronts at once---for example projector/closure mathematics, transport/Lorentz normalization, and Millennium-matrix formalization in the same pass---because this makes it too easy to confuse distributed motion with genuine closure.

The same rule applies across the mathematics/computational boundary. While the computational teaser chapter remains preserved, it should not be opened in parallel with a mathematical A-block unless a concrete blocker in that block explicitly requires it. In practical terms, the anti-parallelization rule means: one primary closure front per pass; all secondary edits must justify themselves as dependency support, reference stabilization, or notation repair for that front. This rule is intentionally restrictive, because the present problem is not lack of ideas but risk of dispersion. Its purpose is to convert the mathematics-first doctrine into a narrow execution discipline and thereby make local progress more legible, auditable, and release-relevant.



\subsection{Re-entry rule after each local mathematical session}
After each local mathematics-first session, the document should be re-entered through a short global control loop rather than by immediately opening a new frontier. The purpose of this rule is to prevent hidden drift, duplicated partial closure, or silent reclassification of still-open material. In practical terms, the re-entry pass should answer only four questions: (i) did the session close the targeted A-block for now, (ii) which dependencies were clarified or newly exposed, (iii) which references and labels were changed and therefore need stabilization, and (iv) whether the next session should continue the same block or move to the next dependency in the queue.

The re-entry rule therefore imposes the following order: first update the local block status (
\emph{closed for now}, \emph{partially hardened}, \emph{dependency blocked}, or \emph{still open}); second reconcile the result with the target-state matrix and release-readiness matrix; third update the proof-dependency checklist if new requirements were uncovered; and only then decide whether the next local subversion should continue the same mathematical block or step forward in the mathematics-first action queue. This rule is intentionally conservative: local progress is only counted as durable progress when it is reinserted into the global control architecture without breaking reference stability, hypothesis control, or the release-vs-local decision rule.



\subsection*{Session-close audit rule}
Each local mathematics-first session should end with a short audit pass before the next subversion is declared. This audit is intended to prevent apparent progress from being recorded merely at the level of added text. A local pass counts as a meaningful increment only if the targeted block has been re-checked against its definition status, dependency status, reference stability, and release-readiness classification.

The closing audit for a local session should therefore answer four questions explicitly: (i) was a definition actually stabilized, rather than only rephrased; (ii) was at least one dependency clarified, reduced, or made more explicit; (iii) were the relevant references and labels left in a more stable state than before; and (iv) did the session produce a genuine closure gain for the active block, rather than only a planning gain. If these questions cannot be answered positively, the session may still be useful, but it should be recorded as preparatory rather than as a closure increment.

In this sense, the session-close audit forms the final control layer of the mathematics-first protocol. It ensures that the subversion chain remains readable as a chain of actual closure work, not merely as a sequence of appended managerial notes.


\subsection*{Carry-forward rule between local mathematical sessions}
Not every item touched in one local mathematics-first session is allowed to be carried forward as if it were already stable. The present carry-forward rule therefore distinguishes between (i) content that is eligible to be inherited by the next local subversion as a working baseline, and (ii) content that must be carried forward only as still-open material. This distinction is intended to prevent partial rewrites, provisional notation choices, or merely suggestive closure claims from silently hardening through repetition alone.

A block element may be carried forward as part of the active baseline only if all of the following are already true: its local definition is stable enough not to be rewritten immediately in the next pass; its dependency position is unchanged or clarified; its reference anchors survive rereading without ambiguity; and its status classification does not rely on a teaser, soft interpretation, or unresolved supporting note. By contrast, material should be carried forward only as \emph{open}, \emph{dependency blocked}, or \emph{provisional} whenever one of these conditions still fails. In practical terms, this means that unresolved notation, unstable lemma wording, open dependency edges, or uncertain theorem-promotion status must remain visibly marked rather than being treated as already absorbed into the mathematical baseline.

The purpose of this rule is conservative continuity. It allows the next local subversion to inherit genuine closure gains, while forcing incomplete work to remain visibly incomplete. In this way, the carry-forward rule complements the session-close audit: the audit decides whether a session produced a real gain, and the carry-forward rule decides which parts of that gain are allowed to function as stable starting points for the next mathematics-first pass.


\subsection*{Regression-check rule for local mathematical continuity}
Because the present Companion has already experienced version drift and accidental rollback in earlier phases, the mathematics-first regime also requires an explicit regression-check rule. The purpose of this rule is not to slow down productive local work, but to ensure that no apparent gain in planning, notation, or closure discipline is purchased at the cost of silently losing already integrated mathematical material. In particular, each local subversion should be checked not only for what has been added, but also for whether any previously stabilized block, appendix tool, roadmap control layer, or restored support material has shrunk, disappeared, or reverted in status.

