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% ===== TITLE =====
\title{Modal--Liouville Strings in 3+1:\\ Worldsheet Consistency, SM Spectrum and a Single Vacuum}
\author{John Douglas Coulson \\ j.coulson1@proton.me}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We construct a non--critical string theory directly in \(3{+}1\) dimensions. The worldsheet conformal anomaly is cancelled by a physical Modal--Liouville compensator \(\Phi_{\mathrm{modal}}\) with a clean dictionary to Manifold Quantum Gravity (MQG). The internal sector is a rational chiral algebra \(\mathcal{A}_{\mathrm{modal}}\) that realises the Standard Model gauge algebra and chiral matter without extra dimensions or Calabi--Yau moduli. We establish worldsheet consistency (central charge balance, vanishing \(\beta\)--functions on MQG on--shell backgrounds, BRST nilpotency, modular invariance), construct gauge and matter via simple--current RCFT, derive the \(4\)D effective action with \(\alpha'_{\!m}=\Gamma_c^{-1}\), show that continuous moduli are lifted, prove a finite one--loop vacuum set further pruned by \(\Gamma\)--stability, and exhibit standard open--string gauge amplitudes with Regge behaviour and a Yang--Mills limit. The apparent landscape collapses to a \emph{single} vacuum consistent with the Standard Model, uniquely correlated with a recursion depth in MQG. The framework is falsifiable: it predicts no light geometric moduli and only near--equilibrium supersymmetry.
\end{abstract}

\newpage
\tableofcontents
\newpage

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

The origins of string theory lie in the dual--resonance models of the late 1960s and early 1970s: 
Veneziano’s amplitude \citetag{VENEZIANO1968}, the independent worldsheet formulations by 
Nambu, Nielsen, and Susskind \citetag{SUSSKIND1970}, and the Virasoro--Shapiro construction 
\citetag{VIRASHAP1969}. 
These models were later reinterpreted as critical strings in higher--dimensional targets, culminating 
in the ten--dimensional superstring frameworks of the 1980s \citetag{GREENSGHWIT1987} \citetag{POLCHINSKI1}. 
A further conceptual leap was the holographic principle, articulated by ’t~Hooft and Susskind 
\citetag{SUSSKIND1995}, which reshaped our understanding of spacetime/field theory duality.

String theory in its critical form typically enforces worldsheet consistency via higher--dimensional 
targets and special--holonomy compactifications. Here we replace extra dimensions by a physical 
compensator and an internal rational algebra anchored in Manifold Quantum Gravity (MQG). 
We call the compensator the \emph{Modal--Liouville} field \(\Phi_{\mathrm{modal}}\); it cancels the 2D conformal anomaly in \(D=4\). 
The internal sector \(\mathcal{A}_{\mathrm{modal}}\) encodes recursion/braid data and supplies the gauge/matter content.

Our results are organised as \emph{decision gates}:
\begin{itemize}[leftmargin=1.2em]
\item \textbf{Gate A (worldsheet consistency).} Central charge cancellation in \(D=4\), \(\beta\)--functions that vanish on MQG on--shell backgrounds, BRST nilpotency, and one--loop modular invariance.
\item \textbf{Gate B (gauge \& matter).} \(\mathfrak{u}(1)\oplus\mathfrak{su}(2)\oplus\mathfrak{su}(3)\) from \(\mathcal{A}_{\mathrm{modal}}\), chiral families via simple--current projection, anomaly cancellation.
\item \textbf{Gate C (effective action \& finiteness).} \(4\)D string/Einstein dictionary; lifting of continuous moduli; finite one--loop vacua; pruning by \(\Gamma\)--stability.
\item \textbf{Gate D (amplitudes \& dualities).} Open--string gauge vertices and the 4--point tree amplitude with Regge slope \(\alpha'_{\!m}/2\) and a Yang--Mills limit; dualities as internal/simple--current and recursion symmetries.
\end{itemize}

\paragraph{MQG dictionary.}
We use \(\alpha'_{\!m}=\Gamma_c^{-1}\) (coherence cutoff), \(g_s=e^{\Phi_0}\), and interpret ``branes'' as coherence sheets (recursion level sets). 
The internal algebra \(\mathcal{A}_{\mathrm{modal}}\) is rational (finite primaries, modular characters).

\newpage

% ===== STANDING ASSUMPTIONS =====
\section{Standing assumptions}
\label{sec:assumptions}

We fix a \(3{+}1\)D target with metric \(g_{\mu\nu}\) and a spacelike Modal--Liouville background charge \(Q\) in the unitarity domain. 
The internal sector is a unitary, rational, modular chiral algebra \(\mathcal{A}_{\mathrm{modal}}\).

\paragraph{A1. Modal--Liouville compensator.}
\(\Phi_{\mathrm{modal}}\) couples as 
\[
\frac{1}{4\pi}\int_\Sigma \sqrt{h}\,R^{(2)}\Phi .
\]
The background charge \(Q\) is taken spacelike to avoid timelike ghosts and is assumed to lie in the Seiberg unitarity domain, so that all physical vertex operators have bounded conformal weight. Its value is fixed by central--charge balance and the MQG decoherence deficit.

\paragraph{A2. Internal RCFT.}
\(\mathcal{A}_{\mathrm{modal}}\) is unitary and rational with central charge \(c_{\mathrm{int}}\). Its characters form a finite--dimensional representation of \(\mathrm{SL}_2(\mathbb{Z})\). We assume at least one modular invariant partition function exists, and in practice restrict to simple--current extensions and their finite character sets.

\paragraph{A3. MQG dictionary.}
We identify
\[
\alpha'_{\!m}=\Gamma_c^{-1}, \qquad g_s = e^{\Phi_0} .
\]
Spectral gaps \(\Delta_{\mathrm{spec}}>0\) freeze would--be moduli, and recursion stability defines admissible sectors. 
The equivalence between MQG on--shell equations and the \(\sigma\)--model Weyl conditions at \(O(\alpha')\) is not postulated here but proven in Appendix~\ref{app:mqg-sigma} (Theorem~\ref{thm:F-main}).

\paragraph{A4. Stability/GSO projection.}
Physical sectors satisfy a stability constraint \(\Gamma \ge 0\), coinciding with a GSO--like projection that removes tachyons. This ensures unitarity of the physical Hilbert space.

\paragraph{A5. BRST framework.}
We use the RNS formulation with standard ghost sectors. Physical states are defined by BRST cohomology; nilpotency of the BRST charge is checked explicitly in Gate~A.

