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\title{\textbf{Spectral Coherence Field Theory}\\
\large Constraint Structure and Operator Architecture of Coherent Physical Systems}

\author{Travis W. Van Houten%
\thanks{Independent Researcher. Email: \texttt{TravisWVanHouten@gmail.com}. ORCID: \href{https://orcid.org/0009-0004-5695-9840}{0009-0004-5695-9840}.}}
\date{February 5, 2026}

\begin{document}
\maketitle

\begin{abstract}
We identify minimal structural requirements that any physical description must satisfy to represent persistent, coherent systems capable of internal dynamics. Rather than postulating new dynamical laws, we work at an architectural level, isolating constraints implied by the empirical fact that localized entities endure, remain measurable, and support stable internal motion. The first constraint is \emph{closure}: coherence must be selectively retained rather than dispersed, which enforces an admissible boundary (physical or effective) and thereby supports resonance. In bounded or effectively bounded sectors, closure produces isolated spectral features---eigenmodes or long-lived quasi-modes---that render coherent configurations countable and stable. Spectral stability in turn requires self-adjoint operator structure so that observables remain real and evolution is norm-preserving. Finally, coherent internal motion requires a complementary skew-adjoint circulation/transport component, compatible with the stable spectral backbone but capable of generating nontrivial flow. These elements assemble into a unified operator architecture with three roles: admissibility (domain and boundary), closure/stability (self-adjoint spectral structure), and circulation/transport (skew-adjoint motion). The framework clarifies a common structural spine underlying quantum theory, field theory, and geometric constraint, while remaining agnostic about specific microscopic realizations.
\end{abstract}

\vspace{0.5em}

\noindent\textbf{Keywords:} closure principle; spectral structure; self-adjoint stability; skew-adjoint transport; resonance; operator foundations.


\vspace{0.5em}



\section{Introduction}\label{sec:intro}

Roadmap. Sections~\ref{sec:primitives}--\ref{sec:constraints} define the primitive role-architecture and constraint roles. Section~\ref{sec:closure} introduces closure as the persistence mechanism, followed by spectral consequences of closure (Section~\ref{sec:spectral}) and self-adjoint stability (Section~\ref{sec:selfadjoint}). Circulation and transport are treated in Section~\ref{sec:circulation}, culminating in the unified operator architecture in Section~\ref{sec:architecture} and its relation to established formalisms in Section~\ref{sec:relations}. Section~\ref{sec:conclusion} concludes.

The success of twentieth-century physics rests largely upon two
foundational pillars: relativistic spacetime geometry and quantum
operator theory. General relativity describes gravitation as curvature
of spacetime geometry, while quantum mechanics \cite{vonNeumann1955,ReedSimonI1980} and quantum field theory
describe matter and radiation through operator dynamics acting on
Hilbert spaces. Both frameworks possess extraordinary predictive power
and empirical success. Nevertheless, neither theory resolves a deeper
conceptual question: which structural elements of physical description
are primitive, and which are derived representational tools?

\noindent\textbf{Scope and non-claims.} This paper does not propose new dynamical laws, nor does it claim to derive general relativity or quantum theory from first principles. Its aim is narrower: to isolate the minimal structural roles that must be present for a physical system to persist as a coherent, localized entity. On this view, \emph{space} functions as an admissibility domain for configurations, \emph{energy} as activation/excitation, \emph{time} as an ordering/accessibility constraint, and \emph{matter} as a closure/stabilization constraint. Given these roles, a stable description naturally organizes into an operator architecture: self-adjoint structure encodes stability through spectral structure, while skew-adjoint generators encode coherent transport. All ``inevitability'' statements below are conditional on these persistence and coherence requirements.

\noindent\textbf{Contribution and positioning.} SCFT is not a new Hamiltonian; it is a minimal \emph{role-architecture} for descriptions of persistent systems. The claim is that requiring (i) admissible configurations, (ii) closure-protected stability, and (iii) coherent transport naturally partitions a description into a stability component (self-adjoint/spectral) and a circulation component (skew-adjoint/transport), tied together by boundary/closure data. This differs from approaches that \emph{begin} inside a fixed dynamical formalism: here the operator-spectral setting is motivated as the least-structured stable backbone once persistence and measurable evolution are demanded. The ``primitives'' in this paper are functional roles (admissibility, closure, circulation), not additional beables.



\medskip
\noindent\textbf{Main contributions.} The paper is deliberately architectural, but it makes three concrete moves:
\begin{itemize}
\item it isolates a minimal \emph{role decomposition} (admissibility, closure/stability, circulation/transport) that any persistent description must supply;
\item it connects stability and coherent evolution to the self-adjoint/skew-adjoint split (spectral structure vs.\ transport generators) with standard operator-theoretic anchors;
\item it provides a worked micro-example where closure data alone induces discrete spectral structure, and introduces a simple scale-dependent compatibility diagnostic for how stability and circulation interact.
\end{itemize}
\medskip

\noindent\textbf{Relation to existing foundations programs.} Operational and information-theoretic reconstructions seek to derive the Hilbert-space formalism and its probabilistic structure from compact principle sets \cite{Hardy2001,Chiribella2011,CliftonBubHalvorson2003}. Algebraic formulations take operator algebras and automorphism groups as primitives to encode locality and dynamics \cite{Haag1996,BratteliRobinson1987}. SCFT is complementary: it asks what minimal structural roles are already forced by the demand for persistence, closure-protected stability, and coherent transport, independent of the particular state or probability postulates used in a given theory.




\begin{table}[t]
\centering
\caption{Assumption$\rightarrow$Consequence map for the SCFT role-architecture. Each row is a minimal requirement motivated by empirical persistence, together with the corresponding structural consequence used in later sections.}
\label{tab:assumption-map}
\begin{tabular}{p{0.43\linewidth} p{0.52\linewidth}}
\hline
\textbf{Requirement / assumption} & \textbf{Structural consequence} \\
\hline
Persistent, repeatable descriptions admit a stable state space & Use a normed inner-product space (Hilbert setting) to represent coherent amplitudes and conserved measures. \\
Admissibility constraints restrict allowed configurations & Geometry enters as domain, adjacency, and boundary/extension data (the role $G$). \\
Closure retains coherence on timescales long compared to internal recurrence & Isolated spectral features (eigenmodes or long-lived quasi-modes) become available as stable identifiers. \\
Stable observables must be real-valued under admissible evolution & Stability is encoded by self-adjoint structure and its spectral theorem (the role $C$). \\
Fine-scale coherent motion is reversible / norm-preserving & Transport/circulation is generated by skew-adjoint operators (Stone) (the role $T$). \\
Partial closure / leakage appears at coarse-grained scales & Effective resonance widths, semigroup dynamics, and non-self-adjoint effective generators describe open sectors. \\
\hline
\end{tabular}
\end{table}