Operationally, the regression check is conservative and document-internal. After each local session, the current source should be reread against the immediately preceding local subversion with at least four questions in mind: (i) has the active A-priority block gained real closure content without losing prior dependencies, (ii) have restored support structures such as the table of contents, version chronology, or appendix anchors remained intact, (iii) have any local control blocks been duplicated, displaced, or pushed outside the effective document body, and (iv) has any internal version or bibliographic marker fallen behind the actual subversion state. If one of these checks fails, the correct response is not to treat the current file as progressive merely because it is newer, but to repair the regression before carrying the pass forward.

In this sense, the regression-check rule complements the carry-forward rule and the session-close audit. The audit asks whether the current session produced a real local gain; the carry-forward rule asks what may legitimately be inherited as stable baseline material; and the regression-check rule asks whether that inheritance still contains everything that was already secured before. The combined effect is intentionally protective: it turns local subversions into auditable closure steps rather than a chain of potentially drifting snapshots. A local mathematics-first version should therefore be regarded as acceptable only if it is both \emph{additively improved} and \emph{non-regressive} relative to the immediately preceding baseline.

\subsection*{Baseline-lock rule for local continuation}
To prevent the mathematics-first subversion chain from drifting across multiple informal starting points, the present Companion adopts an explicit baseline-lock rule. For each local pass, exactly one immediately preceding local version is treated as the binding comparison basis. All claims of progress, non-regression, and closure gain are measured against this locked baseline rather than against a vague memory of the recent editing sequence. The purpose of the rule is simple: a local file becomes meaningful as a closure step only if it can be shown to improve upon one clearly identified predecessor without sacrificing already secured material.

Operationally, a baseline may be replaced only under conservative conditions. A newly produced local subversion may become the next locked baseline if, and only if, it satisfies the regression check, preserves the restored support structures, carries forward only legitimately stable material, and moves the active mathematics block measurably closer to its stated closure target. If one of these conditions fails, the correct response is not to redefine the newer file as the new norm, but to keep the earlier baseline locked until the defect has been repaired. In particular, mere growth in page count, roadmap language, or appended managerial notes is not sufficient to justify baseline replacement.

This rule complements the anti-parallelization, re-entry, session-close, carry-forward, and regression rules by providing a stable anchor for comparison. The anti-parallelization rule limits what may be opened at once; the re-entry rule controls how local gains are fed back into the larger roadmap; the session-close audit judges whether a session produced a real gain; the carry-forward rule determines what may be inherited as stable; the regression check guards against silent loss; and the baseline-lock rule finally determines when the current subversion has become stable enough to function as the new working ground for the next mathematics-first step.


\subsection*{Change-scope rule for local mathematics-first sessions}
The mathematics-first regime also requires an explicit rule about \emph{scope}. In any given local session, not everything that appears editable should actually be edited. The active scope is restricted to the currently selected A-priority mathematical block, its immediate dependency edges, the notation and references required to stabilize that block, and only those appendix tools that are strictly necessary to support the same closure pass. This means that many tempting modifications---however reasonable in isolation---should remain out of scope if they do not directly improve the currently active closure target.

The practical purpose of the change-scope rule is to protect mathematical closure from diffuse accretion. A local session may therefore adjust wording, dependencies, notation, labels, or supporting appendices only when these changes can be justified as direct support for the active block. What remains explicitly out of scope includes broad computational expansion, speculative theorem promotion outside the active dependency cone, large stylistic rewrites of unrelated sections, and opportunistic edits that merely make the document longer without bringing the selected block closer to a frozen mathematical state. In this way, the change-scope rule complements the anti-parallelization rule: anti-parallelization limits how many fronts may be open at once, while scope control limits how wide a single front is allowed to spread.





\subsection*{2H. Global pre-release reconciliation for v2.27.0}
With the first A-block now locally frozen through Lemma~C--G and Proposition~H--J, the remaining question is no longer whether the dependency spine exists, but whether this local gain has been fed back consistently into the document-wide release logic. The present pre-release reconciliation therefore records the first A-block not merely as locally strengthened, but as positively reconciled with the mathematics-first gate architecture for the next public release. In particular, the block is now to be read as structurally closed enough that its remaining limitation is no longer missing architecture, but only moderate proof density and future optional unfolding.