\paragraph{A6. Modularity.}
The torus partition function with multiplicity matrix \(N\) is required to be modular invariant under \(\mathrm{SL}_2(\mathbb{Z})\). Modular invariance after simple--current extension or projection is assumed as a standing condition.

\paragraph{A7. Anomaly cancellation.}
Chiral spectra are required to satisfy gauge, mixed, and gravitational anomaly constraints. Hypercharge normalisation is compatible with field--theory conventions. Explicit anomaly cancellation is verified at Gate~B; spectra failing this condition are discarded.

\begin{remark}[Falsifiability]
If BRST or modular invariance fails in \(D=4\), Gate~A fails. 
Detection of light geometric moduli falsifies Gate~C. 
Discovery of additional long--range \Uone{} factors falsifies Gate~B.
\end{remark}

\newpage

% =========================
% GATE A — WORLDSHEET CONSISTENCY
% =========================
\section{Gate A: Worldsheet consistency in 3+1}
\label{sec:gateA}

We work in RNS (the bosonic case can be treated in parallel; we record the formulas for both where illuminating).
The worldsheet action is
\begin{equation}
S \;=\; \frac{1}{4\pi \alpha'_{\!m}}\!\int_\Sigma\!\!\sqrt{h}\,h^{ab}\,g_{\mu\nu}(X)\,\partial_a X^\mu \partial_b X^\nu 
\;+\;\frac{1}{4\pi}\!\int_\Sigma\!\!\sqrt{h}\,R^{(2)}\,\Phi(X)
\;+\; S_{\mathrm{int}}[\mathcal{A}_{\mathrm{modal}}]
\;+\; S_{\mathrm{gh}} ,
\end{equation}
with \(\alpha'_{\!m}=\Gamma_c^{-1}\) (Assumption~A3), \(\Phi\equiv \Phi_{\mathrm{modal}}\) the spacelike Modal--Liouville field (Assumption~A1), and \(\mathcal{A}_{\mathrm{modal}}\) a unitary rational internal CFT (Assumption~A2). The Liouville/linear--dilaton coupling implements a background charge \(Q\), shifting the stress tensor and central charge in the standard way \citetag{SEIBERG90}, \citetag{POLCHINSKI1}.

\subsection*{A.1 Central charge cancellation in \(D=4\)}
\begin{lemma}[Central charge bookkeeping]
\label{lem:A-cc}
Let \(c_{\mathrm{m}}\) be the matter central charge from \((X^\mu,\psi^\mu)\), \(c_{\mathrm{gh}}\) the ghost central charge, \(c_{\mathrm{int}}\) that of \(\mathcal{A}_{\mathrm{modal}}\), and \(c_L(Q)\) the (super-)Liouville contribution induced by the spacelike background charge \(Q\).
Then
\begin{equation}
c_{\mathrm{tot}}\;=\;c_{\mathrm{m}}+c_{\mathrm{int}}+c_L(Q)+c_{\mathrm{gh}} .
\end{equation}
For the bosonic string:
\(\;c_{\mathrm{m}}=D,\; c_{\mathrm{gh}}=-26,\; c_L(Q)=1+6Q^2\).
For RNS:
\(\;c_{\mathrm{m}}=\tfrac{3}{2}D,\; c_{\mathrm{gh}}=-15,\; c_L(Q)=\tfrac{3}{2}+3Q^2\).
\end{lemma}

\begin{proof}
Standard: add the contributions of free fields, ghosts, and the background-charge shift of the stress tensor; see \citetag{FRIEDAN84}, citetag{SEIBERG90}, \citetag{POLCHINSKI1}. For the spacelike linear dilaton, the central charge shift is \(+6Q^2\) (bosonic) and \(+3Q^2\) (RNS) in addition to the free scalar/fermion baseline \(1\) and \(\tfrac{1}{2}\) per target dimension, respectively.
\end{proof}

\begin{proposition}[Existence of admissible \((Q,c_{\mathrm{int}})\) in \(D=4\)]
\label{prop:A-cc-sol}
In \(D=4\) there exist spacelike \(Q\) and rational \(c_{\mathrm{int}}\ge 0\) with \(c_{\mathrm{tot}}=0\).
Explicitly, in RNS:
\begin{equation}
c_{\mathrm{tot}}=0
\quad\Longleftrightarrow\quad
Q^2 \;=\; \frac{15-\tfrac{3}{2}D-\tfrac{3}{2}-c_{\mathrm{int}}}{3}
\;=\; \frac{9-c_{\mathrm{int}}}{3}\qquad (D=4),
\end{equation}
so \(c_{\mathrm{int}}=4\) gives \(Q^2=\tfrac{5}{3}\), and \(c_{\mathrm{int}}=0\) gives \(Q^2=3\).
In the bosonic case:
\begin{equation}
c_{\mathrm{tot}}=0
\quad\Longleftrightarrow\quad
Q^2 \;=\; \frac{25-D-c_{\mathrm{int}}}{6}
\;=\; \frac{21-c_{\mathrm{int}}}{6}\qquad (D=4).
\end{equation}
\end{proposition}

\begin{proof}
Insert the values of Lemma~\ref{lem:A-cc} and solve \(c_{\mathrm{tot}}=0\). Spacelike \(Q\) is compatible with the Seiberg unitarity domain \citetag{SEIBERG90}.
\end{proof}

\subsection*{A.2 \(\beta\)–functions and MQG on–shell}
\begin{lemma}[Sigma–model \(\beta\)–functions to \(O(\alpha')\)]
\label{lem:A-beta}
For vanishing \(B\)–field and slowly varying backgrounds, the Weyl conditions through \(O(\alpha')\) read
\begin{align}
\beta^g_{\mu\nu} &= \alpha'_{\!m}\Big(R_{\mu\nu} + 2\nabla_\mu\nabla_\nu \Phi\Big) + O(\alpha'^2), \label{eq:beta-g}\\
\beta^\Phi &= \frac{c_{\mathrm{tot}}}{6} + \alpha'_{\!m}\Big(\Box\Phi - (\nabla\Phi)^2 + \tfrac{1}{2}R\Big) + O(\alpha'^2), \label{eq:beta-phi}
\end{align}
with \(c_{\mathrm{tot}}\) as in Lemma~\ref{lem:A-cc}. 
\end{lemma}

\begin{proof}
These are the standard sigma–model \(\beta\)–functions for the metric and dilaton (up to scheme choices that reshuffle \(\Box\Phi\) and \((\nabla\Phi)^2\) terms), see \citetag{CALLAN85}, \citetag{FRADKINTSEYTLIN85}, \citetag{POLCHINSKI1}. We adopt the scheme consistent with the string–frame action \(S\sim\int e^{-2\Phi}(R+4(\nabla\Phi)^2+\cdots)\).
\end{proof}

\begin{proposition}[MQG on–shell \(\Rightarrow \beta=0\)]
\label{prop:A-mqg-onshell}
Assume \(c_{\mathrm{tot}}=0\) (Proposition~\ref{prop:A-cc-sol}) and that the MQG on–shell equations coincide with \eqref{eq:beta-g}–\eqref{eq:beta-phi} through \(O(\alpha')\) (Assumption~A3). Then \(\beta^g=\beta^\Phi=0\) and the background is Weyl–invariant to this order.
\end{proposition}

\begin{proof}
Immediate from Lemma~\ref{lem:A-beta} once \(c_{\mathrm{tot}}=0\) and the MQG field equations match the sigma–model tensors. 
\end{proof}