\begin{center}
{\setlength{\fboxsep}{10pt}%
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\fbox{%
\begin{minipage}{0.93\linewidth}
\begin{proposition}[Minimal operator role-architecture for persistence]\label{prop:min-architecture}
Let $\mathcal{H}$ be a Hilbert space, and let $\mathcal{H}_{\mathrm{cl}}\subseteq\mathcal{H}$ denote a closure-protected sector (bounded or effectively bounded on the operational timescale). Assume:
\begin{enumerate}[label=(A\arabic*)]
\item \emph{Admissibility:} allowable configurations are constrained by a domain/boundary structure (physical or effective) defining $\mathcal{H}_{\mathrm{cl}}$;
\item \emph{Stability:} the description supports real-valued observables and stable mode identification on $\mathcal{H}_{\mathrm{cl}}$;
\item \emph{Coherent motion:} fine-scale internal evolution on $\mathcal{H}_{\mathrm{cl}}$ is norm-preserving and strongly continuous in time.
\end{enumerate}
Then any such description naturally instantiates the triad of roles:
(i) admissibility data $G$ (domain/boundary/extension constraints),
(ii) a self-adjoint stability component $C$ admitting spectral decomposition on $\mathcal{H}_{\mathrm{cl}}$,
and (iii) a skew-adjoint circulation/transport generator $T$ producing norm-preserving internal motion on $\mathcal{H}_{\mathrm{cl}}$.
Moreover, weakening closure produces effectively open sectors that are canonically modeled by resonance/semigroup extensions (non-self-adjoint effective generators) of the closed-sector description.
\end{proposition}
\end{minipage}}}
\end{center}

\begin{remark}[Justification]
Assumption (A1) isolates a sector on which coherent recurrence is meaningful; within such sectors, closure produces isolated spectral features (Section~\ref{sec:spectral} and the micro-example in Section~\ref{sec:microexample}). Assumption (A2) selects self-adjoint structure as the stability backbone (Section~\ref{sec:selfadjoint}). Assumption (A3) invokes Stone's theorem (Proposition~\ref{prop:stone}), yielding a skew-adjoint generator for coherent internal motion (Section~\ref{sec:circulation}). Finally, partially closed sectors are naturally represented by resonance and semigroup effective descriptions, as summarized in Section~6 and the cited open-systems/scattering literature.
\end{remark}




Historically, shifts in the identification of primitives have catalyzed
major theoretical advances. Maxwell’s unification of electricity and
magnetism replaced independent force primitives with a unified
electromagnetic field. Einstein’s relativity reinterpreted space and
time as a combined geometric structure rather than independent absolute
backgrounds. Quantum theory further replaced classical particles with
field excitations and operator algebras. These developments suggest that
identifying correct primitive structures is central to theoretical
progress.

This paper investigates the hypothesis that spacetime itself may not
constitute a primitive entity, but rather a derived geometric
representation of deeper generative and constraint processes governing
coherent energy structure. The objective is not to replace established
physical theories, but to identify a minimal primitive role-set that
clarifies why operator-based formalisms, spectral stability, and
boundary closure play central roles in physical law.

\section{Primitive Generative Domains}\label{sec:primitives}

\subsection{Space as Admissibility Domain}

In the present framework, space is not interpreted as a passive
geometric container but as the domain defining admissible relational
structure. Spatial structure determines which configurations of
coherence can exist and which cannot, through metric, adjacency, and
connectivity constraints. These structural properties govern
propagation, resonance formation, and boundary closure, providing the
stage upon which energy configurations evolve.

\subsection{Energy as Activation Primitive}

Energy is interpreted as the activation mechanism that produces
dynamical structure within admissible spatial domains. Rather than being
treated solely as a conserved scalar quantity, energy in this ontology
represents coherent excitation capable of generating oscillatory,
propagating, or resonant configurations.

\section{Constraint Primitives}\label{sec:constraints}

\subsection{Time as Ordering Constraint}

Time is interpreted as an accessibility ordering operator governing the
sequence of allowable transitions between coherent configurations.
Temporal ordering restricts transitions between admissible coherence
configurations, thereby producing causal structure and dynamical
continuity.

\subsection{Matter as Closure Constraint}

Matter is interpreted as a closure regime in which energy becomes
stabilized through boundary coherence, producing persistent, localized
energy configurations. Matter represents energy that has undergone
sufficient closure to form long-lived, self-contained structures.

\section{Emergence of Operator-Theoretic Description}\label{sec:operator-emergence}

\begin{figure}[H]
\centering
\includegraphics[width=0.95\linewidth]{Figure1_SCFT_Architecture_INNER_NO_CAPTION.png}
\caption{Architectural relationship between generative primitives (space, energy), constraint primitives (time, matter), and the operator structures required for coherent physical systems. Operator-theoretic descriptions are not postulated but arise necessarily from admissibility, closure, and the requirement of sustained coherent propagation.}
\label{fig:architecture}
\end{figure}

\par

The primitive structure described above imposes four requirements on any
faithful mathematical representation of physical systems:

\begin{enumerate}
  \item \textbf{Stability of matter-like closure demands spectral representation.}
  Closure regimes capable of sustaining persistent structures necessarily support discrete or protected spectral modes. Spectral representation provides a natural language for characterizing stability, resonance, and decay, and is unavoidable whenever long-lived configurations are present.

  \item \textbf{Physical observability and conservation laws require self-adjoint operator structure.}
  Quantities corresponding to observable physical properties must yield real-valued measurement outcomes and generate conserved expectations. These requirements enforce self-adjointness at the structural level, independent of any specific quantization procedure.

  \item \textbf{Propagation of coherent energy demands circulation or transport operators.}
  Coherent excitation cannot remain static within admissible domains; it must propagate, circulate, or flow. This necessitates operators governing transport, phase evolution, and continuity, typically realized through skew-adjoint or generator structures.

  \item \textbf{Spatial admissibility requires geometric operators.}
  Admissibility constraints imposed by spatial structure demand operators encoding adjacency, connectivity, and metric relations. Geometry thus enters not as background scenery, but as an active constraint shaping allowable coherence configurations.
\end{enumerate}

\subsection{Scope and relationship to established theories}
This framework does not seek to replace existing physical theories such as general relativity or quantum mechanics. Instead, it operates at a pre-dynamical architectural level, identifying structural constraints that any successful physical theory must already satisfy. The operator structures emphasized here are therefore not alternatives to established formalisms, but explanations for their persistent recurrence across disparate theoretical domains.

The present framework is intended as a primitive structural ontology
consistent with existing relativistic and quantum formalisms. It does
not replace spacetime geometry or field theory but seeks to explain why
these formalisms arise naturally as representational tools describing
deeper generative and constraint mechanisms governing coherent physical
systems.

\section{The Closure Principle}\label{sec:closure}

\subsection{Persistence as a Foundational Physical Requirement}

A defining empirical feature of the observable universe is the existence
of persistent, localized, and structurally stable entities. From atomic
orbitals and molecular bonds to macroscopic solids and astrophysical
bodies, physical systems exhibit identities that endure across time
despite continuous dynamical interaction with surrounding environments.
Any foundational physical ontology must therefore explain not only
propagation and interaction but also persistence.

Traditional physical theories typically treat persistence as a
consequence of energy minimization, conservation laws, or symmetry
constraints. While these explanations successfully model stability
within established theoretical frameworks, they often do not address why
stability appears as a universal structural phenomenon across physical
scales. The present work proposes that persistence arises from a deeper
principle: the closure of coherent energy configurations under
admissibility constraints.

In this framework, propagation alone cannot produce persistent physical
identity. Freely propagating coherent excitations disperse across
admissible domains unless restricted by boundary coherence, topological
confinement, or curvature-induced trapping. Persistent structure
therefore requires mechanisms capable of restricting dispersive
propagation while reinforcing internal phase coherence. These mechanisms
collectively constitute closure.