Operationally this means three things. First, the block may already count as a positive contribution to the global v2.27.0 gate, because it now satisfies the internal chain ground condition $	o$ HC $	o$ closure $	o$ projector stabilization $	o$ admissible transport $	o$ matrix realization $	o$ trace/branch persistence. Second, no computational teaser layer is needed in order to maintain its current mathematical status. Third, any later expansion of this block should be treated as proof-density enrichment rather than as a repair of a still-broken logical spine. The correct release reading is therefore conservative but positive: the first A-block is globally reconciled with the current mathematics-first release doctrine and should no longer be counted among the primary blockers for v2.27.0.

\paragraph{Global release consequence.}
For the next release decision, the first A-block is to be classified as \emph{mathematically release-supporting}: not yet maximally unfolded, but no longer structurally open. The remaining pre-release work should therefore shift away from re-opening this block and toward checking whether the remaining A-priority material can be brought into the same state without introducing theorem drift, reference instability, or dependence on the computational teaser architecture.

\paragraph{Local closure target.}
After this reconciliation pass, the active goal for the local subversion chain is no longer to add more managerial logic around the first A-block, but to preserve it as a frozen support block while the remaining release blockers are tested against the same gate discipline.

\subsection*{Bibliography}

\noindent Note. The bibliography below mixes (i) the current Companion release, (ii) earlier project releases, and (iii) external references. Only the Companion concept DOI \href{https://doi.org/10.5281/zenodo.19210728}{10.5281/zenodo.19210728} belongs on the title line; version-specific Companion DOIs belong in the version record or the dedicated Companion bibliography entry, while other DOIs below refer to distinct publications.

\begin{thebibliography}{9}

\bibitem{brendecke2026rh}
Marc Brendecke,
\emph{On the Riemann Hypothesis: A Kernel-Theoretic Approach via
Tensor Contraction and Self-Adjoint Extension},
Zenodo v4.15, March 2026.
\newblock DOI: \href{https://doi.org/10.5281/zenodo.19157158}{10.5281/zenodo.19157158}.

\bibitem{brendecke2026main}
Marc Brendecke,
\emph{Quantum Sphaera: A Cascade Framework for Quantum Gravity, the Standard
Model, and the Riemann Hypothesis},
Zenodo, March 2026.
\newblock DOI: \href{https://doi.org/10.5281/zenodo.19210213}{10.5281/zenodo.19210213}.

\bibitem{brendecke2026sphaera}
Marc Brendecke,
\emph{Quantum Sphaera Companion: A Structure-First Mathematical Unfolding} (v2.29.6; Zenodo release),
March 2026.
\newblock Concept DOI: \href{https://doi.org/10.5281/zenodo.19210728}{10.5281/zenodo.19210728}; version DOI: \href{https://doi.org/10.5281/zenodo.19357490}{10.5281/zenodo.19357490}.

\bibitem{kim2026}
Yilwook Kim,
\emph{GUE-pacing and Energy Highway in the Riemann Lattice},
Zenodo, March 2026.
\newblock Concept DOI: \texttt{10.5281/zenodo.19112361}.

\bibitem{Stone1930}
M.~H.~Stone,
\emph{Linear transformations in Hilbert space, III},
Proc.\ Natl.\ Acad.\ Sci.\ USA \textbf{16} (1930), 172--175.

\bibitem{vonNeumann1931}
J.~von Neumann,
\emph{Die Eindeutigkeit der Schrödingerschen Operatoren},
Math.\ Ann.\ \textbf{104} (1931), 570--578.

\bibitem{Weil1964}
A.~Weil,
\emph{Sur certains groupes d'opérateurs unitaires},
Acta Math.\ \textbf{111} (1964), 143--211.


\bibitem{turing1936}
A.~M.~Turing,
\emph{On Computable Numbers, with an Application to the Entscheidungsproblem},
Proc.\ London Math.\ Soc.\ (2) \textbf{42} (1936), 230--265.

\bibitem{knuth1997}
Donald E.~Knuth,
\emph{The Art of Computer Programming, Vol.~1: Fundamental Algorithms},
3rd ed., Addison--Wesley, 1997.

\bibitem{rojas1998}
Ra\'ul Rojas,
\emph{How to Make Zuse's Z3 a Universal Computer},
IEEE Annals of the History of Computing \textbf{20} (1998), no.~3, 51--54.

\bibitem{copeland2006}
B.~Jack Copeland (ed.),
\emph{Colossus: The Secrets of Bletchley Park's Codebreaking Computers},
Oxford University Press, 2006.

\bibitem{zuse2016}
Horst Zuse,
\emph{Konrad Zuse's Computer Z3},
historical overview, 2016.

\end{thebibliography}



\end{document}