\subsection*{A.3 BRST nilpotency and no–ghost domain}
\begin{theorem}[BRST nilpotency in \(D=4\)]
\label{thm:A-brst}
Under \(c_{\mathrm{tot}}=0\) and \(\beta=0\) of Proposition~\ref{prop:A-mqg-onshell}, the quantum BRST charge satisfies \(Q_{\mathrm{BRST}}^2=0\).
\end{theorem}

\begin{proof}
In the BRST formalism, \(Q_{\mathrm{BRST}}^2\) probes (i) the Virasoro central term and (ii) Weyl non–invariance of composite operators; see \citetag{FRIEDAN84}, \citetag{POLCHINSKI1}. With \(c_{\mathrm{tot}}=0\) the Virasoro anomaly cancels, and with \(\beta=0\) the renormalisation of the background couplings vanishes; hence the nilpotency obstruction is absent. 
\end{proof}

\begin{proposition}[No–ghost domain (spacelike Liouville)]
\label{prop:A-noghost}
With \(Q\) spacelike and in the Seiberg unitarity domain (Assumption~A1), the physical spectrum defined by BRST cohomology is ghost–free.
\end{proposition}

\begin{proof}
The no–ghost argument extends from the critical case (see \citetag{POLCHINSKI1} and the classic no–ghost theorems \citetag{GODDARDTHORN}) to the linear–dilaton background provided the background charge is spacelike so that norms remain positive and the Liouville momenta obey the Seiberg bound \citetag{SEIBERG90}. 
\end{proof}

\subsection*{A.4 One–loop modular invariance}
\begin{proposition}[Modular invariant torus partition function]
\label{prop:A-modZ}
Let \(Z\) factorise as
\begin{equation}
Z(\tau,\bar\tau)= Z_X\, Z_{\Phi}\, Z_{\mathrm{int}}\, Z_{\mathrm{gh}},
\end{equation}
with
\(
Z_X=(\mathrm{Im}\,\tau)^{-D/2}\,|\eta(\tau)|^{-2D}
\) for \(D=4\),
\(Z_{\mathrm{gh}}\) the standard ghost factor,
\(Z_{\mathrm{int}}=\sum_{i,\bar\imath}N_{i\bar\imath}\,\chi_i(\tau)\,\overline{\chi_{\bar\imath}(\tau)}\) a rational modular invariant (Assumption~A2/A6), and
\begin{equation}
Z_{\Phi} \;=\; (\mathrm{Im}\,\tau)^{-1/2}|\eta(\tau)|^{-2}
\int_{\mathbb{R}} \! dp\, \exp\!\Big[-\pi\,\mathrm{Im}\,\tau\,\big(p^2+Q^2\big)\Big],
\end{equation}
the spacelike linear–dilaton (Liouville zero–mode) factor \citetag{SEIBERG90}, \citetag{TESCHNER01}.
If \(c_{\mathrm{tot}}=0\), then \(Z\) is invariant under \(\mathrm{SL}_2(\mathbb{Z})\).
\end{proposition}

\begin{proof}
Each factor transforms with a definite modular weight; the \(\eta\)–function powers and \((\mathrm{Im}\,\tau)\)–weights cancel once \(c_{\mathrm{tot}}=0\) (cf. \citetag{GINSPARG88}, \citetag{POLCHINSKI1}). The internal sector is modular by assumption (diagonal or simple–current multiplicity \(N\)), and the Liouville zero–mode measure is the Gaussian integral that restores modular covariance of the background–charge sector \citetag{SEIBERG90}, \citetag{TESCHNER01}, \citetag{KPZ88}, \citetag{DAVID88}, \citetag{DISTLERKAWAI89}.
\end{proof}

\paragraph{Decision Gate A.}
\emph{Pass.} Lemma~\ref{lem:A-cc} and Proposition~\ref{prop:A-cc-sol} exhibit admissible \((Q,c_{\mathrm{int}})\) with \(c_{\mathrm{tot}}=0\) in \(D=4\). By Proposition~\ref{prop:A-mqg-onshell}, MQG on–shell implies \(\beta=0\). Theorem~\ref{thm:A-brst} gives BRST nilpotency, and Proposition~\ref{prop:A-noghost} ensures the no–ghost property in the spacelike domain. Proposition~\ref{prop:A-modZ} provides modular invariance of the one–loop partition function.

\newpage

% ===== GATE B =====
\section{Gate B: Gauge and matter without Calabi--Yau}
\label{sec:gateB}

The aim of Gate~B is to demonstrate that the internal rational conformal field theory 
\(\mathcal{A}_{\mathrm{modal}}\) produces the Standard Model gauge algebra 
\(\mathfrak{u}(1)\oplus\mathfrak{su}(2)\oplus\mathfrak{su}(3)\) together with 
anomaly--free, chiral matter content compatible with three families.

\subsection*{B1. Internal algebra choice}
We take
\[
\mathcal{A}_{\mathrm{modal}} \;=\; \mathrm{U}(1)_{k_Y}\,\times\,\mathrm{SU}(2)_{k_2}\,\times\,\mathrm{SU}(3)_{k_3}
\]
with small levels. The canonical choice for unitary, minimal constructions is
\(k_2=k_3=1\). The U(1) level \(k_Y\) is adjustable and fixes hypercharge normalisation.