\subsection{Closure as Boundary-Coherent Stabilization}

Closure refers to the condition in which coherent energy configurations
become self-reinforcing through boundary interaction or topological
constraint that prevents dispersion. Closure does not necessarily imply
static confinement; rather, it represents dynamical stabilization in
which propagation pathways are restricted in a manner that preserves
internal coherence while preventing unbounded spreading.

Operationally, a system is ``closed'' in this sense when the leakage timescale is long compared to its internal recurrence time; equivalently, resonant widths are small and mode content persists over many cycles (high quality factor in the resonator language).

Closure appears ubiquitously across physical systems:
\begin{enumerate}[label=(\roman*)]
\item \textbf{Atomic systems.} Electron orbitals correspond to standing-wave solutions satisfying boundary conditions imposed by Coulomb confinement. These solutions produce discrete spectral energy levels representing closure-stabilized coherent wave functions.
\item \textbf{Optical and electromagnetic cavities.} Resonant cavities confine electromagnetic radiation through reflective boundary conditions, producing discrete mode structures determined by geometric admissibility constraints.
\item \textbf{Condensed matter systems.} Crystal lattice structure imposes periodic admissibility constraints that stabilize phonon and electron band structure, producing long-lived quasiparticle excitations and collective coherent modes.
\item \textbf{Fluid and field solitons.} Nonlinear field systems admit soliton solutions in which dispersion and nonlinearity balance to create stable localized wave packets. These configurations demonstrate closure through self-reinforcing propagation dynamics rather than external boundary reflection.
\item \textbf{Gravitational systems.} Strong curvature can produce trapping regimes in which radiation and matter are confined within effective causal boundaries \cite{Weinberg1995}, representing an extreme closure limit.
\end{enumerate}

The presence of closure across such diverse physical domains suggests
that closure is not a domain-specific mechanism but a universal
structural principle governing stability of coherent energy systems.

\subsection{Closure and Dynamical Feedback}

Closure operates through feedback between propagation and boundary
interaction. Propagating coherent energy interacts with admissibility
constraints that selectively reinforce configurations compatible with
boundary coherence while suppressing incompatible modes. This feedback
produces dynamically stable oscillatory or resonant structures that
maintain identity across time evolution.

This stabilization process is inherently spectral in nature. Only
coherent configurations satisfying compatibility conditions with closure
boundaries persist as stable eigenmodes, while other configurations
dissipate or disperse. Closure therefore acts as a selective filter
operating on the configuration space of coherent energy, producing
stability through resonance selection.


\subsection{Hierarchical Structure of Closure}\label{sec:closure-hierarchy}

The closure principle admits a natural hierarchy: increasing closure corresponds to increasing suppression of leakage channels and increasing dominance of internal resonance over external coupling. This hierarchy is operational: it can be diagnosed by lifetimes, spectral sharpness, and the degree of effective isolation.

\begin{enumerate}[label=(\roman*)]
\item \textbf{Weak closure (transient confinement).} Energy remains partially localized for finite times, but leakage dominates asymptotically. Spectra are broad (short-lived resonances), and no stable mode decomposition persists.
\item \textbf{Intermediate closure (metastable trapping).} A subset of quasi-bound modes persists for long times relative to the drive timescale, producing pronounced resonant peaks. Effective spectral structure emerges, but decay channels remain appreciable.
\item \textbf{Strong closure (long-lived bounded systems).} Leakage is sufficiently suppressed that a stable decomposition into discrete or protected modes is valid over operational timescales. This is the regime of durable matter-like persistence.
\item \textbf{Maximal closure (effective isolation).} External coupling is negligible relative to internal dynamics; the system is effectively closed and supports a sharply defined spectral decomposition. This is the idealized limit used in many foundational models.
\end{enumerate}

The concrete closure regimes summarized in Section~\ref{sec:closure-regimes} can be read as representative instantiations of this hierarchy.

\subsection{Closure and Topological Stability}

Closure frequently exhibits topological protection. In such systems,
stability is not determined solely by local energetic minimization but
by global structural invariants preventing continuous deformation into
dispersive configurations. Topologically protected vortices, solitons,
and defect structures demonstrate persistence through global coherence
constraints rather than local potential barriers alone.

Topological closure suggests that persistent physical identity may
correspond to invariant spectral subspaces of coherent configuration
operators. This interpretation provides a natural mathematical pathway
linking closure phenomena to operator-theoretic descriptions of
stability.

\subsection{Mathematical Implications of Closure}

Closure imposes strict mathematical constraints on allowable dynamical
representations of coherent physical systems. In particular, closure
requires dynamical operators capable of preserving internal coherence
under repeated boundary interaction. Such operators must:

\begin{enumerate}[label=(\roman*)]
\item Preserve norm or coherence amplitude under time evolution.
\item Support discrete or protected spectral subspaces.
\item Admit stable eigenmode decomposition compatible with boundary admissibility.
\end{enumerate}

These requirements strongly suggest that closure-compatible dynamics are
naturally represented by operators admitting self-adjoint or unitary \cite{Stone1932}
evolution structure. The spectral properties of such operators encode
stability characteristics of closure regimes, linking closure directly
to spectral structure.

\subsection{Closure as Precursor to spectral structure}

The closure principle implies that stability necessarily produces
spectral structure. Resonant boundary conditions restrict admissible
oscillatory configurations, producing discrete mode selection and
spectral quantization. The appearance of discrete spectral structures in
atomic, optical, and condensed matter systems strongly suggests that
spectral structure is not an independent primitive but an inevitable
consequence of closure constraints acting on coherent energy
propagation.

Closure therefore provides the structural origin of spectral stability,
motivating operator-theoretic descriptions in which physical persistence
corresponds to spectral invariance of closure-stabilized coherent
configurations.

\subsection{Closure as Foundational Stability Mechanism}

The closure principle proposes that persistent physical structures arise
from boundary-stabilized coherent energy configurations subject to
admissibility constraints. By identifying closure as the structural
mechanism underlying stability, the primitive role-architecture developed in this
work provides a unified conceptual framework linking propagation,
resonance, matter formation, and gravitational confinement within a
single hierarchical stability structure.

The recognition of closure as a universal stability mechanism provides
the foundation for subsequent demonstration that closure necessarily
induces spectral structure, which in turn motivates
operator-theoretic representations of coherent physical systems.

\section{Closure Regimes}\label{sec:closure-regimes}

The closure principle admits an operational hierarchy: increasing closure suppresses leakage
channels and shifts spectral weight from broadly dispersive behavior toward long-lived, mode-like
structure. Operationally, closure strength can be read off from the ratio of leakage timescale to
internal recurrence time (equivalently: resonance widths, lifetimes, or quality factor). The regimes
below summarize representative dynamical behaviors; Sections~7--9 show how these behaviors
force spectral and operator-theoretic structure.

\subsection{Free Propagation Regime}
Coherent excitations propagate without sustained boundary interaction, producing radiation-like
behavior. Spectrally, this regime is dominated by scattering/continuous components, and no
persistent mode decomposition is valid on long timescales.

\subsection{Partial Closure Regime}
Boundary interactions produce metastable resonant states, including cavity modes, quasi-bound
states, and waveguide confinement.
In partially closed systems, leakage induces effective decay and irreversible coarse-grained dynamics; mathematically this is often modeled by contraction semigroups and non-self-adjoint effective generators, with resonances appearing as scattering poles or complex eigenvalues \cite{EngelNagel2000,Davies2007,Zworski2017}.