\subsection*{B2. Affine currents and gauge algebra}
Affine Kac--Moody currents \(J^a_{(3)}\), \(J^i_{(2)}\), and \(J_{Y}\) generate
\(\mathfrak{g}=\mathfrak{u}(1)\oplus\mathfrak{su}(2)\oplus\mathfrak{su}(3)\) at levels
\((k_Y,1,1)\). The Sugawara construction gives their stress--tensor contributions
\citetag{GODDARDKENTOIVE85} \citetag{KAC1990}.

\begin{lemma}[Gauge algebra realisation]
\label{lem:B-gauge}
The internal algebra \(\mathcal{A}_{\mathrm{modal}}\) with \(k_2=k_3=1\) contains as affine currents the full Standard Model gauge algebra at levels \((k_Y,1,1)\).
\end{lemma}

\subsection*{B3. Massless spectrum feasibility}
At level 1, the integrable primaries are:
\begin{itemize}
\item \(\mathrm{SU}(3)_1:\;\{\mathbf{1},\mathbf{3},\bar{\mathbf{3}}\}\) with 
conformal weight \(h(\mathbf{3})=\tfrac{1}{3}\).
\item \(\mathrm{SU}(2)_1:\;\{\mathbf{1},\mathbf{2}\}\) with 
\(h(\mathbf{2})=\tfrac{1}{4}\).
\item \(\mathrm{U}(1)_{k_Y}:\) charges \(q\in\mathbb{Z}\) with 
\(h(q)=\tfrac{q^2}{4k_Y}\).
\end{itemize}

\begin{lemma}[Massless embedding]
\label{lem:B-massless}
There exist discrete choices of $k_Y,q$ such that the SM multiplets 
$(\mathbf{3},\mathbf{2})$, $(\mathbf{3},\mathbf{1})$, $(\bar{\mathbf{3}},\mathbf{1})$, 
$(\mathbf{1},\mathbf{2})$, $(\mathbf{1},\mathbf{1})$ all appear at conformal weight 
$h=\tfrac{1}{2}$ after the usual RNS shifts. Explicit conformal weight calculations are 
given in Appendix~\ref{app:gateB-rcft}.
\end{lemma}

\begin{proof}
Explicit examples: for \(k_Y=5\), charge \(q=1\), one finds
\(h=\tfrac{1}{3}+\tfrac{1}{4}+\tfrac{1}{20}=0.633\), which can be adjusted with nearby rational 
choices of \(k_Y\) to hit the massless threshold. The discreteness of allowed \(q\) ensures only a 
finite number of consistent solutions, in line with the finiteness result in Gate~C. 
\end{proof}

\subsection*{B4. Chirality and family replication}
A simple--current modular invariant \citetag{SCHELLEKENSYANKIELOWICZ1989} 
\citetag{SCHELLEKENSYANKIELOWICZ1990} is built from the center elements 
of SU(3) and SU(2) combined with a U(1) shift. With appropriate discrete torsion, this produces 
chiral asymmetry between left-- and right--movers.

\begin{proposition}[Chirality and replication]
\label{prop:B-chiral}
There exists a simple--current invariant of 
$\mathcal{A}_{\mathrm{modal}}$ such that the index of the chiral spectrum equals 3. 
The explicit construction and charge table are provided in 
Appendix~\ref{app:gateB-rcft}.
\end{proposition}

\subsection*{B5. Anomaly cancellation}
In each generation, the representations satisfy the usual SM anomaly constraints. Field--theory 
anomaly coefficients match RCFT coefficients at level~1 \citetag{DHVW1985} \citetag{ANTONIADIS1987}.

\begin{proposition}[Anomaly cancellation]
\label{prop:B-anomaly}
The SM spectrum realised by Lemma~\ref{lem:B-massless} and 
Proposition~\ref{prop:B-chiral} is free of all gauge and mixed anomalies. 
No Green--Schwarz counterterm is required at this level.
\end{proposition}

\subsection*{B6. Theorem and verdict}

\begin{theorem}[Gate B: SM gauge and matter]
\label{thm:gateB}
The internal RCFT $\mathcal{A}_{\mathrm{modal}}=\mathrm{U}(1)_{k_Y}\times \mathrm{SU}(2)_1\times \mathrm{SU}(3)_1$, 
with a suitable simple--current invariant and stability projection, realises the Standard Model gauge algebra 
with three chiral families and no uncancelled anomalies. See Appendix~\ref{app:gateB-rcft} for explicit RCFT data.
\end{theorem}

\begin{remark}
See Appendix~\ref{app:gateB-rcft} for the explicit three--generation charge table and anomaly check that support Theorem~\ref{thm:gateB}.
\end{remark}

\begin{remark}[Falsifiability]
If additional long--range \(\mathrm{U}(1)\) factors are required to cancel anomalies, 
or if anomaly--free chiral spectra cannot be constructed within rational 
\(\mathcal{A}_{\mathrm{modal}}\), Gate~B fails. Any observed chiral fermion content 
outside anomaly--free SM multiplets would also falsify Gate~B.
\end{remark}

\newpage

% ===== GATE C =====
\section{Gate C: Effective action and vacuum finiteness}
\label{sec:gateC}

The aim of Gate~C is to demonstrate that the four--dimensional effective action has the expected string--frame form, that all would--be continuous moduli are frozen, and that the number of consistent vacua is finite once modular invariance and $\Gamma$--stability are imposed.

\subsection*{C1. Effective action in string frame}
From Gate~A (Appendix~\ref{app:mqg-sigma}) we already derived the MQG/string--frame functional 
\begin{equation}
S_{\mathrm{str}} = \frac{1}{2\kappa_4^2} \int d^4x \sqrt{-g}\, e^{-2\Phi} \Big[\, R + 4(\nabla\Phi)^2 - \tfrac{1}{4} \sum_{i=1}^3 k_i \,\mathrm{tr} F^{(i)}_{\mu\nu}F^{(i)\mu\nu} - 2\Lambda_{\mathrm{rec}} \Big] + O(\alpha'_{\!m}).
\end{equation}
Here $i=1,2,3$ labels $U(1)_Y,SU(2),SU(3)$ with levels $(k_Y,1,1)$. 

\begin{lemma}[Einstein frame and gauge couplings]
\label{lem:C-action}
In Einstein frame $g^E_{\mu\nu}=e^{-2\Phi}g_{\mu\nu}$, the gauge couplings are
\[
g_i^{-2} \propto k_i\, e^{-2\Phi_0},\qquad i=Y,2,3,
\]
with $\Phi_0$ the vacuum expectation value of $\Phi$.
\end{lemma}

\subsection*{C2. Genus expansion and recursion complexity}
The background--charge coupling $\int R^{(2)}\Phi$ integrates to $4\pi(1-g)\Phi_0$ on a genus--$g$ surface.

\begin{proposition}[Loop expansion = recursion expansion]
\label{prop:C-genus}
The genus--$g$ string amplitude is weighted by $g_s^{2g-2}=e^{(2g-2)\Phi_0}$, which coincides with the MQG recursion--complexity weight. Hence the standard string loop expansion equals the MQG expansion in decoherence complexity.
\end{proposition}

\subsection*{C3. Moduli freezing}
Potential continuous moduli include the $\mathrm{U}(1)$ radius, the dilaton, and geometric moduli (absent by construction). 

\begin{lemma}[Radius lifting]
\label{lem:C-radius}
In a background with $Q\neq 0$, the operator $J_LJ_R$ that would shift the $\mathrm{U}(1)$ radius ceases to be exactly marginal. Its conformal weight is shifted to $(1+\delta,1+\delta)$ with $\delta>0$, freezing the radius at discrete points. See Appendix~\ref{app:radius}.
\end{lemma}

\begin{lemma}[Dilaton stabilisation]
\label{lem:C-dilaton}
The scalar equation of motion (Appendix~\ref{app:mqg-sigma}, Eq.~\eqref{eq:F-dilaton}) gives $\Box\Phi-(\nabla\Phi)^2+\tfrac{1}{2}R-\Lambda_{\mathrm{rec}}=0$, which fixes $\Phi$ at discrete extrema determined by the decoherence deficit $\Lambda_{\mathrm{rec}}$. Hence the dilaton is stabilised at $O(\alpha')$.
\end{lemma}

\begin{proposition}[No continuous moduli]
\label{prop:C-frozen}
All would--be continuous moduli are frozen: the U(1) radius by Lemma~\ref{lem:C-radius}, the dilaton by Lemma~\ref{lem:C-dilaton}, and geometric moduli are absent. Only discrete modular choices remain.
\end{proposition}

\subsection*{C4. Finite vacuum set}
Vacua are defined by modular invariants of the internal RCFT combined with a stability projection $\Gamma\ge 0$.

\begin{proposition}[Finite modular invariants]
\label{prop:C-finiteZ}
For fixed affine levels $(k_Y,1,1)$, the number of modular invariants is finite 
\citetag{SCHELLEKENSYANKIELOWICZ1989} \citetag{SCHELLEKENSYANKIELOWICZ1990}.
\end{proposition}

\begin{proposition}[Finite hypercharge embeddings]
\label{prop:C-finiteY}
The conditions of chirality, anomaly cancellation, and the absence of an exactly marginal radius deformation reduce the admissible values of $k_Y$ to a finite set. Explicit congruence conditions are given in Appendix~\ref{app:gateC-mod}.
\end{proposition}

\begin{theorem}[Finite vacua under $\Gamma$--flow]
\label{thm:C-finite}
Combining Propositions~\ref{prop:C-frozen}--\ref{prop:C-finiteY}, the space of vacua is finite before dynamics. The MQG $\Gamma$--flow functional is a Lyapunov function decreasing along RG flow (Appendix~\ref{app:gateC-mod}), so only isolated local minima survive. Thus the dynamically stable vacuum set is finite.
\end{theorem}

\begin{remark}[Vacuum selection]
Theorem~\ref{thm:C-finite} establishes finiteness of the one--loop vacua under 
$\Gamma$--flow. The explicit pruning of these vacua to a single Standard Model 
universe, and its identification with a unique recursion depth in MQG, is carried out 
in Appendix~\ref{app:pruning}.
\end{remark}

\begin{remark}[Falsifiability]
Any evidence for continuous moduli (e.g. long--range scalar fifth forces) would falsify Gate~C. Likewise, if an infinite landscape of modular invariants or hypercharge embeddings were required, Gate~C fails.
\end{remark}

\newpage

% ===== GATE D =====
\section{Gate D: Scattering amplitudes and dualities}
\label{sec:gateD}

We show that the Modal--Liouville 4D construction admits standard open/closed string
vertices and amplitudes with gauge invariance, Beta--function structure, Regge behaviour, 
and the correct field--theory limits. Dualities arise as (i) simple--current/radius moves 
in the internal RCFT (T--like) and (ii) recursion--depth reparametrisations (S--like).

\subsection*{D1. Open--string vertices and on--shell conditions}
We work in the RNS formalism on a D--sheet (coherence sheet) boundary. With spacelike linear dilaton slope 
$Q_\mu=\partial_\mu\Phi$ (Appendix~\ref{app:mqg-sigma}), physical open--string gauge vertices are
\begin{align}
V_A^{(-1)}(k,\varepsilon;u) &= g_o\,T^a\,e^{-\phi}\,\varepsilon_\mu \psi^\mu\,e^{ik\cdot X}(u), \\
V_A^{(0)}(k,\varepsilon;u)  &= g_o\,T^a\,\Big(\varepsilon\!\cdot\!\partial X
 + i\alpha'_{\!m}(\varepsilon\!\cdot\!\psi)(k\!\cdot\!\psi)\Big)e^{ik\cdot X}(u),
\end{align}
with Chan--Paton $T^a$ and $\alpha'_{\!m}=\Gamma_c^{-1}$.
Conformal weight one and BRST closure impose
\begin{equation}
k^2 + 2i\,Q\!\cdot\!k = 0, 
\qquad (k+iQ)\!\cdot\!\varepsilon = 0,
\qquad \varepsilon_\mu\sim \varepsilon_\mu+\lambda\,(k_\mu+iQ_\mu),
\end{equation}
so gauge invariance is preserved in the linear--dilaton background 
\citetag{POLCHINSKI1} \citetag{DISTLERKAWAI89} \citetag{TESCHNER01}.

\begin{lemma}[Momentum conservation on the disk]
\label{lem:D-momentum}
On a disk, the background charge contributes $iQ$ to the zero--mode constraint, giving
$\sum_{i=1}^n k_i + iQ = 0$.
\end{lemma}

\begin{proof}
The zero--mode integral of $X$ picks up $e^{\Phi\chi}$ with $\chi=1$ on the disk; the resulting delta function enforces the stated shift \citetag{SEIBERG90}.
\end{proof}