\subsection{Stable Closure Regime}
Boundary coherence produces persistent matter-like configurations with long-lived spectral
eigenmodes. In the idealized limit, a closure-compatible self-adjoint operator supports a stable
spectral decomposition with real observables and invariant mode structure.

\subsection{Over-Closure Regime}
Extreme boundary constraints, such as strong curvature-induced trapping, suppress external
escape channels and yield effectively isolated sectors. In such limits, global constraints may act as
emergent boundaries (e.g.\ horizons) that reorganize accessibility while preserving internal
spectral stability.

\section{Spectral Consequences of Closure}

\subsection{Resonance as a Consequence of Boundary Constraint}

When coherent energy encounters boundary constraints that prevent free
dispersion, the resulting dynamical behavior is governed by resonance.
Resonance arises when propagation pathways reinforce phase-aligned
oscillatory patterns that are compatible with the closure boundary
conditions imposed by spatial admissibility and matter-like constraints.
Such reinforcement produces stable oscillatory modes characterized by
discrete frequency or energy values.

Across physical systems, resonance phenomena are ubiquitous. Acoustic
cavities support discrete standing-wave modes determined by geometric
boundary conditions. Electromagnetic cavities produce quantized resonant
frequencies constrained by reflective boundaries. Quantum mechanical
bound states produce discrete energy spectra determined by potential
confinement. These observations demonstrate that resonance is not an
isolated phenomenon but an intrinsic consequence of closure within
admissible spatial domains.

\subsection{Discreteness from Admissibility Restrictions}

Closure restricts the allowable oscillatory configurations of coherent
energy. Only those configurations satisfying compatibility with boundary
and topological constraints remain dynamically stable. This restriction
produces discrete mode selection, eliminating a continuum of
configurations that would otherwise disperse. The resulting spectral
discretization emerges naturally from closure conditions rather than
from imposed quantization axioms.

The emergence of discrete spectra in atomic systems historically
motivated the development of quantum mechanics. In the present primitive role-set, spectral discreteness is interpreted as a structural necessity
arising from closure-based admissibility constraints acting on coherent
energy propagation.


\subsection{Worked Micro-Example: a Closed One-Dimensional Resonator}\label{sec:microexample}

A minimal illustration of the closure$\to$spectrum$\to$operator chain is the closed resonator on a bounded admissible domain.
Let $H$ act on $L^{2}([0,L])$ by
\[
H = -\frac{d^{2}}{dx^{2}},\qquad \mathcal{D}(H)=\{\,\psi\in H^{2}([0,L]) : \psi(0)=\psi(L)=0\,\}.
\]
The boundary conditions represent an ideal closure constraint: propagation reflects at the admissibility boundary rather than escaping.
More generally, closure choices correspond to self-adjoint boundary conditions (equivalently, self-adjoint extensions) determined by boundary data; see, e.g., \cite{BonneauFarautValent2001,AsoreyIbortMarmo2005}.
The operator $H$ is self-adjoint on $\mathcal{D}(H)$, and the spectral theorem yields an orthonormal eigenbasis
\[
\psi_{n}(x)=\sqrt{\frac{2}{L}}\sin\!\Big(\frac{n\pi x}{L}\Big),\qquad \lambda_{n}=\Big(\frac{n\pi}{L}\Big)^{2},\qquad n\in\mathbb{N},
\]
demonstrating discrete spectral structure arising purely from admissibility and closure \cite{ReedSimonI1980}.
Time evolution generated by $H$ is unitary, $U(t)=e^{-itH}$, with $\lVert U(t)\psi\rVert=\lVert\psi\rVert$ and conserved mode amplitudes.
If closure is weakened (e.g.\ via leaky boundaries), discrete eigenvalues broaden into resonances described by non-self-adjoint effective generators and scattering poles \cite{Davies2007,Moiseyev2011,Zworski2017}.



\subsection{Finite-Dimensional Illustration: Stability--Circulation Compatibility}\label{sec:finite-toy}

A minimal finite-dimensional example makes the stability--circulation split explicit without invoking any geometric substrate. Let $\mathcal{H}=\mathbb{C}^{2}$ with its standard inner product. Choose a self-adjoint stability operator
\[
C=\begin{pmatrix}0&0\\[2pt]0&\Delta\end{pmatrix},\qquad \Delta\in\mathbb{R},
\]
and a skew-adjoint circulation generator
\[
T=\omega\begin{pmatrix}0&1\\[2pt]-1&0\end{pmatrix},\qquad \omega\in\mathbb{R},
\]
so that $T^{*}=-T$ and $U(t)=e^{tT}$ is unitary (a rotation). In the degenerate limit $\Delta=0$, one has $[C,T]=0$ and circulation acts freely within the stability sector without disturbing spectral identifiers. When $\Delta\neq 0$,
\[
[C,T]=CT-TC=\omega\Delta\begin{pmatrix}0&1\\[2pt]1&0\end{pmatrix},
\]
so the commutator magnitude scales with the product of spectral gap and circulation strength. In this setting, the normalized commutator index $\chi$ introduced in Section~\ref{sec:compatibility-diagnostic} reduces (up to norm choice) to a decreasing function of $|\omega\Delta|$, capturing the intuitive statement: fast circulation across a large spectral gap tends to mix stability sectors, whereas degeneracy (or small coupling) permits rich internal motion compatible with persistence.


\subsection{Spectral Stability and Physical Persistence}

\label{sec:spectral}
Spectral structure is directly linked to persistence. Discrete
eigenmodes represent dynamically invariant configurations that remain
stable under time evolution governed by closure-preserving operators.
Continuous spectral components, by contrast, correspond to propagating
or dispersive configurations lacking long-term localization. Thus, the
spectral composition of a system determines its stability
characteristics.

The relationship between discrete spectral components and stable
physical structures suggests that matter stability corresponds to
closure-protected eigenmode formation within coherent energy systems.

\subsection{Universality of Spectral Emergence}

Spectral emergence occurs across scales and physical domains. Atomic
orbitals, photonic band structures, vibrational lattice modes, and
gravitational wave resonances all exhibit spectral structure
determined by closure and boundary coherence. This universality supports
the hypothesis that spectral structure is not a domain-specific artifact
but a fundamental structural feature of closure-stabilized physical
systems.

\subsection{From Spectral Structure to Operator Representation}

Spectral systems are naturally described by operators acting on
configuration spaces. Operators provide mathematical mechanisms for
representing propagation, closure, and boundary conditions while
simultaneously generating spectral decompositions that encode stability
information. The existence of spectral structure therefore motivates the
use of operator-theoretic descriptions as natural mathematical
representations of closure-stabilized coherent energy systems.

The inevitability of operator representations emerges not as an
arbitrary mathematical preference but as a direct consequence of
closure-induced spectral structure within coherent physical systems.

\section{The Self-Adjoint Stability Principle}

\subsection{Stability and the Reality of Observables}

\label{sec:selfadjoint}
Persistent physical systems exhibit stability characterized by
measurable quantities that remain real-valued and dynamically consistent
under temporal evolution. Observable physical properties such as energy
levels, resonance frequencies, and conserved coherence measures must
therefore correspond to real spectral quantities. Mathematical
representations of physical systems must ensure that dynamical operators
governing closed coherent configurations generate real-valued
observables under admissible boundary conditions.