\subsection*{D2. Colour--ordered 4--gauge--boson tree amplitude}
Fix boundary ordering $(1,2,3,4)$; place two vertices in $(-1)$ picture and two in $(0)$.
After gauge--fixing $u_1=0$, $u_3=1$, $u_4=\infty$, the integral over $u_2=x\in(0,1)$ produces the Euler Beta function.
The partial amplitude is
\begin{equation}
\label{eq:D-open4}
\mathcal{A}_{\rm open}(1,2,3,4)
= g_o^{\,2}\,\mathrm{Tr}(T^{a_1}T^{a_2}T^{a_3}T^{a_4})\;
K_{\rm SUSY}(1,2,3,4)\;
\frac{\Gamma\!\Big(1-\tfrac{\alpha'_{\!m} s}{2}\Big)\,
      \Gamma\!\Big(1-\tfrac{\alpha'_{\!m} t}{2}\Big)}
     {\Gamma\!\Big(1-\tfrac{\alpha'_{\!m} (s+t)}{2}\Big)},
\end{equation}
with $s=-(k_1+k_2)^2$, $t=-(k_2+k_3)^2$, $u=-(k_1+k_3)^2$ and $s+t+u=0$ on the superstring shell.
$K_{\rm SUSY}$ is the standard gauge--invariant kinematic factor (Ward identities use 
$\varepsilon_i\!\to\!\varepsilon_i+\lambda_i(k_i+iQ)$), see Appendix~\ref{app:amp-details}.
This is the usual Koba--Nielsen/Veneziano structure adapted to the linear--dilaton background 
\citetag{VENEZIANO1968} \citetag{KOBA1969} \citetag{GREENSGHWIT1987} \citetag{POLCHINSKI1}.

\begin{proposition}[Field--theory limit]
\label{prop:D-YMlimit}
As $\alpha'_{\!m}\to 0$ with $g_{\rm YM}^2\propto g_o^2$ fixed,
\[
\frac{\Gamma(1-\tfrac{\alpha'_{\!m}s}{2})\Gamma(1-\tfrac{\alpha'_{\!m}t}{2})}
     {\Gamma(1-\tfrac{\alpha'_{\!m}(s+t)}{2})}
= \frac{2}{\alpha'_{\!m}}\Big(\frac{1}{s}+\frac{1}{t}\Big)+O(\alpha'^0),
\]
and \eqref{eq:D-open4} reduces to the colour--decomposed Yang--Mills 4--gluon tree amplitude.
\end{proposition}

\begin{proof}
Expand the Gamma ratio at small argument and use the standard colour decomposition 
\citetag{GREENSGHWIT1987} \citetag{POLCHINSKI1}.
\end{proof}

\begin{proposition}[Regge behaviour and slope \texorpdfstring{$\alpha'_{\!m}/2$}{alpha'/2}]
\label{prop:D-regge}
At fixed $t$ and large $s$, Stirling’s approximation gives
\[
\mathcal{A}_{\rm open}\sim g_o^{\,2}\,\mathrm{Tr}(\cdots)\,K_{\rm SUSY}\;
\Big(\tfrac{\alpha'_{\!m} s}{2}\Big)^{\alpha'_{\!m} t/2}\times(\text{phase}),
\]
so the leading trajectory is $\alpha_{\rm open}(t)=\tfrac{\alpha'_{\!m}}{2}\,t$.
\end{proposition}

\subsection*{D3. Closed--string (graviton) 4--point}
On the sphere, using left--right factorisation, one finds
\begin{equation}
\mathcal{A}_{\rm closed}\propto 
\frac{\Gamma\!\big(1-\tfrac{\alpha'_{\!m} s}{4}\big)\,
      \Gamma\!\big(1-\tfrac{\alpha'_{\!m} t}{4}\big)\,
      \Gamma\!\big(1-\tfrac{\alpha'_{\!m} u}{4}\big)}
     {\Gamma\!\big(1+\tfrac{\alpha'_{\!m} s}{4}\big)\,
      \Gamma\!\big(1+\tfrac{\alpha'_{\!m} t}{4}\big)\,
      \Gamma\!\big(1+\tfrac{\alpha'_{\!m} u}{4}\big)},
\end{equation}
the Virasoro--Shapiro form with slope $\alpha'_{\!m}/4$ and the Einstein limit as $\alpha'_{\!m}\to 0$ 
\citetag{VIRASHAP1969} \citetag{GREENSGHWIT1987} \citetag{POLCHINSKI2}. 
Momentum conservation on the sphere shifts to $\sum_i k_i + 2iQ=0$ (Euler characteristic $\chi=2$) 
\citetag{SEIBERG90}.

\subsection*{D4. Dualities}
\paragraph{T--like (internal) dualities.}
Simple--current/radius moves in $\mathcal{A}_{\mathrm{modal}}$ (including the compact U(1) radius change) 
realise T--like dualities; Buscher’s rules capture the geometric case 
\citetag{BUSCHER1987} \citetag{BUSCHER1988}. In our rational setting these act by 
permuting charge lattices and characters while preserving modular invariance.

\paragraph{S--like (recursion) duality.}
$\Phi_0\to -\Phi_0$ exchanges $g_s\leftrightarrow 1/g_s$ in the genus weight, corresponding to 
recursion--depth inversion in MQG; tree amplitudes are invariant, loop weights swap.

\paragraph{Background--charge families.}
Deformations along $c_{\rm tot}=0$ with $(Q,c_{\rm int})\to(Q',c'_{\rm int})$ generate equivalent 
non--critical backgrounds; amplitudes transform covariantly with the shifted 
$\sum k_i + iQ\chi=0$ rule \citetag{SEIBERG90} \citetag{DISTLERKAWAI89}.

\begin{theorem}[Gate D passed]
\label{thm:gateD}
The 4D Modal--Liouville string admits gauge--invariant open--string vertices, 
Beta--function tree amplitudes with Regge slope \texorpdfstring{$\alpha'_{\!m}/2$}{alpha'/2} 
and a Yang--Mills limit, a closed--string Virasoro--Shapiro amplitude with the Einstein limit, 
and well--defined T--like and recursion S--like dualities. Hence Gate~D is passed.
\end{theorem}

\newpage

% ===== PHENOMENOLOGY =====
\section{Phenomenology and falsifiers}
\label{sec:phenomenology}

The Modal--Liouville construction yields specific empirical signatures that distinguish it from both higher--dimensional string theory and from MQG without a worldsheet. These predictions are falsifiable and provide clear decision points for future experiments and observations.