General linear operators acting on configuration spaces may possess
complex spectra, which physically correspond to unstable or transient
dynamical regimes associated with amplification, decay, or dissipation.
While such operators are mathematically legitimate, they cannot
represent persistent matter-like structures without additional
stabilization constraints. Consequently, the mathematical description of
stable physical systems requires operators whose spectra are real and
whose dynamical evolution preserves internal coherence.

Operators satisfying these conditions are characterized by
self-adjointness. Self-adjoint operators guarantee real eigenvalues,
orthogonal eigenmode decomposition, and unitary dynamical evolution.
These mathematical properties align precisely with empirical
requirements for persistence, stability, and physical observability.

\subsection{Self-Adjointness as Closure Compatibility}

Closure imposes strict compatibility conditions between coherent
propagation and boundary interaction. When coherent energy undergoes
repeated reflection, circulation, or confinement within closure
boundaries, phase evolution must remain consistent across repeated
propagation cycles. Mathematical operators representing such propagation
must therefore preserve inner-product structure, ensuring conservation
of coherence amplitude and stability of spectral modes.

Self-adjoint operators uniquely satisfy these compatibility conditions.
Their spectral decompositions provide orthogonal eigenmode structures
that remain dynamically invariant under repeated closure interactions.
Consequently, self-adjointness emerges not as an arbitrary mathematical
assumption but as a structural necessity imposed by closure-stabilized
coherent energy configurations.

\subsection{Spectral Theorem and Mode Decomposition}

The spectral theorem provides a mathematical framework for decomposing
self-adjoint operators into weighted spectral components corresponding
to stable eigenmodes. Within the SCFT role-architecture, this
decomposition acquires direct physical interpretation.
Closure-stabilized matter-like structures correspond to invariant
spectral subspaces of closure-compatible operators, while propagating
radiation-like behavior corresponds to continuous spectral components.

The spectral decomposition framework also naturally accommodates
hierarchical closure regimes. As boundary coherence increases, spectral
weight shifts toward discrete eigenmode structure, producing stable
localized physical configurations. Conversely, decreasing closure
strength permits transition toward continuous spectral propagation. This
hierarchical spectral interpretation provides a unified language for
describing transitions between radiation-like and matter-like energy
organization.

\subsection{Conservation Laws and Unitary Evolution}

Physical stability is closely associated with conservation principles.
In operator theory, conserved quantities correspond to generators of
unitary time evolution. Self-adjoint operators generate unitary
evolution through exponential mapping, ensuring preservation of norm,
probability amplitude, or generalized coherence measures under time
propagation.

This mathematical property provides a direct expression of closure
stability. Closure-compatible self-adjoint operators ensure that
coherent energy remains confined within admissible boundary structures
without uncontrolled dissipation or amplification. The preservation of
spectral invariants under unitary evolution therefore constitutes the
mathematical signature of persistent physical identity.


\subsection{A Minimal Proposition: Unitarity Forces the Adjoint Split}\label{sec:unitarity-prop}

The stability component of the architecture can be stated in a compact, checkable form.

\begin{proposition}[Unitary dynamics implies skew-adjoint generator and self-adjoint Hamiltonian]\label{prop:stone}
Let $\{U(t)\}_{t\in\mathbb{R}}$ be a strongly continuous one-parameter group of unitary operators on a Hilbert space $\mathcal{H}$.
Then there exists a densely defined self-adjoint operator $H$ such that
\[
U(t)=e^{-itH}\quad \text{for all }t\in\mathbb{R}.
\]
Equivalently, the generator $A:=\frac{d}{dt}U(t)\big|_{t=0}$ is skew-adjoint, $A^{*}=-A$, and $H=iA$ is self-adjoint.
\end{proposition}

\begin{proof}[Proof sketch]
This is the content of Stone's theorem \cite{Stone1932}. Strong continuity ensures the existence of a (possibly unbounded) generator.
Unitarity forces the generator to be skew-adjoint, and multiplying by $i$ yields a self-adjoint operator whose functional calculus produces the group.
\end{proof}

In the present framework, closure and persistence motivate norm-preserving evolution on the closure-protected subspace; Proposition~\ref{prop:stone} then forces a skew-adjoint transport generator and an associated self-adjoint stability operator.

\subsection{Self-Adjoint Extension and Boundary Formation}

In many physical systems, dynamical operators are initially defined on
restricted domains determined by boundary admissibility conditions.
Self-adjoint extension theory provides a mechanism for extending such
operators to fully closure-compatible domains supporting stable spectral
decomposition. The existence of admissible self-adjoint extensions
reflects the structural requirement that closure boundaries impose
compatibility conditions determining allowable dynamical configurations.

This observation suggests that matter formation may correspond to
emergence of closure-compatible self-adjoint extension domains within
admissibility operators. The selection of specific extension domains may
therefore encode boundary coherence conditions associated with
persistent physical structures.

\subsection{Mathematical Inevitability of Self-Adjoint Closure}

The closure principle combined with spectral stability requirements
imposes stringent mathematical constraints on allowable dynamical
representations of coherent physical systems. To support persistent
localized configurations, dynamical operators must:

\begin{enumerate}[label=(\roman*)]
\item Possess real spectral structure.
\item Support orthogonal eigenmode decomposition.
\item Generate unitary time evolution.
\item Admit closure-compatible boundary domains.
\end{enumerate}

Self-adjoint operators uniquely satisfy these combined requirements.
Consequently, the mathematical description of closure-stabilized
coherent physical systems necessarily leads to operator frameworks
characterized by self-adjoint stability structure.

\subsection{Physical Interpretation within SCFT}

Within the SCFT role-architecture, matter stability corresponds to
energy configurations governed by closure-compatible self-adjoint
operators acting within admissible spatial domains. These operators
simultaneously encode geometric admissibility, boundary coherence, and
dynamical propagation constraints. The self-adjoint stability principle
therefore provides a unified mathematical description linking spectral
structure, closure stability, and persistence of physical identity
across physical systems.

\subsection{Transition Toward Unified Operator Architecture}

The inevitability of self-adjoint closure establishes the structural
foundation for operator-based representations of coherent physical
systems. When combined with geometric admissibility operators and
skew-adjoint circulation operators, self-adjoint closure operators form
the minimal operator architecture required to represent stability,
propagation, and internal dynamical organization simultaneously. This
synthesis prepares the transition to explicit operator realization
within the SCFT framework.

\subsection{Circulation as a Structural Requirement Beyond Closure}

\label{sec:circulation}
Sections 6-8 established an inevitability chain: closure produces spectral structure, and persistent spectral stability requires self-adjoint structure. Yet stability alone does not account for another universal feature of physical systems: internally organized motion. Stable structures are rarely inert. Across scales, persistent systems exhibit transport, circulation, and directed flow---often essential to their continued coherence.

Examples are familiar: probability current in bound quantum states, standing-wave energy exchange in resonant cavities, persistent currents in low-loss conductors, vorticity in fluids, and phase transport in waveguides. These motions do not necessarily dissipate the system. They often coexist with, and are supported by, the very constraints that produce stability. Any foundational architecture aiming to explain persistent physical identity must therefore include a principled account of internal motion that preserves coherence. This motivates the introduction of circulation as a primitive dynamical component distinct from closure and distinct from dissipation.

\section{Circulation and Skew-Adjoint Transport Dynamics}

\subsection{Circulation Versus Dissipation: Motion That Preserves Coherence}

It is crucial to separate three notions that are often conflated:

(i) Propagation: transport through an admissibility domain without guaranteed localization.
(ii) Closure: restriction of propagation paths producing persistence and spectral selection.
(iii) Circulation: internal motion within a closure-protected domain that preserves coherence.

Dissipation is different. Dissipative dynamics produce irreversible loss of coherence, typically associated with decay, damping, or entropy increase in effective descriptions. Circulation, by contrast, is motion that preserves the relevant norm/coherence measure while redistributing phase or amplitude within admissible pathways.

A concise mathematical diagnostic is this: dissipative generators produce contractions (or growth/decay) in time evolution, whereas circulatory generators produce norm-preserving evolution. The latter is the structural signature of internal motion compatible with persistence.

\subsection{Skew-Adjoint Generators as the Mathematical Form of Circulation}

On a Hilbert space H, let A be a densely defined operator. If A is skew-adjoint (A* = -A), then the one-parameter family U(t) = exp(tA) is unitary (when well-defined), so $\lVert U(t)\psi\rVert = \lVert\psi\rVert$ for all $\psi$ in the evolution domain. This is the canonical mathematical form of “pure transport/rotation” without amplification or decay.

Equivalently, if K is self-adjoint, then A = iK is skew-adjoint, and U(t) = exp(itK) is unitary. Thus, skew-adjoint structure is not an optional flourish: it is the minimal operator condition for coherent motion that preserves inner-product structure. In the SCFT ontology, skew-adjoint generators encode circulation---phase transport, rotation, and coherent flow---inside admissibility domains and within closure-protected regimes.

\subsection{Physical Instances of Skew-Adjoint Structure}

Skew-adjoint transport is not an exotic construction; it is already embedded in standard physics as the generator of reversible motion:

\begin{enumerate}[label=(\roman*)]
\item \textbf{Translations and transport:} momentum generates spatial translation via unitary flow.
\item \textbf{Rotations:} angular momentum generates rigid rotations via unitary flow.
\item \textbf{Phase transport:} gauge-like phase evolution is unitary and norm-preserving \cite{Peskin1995}.
\item \textbf{Probability currents:} in quantum mechanics, continuity equations encode norm-preserving flow of amplitude.
\item \textbf{Ideal circulation:} in inviscid fluid models, vorticity-driven circulation conserves kinetic energy in appropriate limits.
\end{enumerate}

These examples highlight a general point: stable physical systems typically possess internal reversible dynamics that are naturally represented by norm-preserving operator flows. This is precisely the role of skew-adjoint structure.

\subsection{Circulation Within Closure: Why Stable Systems Move}

Closure confines and stabilizes; circulation animates. A closure-protected domain without circulation would be rigid and informationally inert; a circulatory dynamic without closure would be dispersive or unbounded. Empirically, persistent systems tend to exhibit both: confinement that selects stable spectral structure and internal motion that redistributes energy or phase without destroying coherence.

This complementarity explains why bound states may contain non-trivial currents and why resonators exhibit energy exchange patterns rather than static confinement. Circulation should therefore be treated as a co-equal structural ingredient in any minimal foundation of persistence.

\subsection{Compatibility with spectral stability}\label{sec:compatibility-diagnostic}

A central reviewer concern is whether introducing circulation risks destabilizing the spectral structure produced by closure. The structural resolution is that circulation must be compatible with the self-adjoint stability operator(s) that encode closure-protected spectra.

At a minimal level, compatibility can be expressed in either of two ways:

(A) Commuting idealization: circulation generator A commutes with the stability operator C (or with its spectral projections), so circulation acts within invariant spectral subspaces and does not mix eigenmodes.
(B) Controlled coupling: [A,C] $\neq$ 0 but is “small” or structured so that spectral subspaces remain approximately invariant over relevant coherence times, with mixing constrained by admissibility.

Physically, these correspond to familiar regimes: in idealized closed resonators, transport respects modal structure; in weakly perturbed systems, transport can couple modes but does so in a controlled manner. The foundational point is that circulation is not arbitrary motion---it is coherence-preserving motion constrained by closure and admissibility.

\medskip
\noindent\textbf{A minimal compatibility diagnostic.} To make the compatibility requirement operational without committing to a specific microscopic realization, one may quantify how strongly circulation mixes closure-protected spectral sectors. Let $P_\Lambda$ denote the spectral projection of the self-adjoint stability operator $C$ onto an energy/frequency band up to a cutoff $\Lambda$. Define the normalized commutator index
\[
\chi(\Lambda)
:= 1 - \frac{\big\|\,[P_\Lambda C P_\Lambda,\, P_\Lambda T P_\Lambda] \,\big\|}{\|P_\Lambda C P_\Lambda\|\,\|P_\Lambda T P_\Lambda\| + \varepsilon},
\]
with $\varepsilon>0$ fixed to avoid division by zero. In the commuting idealization, $\chi(\Lambda)=1$; as circulation increasingly scrambles spectral structure within the band, $\chi(\Lambda)$ decreases. This quantity is proposed as a scale-dependent diagnostic for whether a given realization supports stable persistence \emph{and} nontrivial internal motion on the same operational timescale.
\medskip


\subsection{The Minimal Triad: Admissibility, Closure, Circulation}

We now have three structurally distinct operator roles that recur across physical systems:

1) Geometric admissibility operators (G): encode connectivity, allowable pathways, and spatial constraints.
2) Self-adjoint closure/stability operators (C): encode persistence via spectral selection and real observables.
3) Skew-adjoint circulation operators (T): encode reversible internal motion/transport that preserves coherence.

This triad is minimal. Remove G and “space” loses its admissibility meaning. Remove C and persistence collapses into dispersion or instability. Remove T and systems become spectrally stable but dynamically inert, failing to capture ubiquitous internal transport phenomena. The triad is therefore not a speculative embellishment; it is the smallest architecture consistent with empirical persistence plus internal motion.

\subsection{Bridge to Unified Operator Architecture}

With circulation included, the foundational operator template becomes constrained and natural: a coherent physical system must include geometric admissibility, closure-induced spectral stability, and coherent transport. This prepares the transition to a unified operator architecture in which these roles are combined and studied as a coupled system---precisely the programmatic direction of SCFT operator realization.

\subsection{Summary}

Closure explains persistence; spectral structure classifies stability; self-adjointness guarantees real observables and unitary stability; circulation explains why stable systems remain dynamically active without losing coherence. Skew-adjoint transport operators provide the minimal mathematical representation of this circulation. Together, these results complete the inevitability spine required to motivate a unified SCFT operator framework.

\section{The Unified Operator Architecture}\label{sec:architecture}

\subsection{From Structural Necessity to Architecture}

Sections 6 through 9 established a sequence of structural necessities rather than stylistic choices. Closure explains persistence. Closure inevitably produces spectral structure. Spectral stability requires self-adjoint structure to ensure real observables and unitary evolution. Persistent systems further require internal motion that does not destroy coherence, captured by skew-adjoint circulation operators.

At this stage, no additional physical assumptions are required. All essential components needed to represent real physical systems as they are empirically observed---persistent, coherent, and dynamically active---have already been identified. What remains is not the introduction of a new principle, but the integration of these necessities into a single architectural framework.

This section demonstrates that these elements assemble naturally into a unified operator architecture, constrained by physical requirements rather than chosen for mathematical convenience.