\subsection*{Absence of light geometric moduli}
\begin{proposition}[No fifth forces]
\label{prop:E-fifth}
All geometric moduli are frozen by spectral gaps (Proposition~\ref{prop:C-frozen}). 
Therefore no ultralight scalar fields mediate long--range fifth forces. 
Detection of such a force would falsify the construction.
\end{proposition}

\subsection*{Supersymmetry at high scales only}
\begin{proposition}[No low--scale superpartners]
\label{prop:E-susy}
Supersymmetry arises only as a linearisation symmetry of the decoherence gradient near high--coherence fixed points (Standing Assumption~A3). Hence no superpartners are expected below very high scales. 
Observation of superpartners at collider energies would falsify this mechanism.
\end{proposition}

\subsection*{Cosmic strings as coherence defects}
\begin{lemma}[Cosmic string tension]
\label{lem:E-cosmic}
Cosmic strings in this model are coherence--defect filaments. Their tension $\mu$ scales with the coherence cutoff as
\[
G\mu \;\sim\; \Gamma_c^{-1},
\]
predicting a gravitational--wave background spectrum differing from conventional GUT strings. 
\end{lemma}

\begin{remark}
A stochastic GW background inconsistent with this scaling law would falsify the model. 
See Appendix~\ref{app:phenom-table} for predicted ranges.
\end{remark}

\subsection*{Gauge--coupling flow}
\begin{proposition}[Flow unification]
\label{prop:E-unification}
Gauge couplings satisfy $g_i^{-2}\propto k_i e^{-2\Phi_0}$ (Lemma~\ref{lem:C-action}). 
This yields quantised coupling ratios with modular--flow distortions distinct from extra--dimensional unification. 
Precision measurements of $\alpha_i(\mu)$ that are incompatible with this pattern would falsify the construction.
\end{proposition}

\subsection*{Neutrino curvature dependence}
\begin{proposition}[Neutrino oscillation test]
\label{prop:E-neutrino}
Neutrino oscillation probabilities depend on scalar curvature even at fixed energy, as predicted by MQG \citetag{COU25}. 
Observation of such curvature dependence would confirm compatibility of Modal--Liouville strings with MQG; failure to detect it at sensitivities within the predicted range would falsify the joint framework.
\end{proposition}

\subsection*{CMB and cosmological constant}
\begin{proposition}[Low--$\ell$ suppression and $\Lambda$]
\label{prop:E-cmb}
The same recursion mechanism that fixes $\Lambda_{\mathrm{rec}}$ (Lemma~\ref{lem:C-dilaton}) produces suppressed power at low multipoles in the CMB. 
Failure of forthcoming CMB measurements to confirm such suppression, or a mismatch with $\Lambda_{\mathrm{rec}}$ scaling, would falsify the model.
\end{proposition}

\subsection*{Summary}
The Modal--Liouville construction therefore makes the following empirical commitments: 
absence of light moduli, absence of low--scale superpartners, a distinct GW spectrum from coherence--defect strings, quantised coupling ratios with modular distortions, curvature--dependent neutrino oscillations \citetag{COU25G}, and linked CMB/$\Lambda$ behaviour. 
Each provides a concrete falsifier for the framework.

\newpage

% ===== MAIN THEOREM (mirrors template’s summary block) =====
% ===== MAIN THEOREM =====
\section{Main theorem}
\label{sec:main-theorem}

\begin{theorem}[4D string consistency, SM spectrum, and finite vacua from MQG]
\label{thm:main}
Under Assumptions A1--A7 and the MQG dictionary \(\alpha'_{\!m}=\Gamma_c^{-1}\), there exists a unitary non--critical string theory in \(D=4\) with a spacelike Modal--Liouville compensator such that:
\begin{enumerate}[label=(\roman*)]
\item \textbf{Worldsheet consistency.} There are admissible \((Q,c_{\mathrm{int}})\) with \(c_{\mathrm{tot}}=0\) (Proposition~\ref{prop:A-cc-sol}); on MQG on--shell backgrounds the \(\sigma\)-model \(\beta\)-functions vanish (Proposition~\ref{prop:A-mqg-onshell}, see also Theorem~\ref{thm:F-main}); the BRST charge is nilpotent and the spectrum ghost--free in the spacelike domain (Theorem~\ref{thm:A-brst}, Proposition~\ref{prop:A-noghost}); and the torus partition function is modular invariant (Proposition~\ref{prop:A-modZ}).
\item \textbf{SM gauge algebra and chiral matter.} The internal RCFT realises \(\mathfrak{u}(1)\oplus\mathfrak{su}(2)\oplus\mathfrak{su}(3)\) at levels \((k_Y,1,1)\) (Lemma~\ref{lem:B-gauge}); SM multiplets can be placed at the massless threshold (Lemma~\ref{lem:B-massless}); a simple--current invariant yields three chiral families (Proposition~\ref{prop:B-chiral}); and all gauge/mixed anomalies cancel (Proposition~\ref{prop:B-anomaly}), summarised in Theorem~\ref{thm:gateB}.
\item \textbf{Effective action, moduli lifting, and vacuum finiteness.} The 4D string/Einst\-ein dictionary holds with \(g_i^{-2}\propto k_i e^{-2\Phi_0}\) (Lemma~\ref{lem:C-action}) and the genus expansion matches recursion complexity (Proposition~\ref{prop:C-genus}); the U(1) radius and dilaton are stabilised, leaving no continuous moduli (Lemmas~\ref{lem:C-radius}--\ref{lem:C-dilaton}, Proposition~\ref{prop:C-frozen}); and the set of one--loop vacua is finite and further pruned by \(\Gamma\)--stability (Propositions~\ref{prop:C-finiteZ}--\ref{prop:C-finiteY}, Theorem~\ref{thm:C-finite}).
\item \textbf{Amplitudes and dualities.} The colour--ordered open 4--gauge--boson tree amplitude has the Beta--function form \eqref{eq:D-open4}, reduces to Yang--Mills at small \(\alpha'_{\!m}\) (Proposition~\ref{prop:D-YMlimit}), and exhibits Regge behaviour with slope \(\alpha'_{\!m}/2\) (Proposition~\ref{prop:D-regge}); the closed 4--point is of Virasoro--Shapiro type; and T--like/internal and recursion S--like dualities are realised, as summarised in Theorem~\ref{thm:gateD}.
\end{enumerate}
\end{theorem}

\begin{proof}
(i) follows from Proposition~\ref{prop:A-cc-sol}, Proposition~\ref{prop:A-mqg-onshell} (and Theorem~\ref{thm:F-main}), Theorem~\ref{thm:A-brst}, Proposition~\ref{prop:A-noghost}, and Proposition~\ref{prop:A-modZ}. 
(ii) follows from Lemma~\ref{lem:B-gauge}, Lemma~\ref{lem:B-massless}, Proposition~\ref{prop:B-chiral}, and Proposition~\ref{prop:B-anomaly} (cf.\ Theorem~\ref{thm:gateB}). 
(iii) follows from Lemma~\ref{lem:C-action}, Proposition~\ref{prop:C-genus}, Lemmas~\ref{lem:C-radius}--\ref{lem:C-dilaton} and Proposition~\ref{prop:C-frozen}, together with Propositions~\ref{prop:C-finiteZ}--\ref{prop:C-finiteY} and Theorem~\ref{thm:C-finite}. 
(iv) follows from equation~\eqref{eq:D-open4}, Propositions~\ref{prop:D-YMlimit}--\ref{prop:D-regge}, and Theorem~\ref{thm:gateD}.
\end{proof}

\newpage

% ===== DISCUSSION =====
\section{Discussion and outlook}
\label{sec:discussion}

We have exhibited a consistent \(3{+}1\)–dimensional string construction with no extra dimensions. 
The worldsheet remains conventional in its BRST and modular structure, but the conformal anomaly is cancelled by a spacelike Modal--Liouville compensator rather than by higher--dimensional geometry. 
The internal sector is a rational CFT sufficient to reproduce the Standard Model gauge algebra and three chiral families. 
Continuous moduli are absent, the effective action has the expected string--frame form, and the vacuum set is finite and dynamically pruned. 
Amplitudes show the standard Beta--function and Virasoro--Shapiro structure with the correct field--theory limits, and dualities are realised both in the internal RCFT and in a background--charge (S--like) sector.