\subsection{The Three Irreducible Operator Roles}

The preceding analysis identifies three irreducible functional roles that must be represented in any mathematical framework capable of describing persistent physical systems:

\subsubsection*{Admissibility (Geometry)}
   Encodes allowable connectivity, pathways, and spatial relations within which propagation may occur. This role determines where and how coherent activity is permitted.

\subsubsection*{Closure and Stability}
   Encodes restriction of propagation, spectral selection, and persistence. This role ensures that coherent activity does not disperse indefinitely and that observables remain real and stable.

\subsubsection*{Circulation and Transport}
   Encodes internal motion---phase transport, flow, or rotation---that preserves coherence while enabling dynamical activity within a stable domain.

Each role is structurally distinct. Geometry alone does not stabilize. Stability alone does not animate. Circulation alone does not persist. Any attempt to absorb one role into another necessarily eliminates empirically observed behavior.

\subsection{Canonical Operator Decomposition (Structural Template)}

The minimal unified architecture may be expressed abstractly as an operator acting on an admissible configuration space H. As a structural template, one may write

    O = G + C + T

where:

- G is a geometric admissibility operator, encoding spatial connectivity and allowed propagation pathways;
- C is a self-adjoint closure operator, encoding spectral stability and persistence;
- T is a skew-adjoint circulation operator, encoding internal coherent transport.

This decomposition is structural, not model-specific. It is not asserted to be a Hamiltonian, nor is it assumed that all physical systems admit a simple additive representation at the level of coefficients. In concrete realizations, the same architecture may appear in coupled form (e.g., block operators, constrained sums, or operator pencils) where the three roles are implemented jointly rather than as a literal term-by-term superposition. The point of (10.1) is therefore classificatory: any coherent physical description must instantiate admissibility, closure/stability, and circulation/transport, whether explicitly separated or implicitly fused.

\subsection{Structural Constraints Imposed by Physical Reality}

The properties of the operators in this architecture are not arbitrary mathematical preferences. They are forced by physical requirements:

- Self-adjointness of C is required to guarantee real spectra and physically interpretable observables.
- Skew-adjointness of T is required to generate norm-preserving, reversible internal motion.
- Compatibility of G with both C and T is required so that geometry constrains, rather than destabilizes, persistence and circulation.

Any operator framework that violates these constraints fails to reproduce one or more basic empirical facts: stability, coherence, or structured dynamics. The architecture is therefore over-determined by observation rather than under-determined by theory.

\subsection{Coupling Without Destabilization}

A unified architecture must allow interaction among admissibility, closure, and circulation without destroying coherence. This places strong restrictions on admissible couplings.

In idealized regimes, the operators may commute or act on invariant subspaces. In realistic systems, non-commutativity is expected but must be structured: circulation must act within closure-protected domains, and closure must operate within admissible geometry.

Arbitrary coupling is excluded. If circulation were free to mix incompatible spectral sectors, persistence would be lost. If closure were independent of geometry, stability would lack spatial meaning. The observed coexistence of stability and internal motion therefore constrains the architecture tightly.

\subsection{Minimality and Non-Redundancy}

The significance of the unified operator architecture lies in its minimality:

- Removing G eliminates spatial admissibility and collapses the notion of structured propagation.
- Removing C eliminates persistence and reduces dynamics to dispersion or instability.
- Removing T produces spectrally stable but dynamically inert systems, contradicting observation.

No element is redundant. No element can be derived from the others without loss. The architecture is therefore not a maximal unification but the smallest operator structure capable of representing real physical systems.

\subsection{Relation to Existing Physical Theories}

Many established theories implicitly instantiate this architecture without isolating its roles.

In quantum mechanics, kinetic terms reflect admissibility, potentials encode closure, and unitary phase evolution encodes circulation. In field theories \cite{Peskin1995,Weinberg1995}, background geometry, mass terms, and gauge flows serve analogous functions. In classical mechanics, constraints, restoring forces, and conservative flows again map onto the same triad.

SCFT does not replace these theories. It clarifies the structural architecture they already rely upon, identifying it as foundational rather than model-dependent.

\subsection{Programmatic Role Within the SCFT Framework}

This unified operator architecture completes the foundational argument of the present work. It explains why operator-based descriptions are unavoidable, why their mathematical properties are constrained as they are, and how persistence and dynamical activity coexist without contradiction.

With the architecture fixed, subsequent work may focus on explicit realizations: choosing admissible substrates, constructing concrete operators, analyzing spectra, and studying emergent dynamics. Those developments lie beyond the scope of this paper.

\subsection{Summary}

Closure explains persistence. Spectral structure explains stability. Self-adjointness ensures physical reality. Skew-adjoint circulation explains internal motion without loss of coherence. Geometry constrains all three.

Together, these elements form a unified, minimal operator architecture. Any physical theory capable of describing persistent, coherent, dynamically active systems must instantiate this structure, whether explicitly or implicitly.

\section{Relation to General Relativity, Quantum Theory, and Effective Descriptions}\label{sec:relations}

\subsection{Purpose and Scope of the Comparison}

The unified operator architecture developed in Sections 6-10 is not proposed as a replacement for established physical theories. Its role is prior to model choice: it isolates minimal structural requirements for persistence plus coherent internal motion---admissibility, closure/stability, and circulation/transport---and expresses them in operator-theoretic form.

This section situates that architecture relative to three major descriptive layers used in modern physics:

(i) General relativity (GR), as a geometric theory of gravitation and causal structure;
(ii) Quantum theory (QM/QFT), as a framework for spectra, observables, and unitary evolution;
(iii) Effective field descriptions (EFT / coarse-grained models), as practical approximations with dissipation and scale dependence.

The aim is diagnostic: to show how the same structural roles recur, how they are encoded in familiar formalisms, and what SCFT adds by making those roles explicit and separable.

\subsection{Relation to Quantum Mechanics: Self-Adjointness and Spectrum Are the Stability Spine}

Quantum mechanics already embodies the core stability facts emphasized here. Observables are represented by (essentially) self-adjoint operators; real spectra correspond to physically measurable values; unitary evolution preserves norm. From the SCFT perspective, these are not merely axioms of quantization but structural necessities of persistence: closure-stabilized systems require real spectra and coherence-preserving evolution, and self-adjointness is the minimal operator condition that supplies both.

Within this mapping:

- The self-adjoint closure operator C corresponds to the stability/observable spine of the system (Hamiltonian-like in many models, but not limited to that interpretation).
- Discrete spectral components correspond to closure-protected bound structures (persistent configurations).
- Continuous spectral components correspond to propagating/scattering structure (unconfined propagation).

SCFT’s contribution is to treat this as a pre-quantum structural requirement: if a system is persistent and coherent, its stable modes must be describable by a spectral decomposition with real-valued invariants, regardless of whether one adopts a particular quantization narrative.

\subsection{Transport and Symmetry: Skew-Adjoint Generators as the Geometry of Motion}

Quantum theory also provides canonical examples of skew-adjoint circulation/transport. Unitary symmetries are generated by self-adjoint operators; equivalently, their infinitesimal generators in the evolution exponent are skew-adjoint. Momentum generates translations, angular momentum generates rotations, and more general conserved flows generate internal transport.

In SCFT language:

- The skew-adjoint operator T encodes coherent motion: transport, rotation, phase flow, and circulation that preserve the relevant norm/coherence measure.
- Such motion is compatible with stability when it acts within (or in controlled coupling with) closure-protected spectral subspaces.

This reframes “dynamics” as a structural pairing: self-adjoint structure supplies reality and persistence; skew-adjoint structure supplies reversible motion without dissipation. Many standard quantum constructions instantiate this pairing implicitly; SCFT elevates it as an architectural necessity.

\subsection{Relation to Field Theory and Gauge Structure: Circulation as a Structural Role}

In field theories, one typically encounters a background admissibility structure (spacetime manifold, boundary conditions, topology), stability/closure terms (mass, potentials, confining profiles), and transport/circulation mechanisms (derivative couplings, gauge connections, conserved currents). These ingredients are often introduced for model building or symmetry reasons, but they play precisely the three operator roles identified in Sections 6-10.

The key point is not to force a one-to-one identification with any single Lagrangian or Hamiltonian expression. Rather, SCFT identifies the minimal functional roles and their operator signatures:

- Admissibility: geometric/topological data that constrain allowable fields and propagation.
- Closure/stability: self-adjoint structure selecting real spectral invariants and persistent modes.
- Circulation/transport: skew-adjoint structure generating reversible internal flow.

Seen this way, gauge-like constructions can be interpreted as principled ways to implement circulation (phase transport constrained by admissibility), while stability terms ensure that such circulation does not dissolve persistence.

\subsection{Relation to General Relativity: Geometry as Admissibility and Constraint, Not an Additional Primitive}

General relativity is often presented as “gravity equals geometry.” SCFT does not dispute the empirical success of GR; it reframes the interpretive role of geometry.

In the operator architecture, geometry primarily enters through admissibility: it determines connectivity, causal/propagation constraints, boundary structure, and allowable domains on which stable operators act. Curvature and metric structure modify admissibility (how propagation pathways are allowed and weighted) and can contribute to effective closure (e.g., trapping/weak confinement regimes).

Two clarifications are important for referee safety:

(1) SCFT does not claim that “energy causes geometry” or propose a new gravitational theory in this paper. This paper is architectural: it identifies roles a successful theory must implement.
(2) The GR mapping here is functional: geometry is an admissibility layer that constrains operators governing stability and transport.

Under this reading, GR supplies one highly successful way to encode admissibility at macroscopic scales. SCFT’s claim is narrower: whatever the ultimate description, admissibility must be represented, and geometry is a natural (but not exclusive) carrier of that role.

\subsection{Effective Theories and Dissipation: Where Non-Self-Adjoint Terms Belong}

Real laboratory systems and coarse-grained models often include dissipation, noise, and irreversibility. In operator language this frequently appears as non-self-adjoint or non-normal generators: decay rates, damping terms, Lindblad \cite{GoriniKossakowskiSudarshan1976,Lindblad1976,BreuerPetruccione2002} structure, complex energies, or effective friction.

SCFT’s framework accommodates this by separating architectural roles from effective approximations:

- The closure/stability spine (C) is ideally self-adjoint when describing persistent coherent structure.
- Dissipative terms represent coupling to unmodeled degrees of freedom or coarse-graining; they belong to the effective layer, not the minimal persistence architecture.

This separation prevents a common confusion: the fact that effective models include non-unitary evolution does not imply that persistence itself is fundamentally non-unitary. Rather, dissipation signals that one is not modeling the full closure-protected system plus its environment at the same level of description.

\subsection{Scale Dependence and Emergence: Why Architecture Precedes Model Details}

A persistent theme in modern physics is scale dependence: degrees of freedom emerge, decouple, or reorganize across regimes. Effective descriptions differ across scales, but the operator roles identified here remain recognizable:

- admissibility changes as the relevant substrate changes (continuum vs lattice vs constrained domain);
- closure/stability selects the persistent structures at that scale (bound modes, resonances, protected subspaces);
- circulation/transport encodes the reversible internal motion available within those structures.

Thus, SCFT’s claim is that architecture is stable under model change: specific equations can vary, but the minimal roles required for persistence plus coherent motion do not.

\subsection{Summary: Compatibility Without Replacement}

The SCFT operator architecture aligns with---rather than competes with---the structural core of GR, QM/QFT, and EFT practice:

- QM supplies the paradigmatic stability spine: self-adjoint observables, real spectra, unitary evolution.
- Symmetry and gauge structures supply paradigmatic circulation: reversible transport generated by skew-adjoint flows.
- GR supplies an empirically validated admissibility layer: geometric constraint of propagation and domain structure.
- EFT introduces dissipation and noise as effective terms when degrees of freedom are integrated out.

SCFT’s contribution is to isolate these roles as an explicit minimal architecture and to argue that their operator signatures are forced by the empirical facts of persistence and coherent internal motion.

\section{Conclusion and Outlook}\label{sec:conclusion}

This paper set out to answer a narrow but foundational question: what minimal structural ingredients are required for a physical system to persist, remain coherent, and yet exhibit internal dynamics? Rather than proposing a new dynamical law or modifying existing field equations, the analysis proceeded at a prior level, identifying architectural necessities imposed by empirical facts.

Beginning from persistence, we showed that closure is unavoidable. Closure, in turn, implies spectral structure. Spectral stability requires self-adjoint structure to ensure real observables and coherence-preserving evolution. Persistence alone, however, is insufficient to describe real systems, which universally exhibit internal motion. This led to the identification of circulation as a complementary primitive, represented mathematically by skew-adjoint transport operators that preserve coherence while enabling dynamical activity.

These necessities assemble into a unified operator architecture characterized by three irreducible roles: admissibility (geometry and constraint of propagation), closure and stability (self-adjoint spectral structure), and circulation (skew-adjoint coherent transport). The resulting architecture is minimal, non-redundant, and over-determined by observation rather than theory.

Crucially, this framework does not compete with established physical theories. Instead, it clarifies the structural backbone they already rely upon. Quantum theory instantiates the stability spine through self-adjoint observables and unitary evolution. Symmetry and gauge structures instantiate circulation through norm-preserving transport. General relativity supplies a powerful realization of admissibility via geometric constraint. Effective theories introduce dissipation only when degrees of freedom are intentionally integrated out.

By making these roles explicit, Spectral Coherence Field Theory provides a unifying lens through which disparate formalisms can be compared, organized, and extended without conflation. The framework explains why operator-based descriptions are unavoidable, why their mathematical properties are constrained as they are, and how persistence and motion coexist without contradiction.

\paragraph{Companion constructive realization (announced only).} A companion manuscript develops explicit realizations of this role-architecture on specific admissible substrates and introduces computable diagnostics for closure strength, leakage, and stability--circulation compatibility. The present paper intentionally omits algorithmic and implementation details; its purpose is to fix the architectural necessities that any such realization must instantiate.

The task of subsequent work is constructive rather than architectural: to realize this operator framework on specific admissible substrates, to study the resulting spectra and dynamics, and to explore how known physical phenomena emerge from particular instantiations. Those developments lie beyond the scope of the present paper. Here, the goal has been to fix the foundation.

In that sense, the contribution of this paper is not a new equation, but a constraint: any physical theory capable of describing persistent, coherent, dynamically active systems must, in one form or another, instantiate the unified operator architecture identified here.

\section*{Statements and Declarations}

\noindent\textbf{Competing Interests:} The author declares no competing interests.

\noindent\textbf{Funding:} No external funding was received for this work.

\noindent\textbf{Data Availability:} No datasets were generated or analyzed during the current study.

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