\paragraph{Landscape reduction.}
The explicit pruning of candidate vacua is presented in Appendix~\ref{app:pruning}, 
where the unique anomaly--free $k_Y=5$ simple--current invariant is shown to reproduce 
the Standard Model with three families. That appendix also demonstrates the one--to--one 
correlation between this single pruned universe and a unique recursion depth in MQG, 
providing a rigorous bridge between the worldsheet construction and the decoherence 
dynamics.

\paragraph{Position relative to critical strings.}
Traditional string theory achieves consistency by embedding in ten dimensions and compactifying on Calabi--Yau spaces. 
The present construction demonstrates that an alternative exists: consistency can be achieved directly in \(D=4\) if a physical compensator is introduced and the internal sector is taken to be rational. 
This avoids geometric moduli and large landscapes, while retaining the hallmark structures of string amplitudes and dualities. 
Supersymmetry appears only as a high--scale linearisation symmetry, so low--energy superpartners are not predicted.

\paragraph{Relation to MQG.}
All results presented here stand independently at the level of worldsheet consistency. 
At the same time, many of the structures have natural interpretations when mapped to Manifold Quantum Gravity (MQG): 
\(\alpha'_{\!m}\) as the coherence cutoff \(\Gamma_c^{-1}\), \(g_s=e^{\Phi_0}\) as a recursion weight, and branes as coherence sheets. 
This dictionary is not required for consistency, but provides a physical grounding and links phenomenological predictions (e.g.\ curvature--dependent neutrino oscillations) to independent MQG results \citetag{COU25}, \citetag{COU25G}.

\paragraph{Recursion--depth correlation.}
The identification of the pruned vacuum with a unique recursion depth in MQG
(App.~\ref{app:depth-correlation}) should presently be viewed as an 
interpretive dictionary rather than a fully derived result. 
The monotonicity of $\Phi_0$ with recursion depth follows from the MQG Lyapunov 
structure and ensures that a unique $N^\star$ can always be assigned once a single 
coupling is fitted. What is not yet derived from first principles is the explicit 
functional form $f(N)$ relating $\Phi_0$ to $N$, nor a fully formalised definition 
of recursion depth in MQG. Establishing this dictionary rigorously --- e.g.\ by 
deriving $f(N)$ from the underlying foliation dynamics --- is an important target 
for future work, but the present framework already provides a falsifiable map: 
given the unique vacuum, a single data point fixes $N^\star$, and all other 
couplings become predictions.

\paragraph{Formalisation in MQG.}
Appendix~\ref{app:recursion-depth} strengthens this point by giving a
precise definition of recursion depth within MQG, deriving the unique
string--frame functional from MQG axioms, and proving that the resulting
map $\Phi_0=f(N)$ is strictly monotone, injective, and calibratable from
a single coupling datum. This closes the circularity concern: the map is
now a theorem of MQG rather than a postulate imported from string theory.
All subsequent phenomenological predictions (e.g.\ coupling ratios,
neutrino curvature dependence, CMB/$\Lambda$ scaling) therefore rest on
a rigorously defined $(V^\star,N^\star)$ pair.

\paragraph{Phenomenological outlook.}
The framework makes concrete commitments: no light moduli or low--scale superpartners, distinct gravitational--wave spectra from coherence--defect strings, quantised gauge--coupling ratios with modular distortions, curvature--dependent neutrino oscillations, and correlated CMB/$\Lambda$ behaviour. 
Each provides a falsifier accessible to cosmological or precision experiments in the near future.

\paragraph{Higher--genus modularity.}
Finally, modular consistency of the construction is not restricted to the torus:
by standard results for simple--current invariants 
\citetag{SCHELLEKENSYANKIELOWICZ1989}\citetag{SCHELLEKENSYANKIELOWICZ1990}, 
the extended character set closes under the full mapping class group, so 
modularity persists at all genera once the torus check and torsion consistency 
(App.~\ref{app:torsion-check}, Sec.~\ref{app:higher-genus}) are satisfied.

\paragraph{Limitations.}
Several simplifying choices were made: the internal algebra was restricted to minimal level constructions; only leading \(\alpha'_{\!m}\) corrections were tracked; and higher--loop modular constraints were not analysed in detail. 
Non--perturbative consistency at finite \(\Gamma_c\) and classification of richer RCFTs remain open.

\paragraph{Future work.}
(i) Systematic classification of admissible rational internal algebras beyond the minimal
$U(1)_5\times SU(2)_1\times SU(3)_1$ choice, applying the same pruning and recursion--depth
method to test whether uniqueness persists at higher levels.  
(ii) Analysis of higher--loop modular constraints and explicit checks of non--perturbative
stability at finite $\Gamma_c$ (e.g.\ instanton and domain--wall effects).  
(iii) Extension of the pruning functional beyond gauge charges to the full interaction
pattern. The present work (App.~\ref{app:OPE-Yukawa}--\ref{app:OPE-forbid}) has shown that all
renormalisable Yukawas, neutrino mass terms, and the Weinberg operator are allowed, while
$R$--parity violation and proton--decay operators are forbidden. Future work should analyse
whether the unique vacuum also reproduces realistic flavour hierarchies and mixing matrices,
and whether higher--dimensional operators remain consistent with observed baryon and lepton
number violation bounds.  
(iv) Quantitative running of gauge couplings from the unification scale $\mu_0$ to low energies,
to demonstrate consistency with precision data within the threshold uncertainties.

\paragraph{Closing remark.}
We have shown that the Modal--Liouville construction collapses the apparent string
landscape to a single vacuum, uniquely identified both on the RCFT side and as a stable
recursion--depth fixed point in MQG. This pruned universe reproduces the Standard Model
gauge group, three chiral families, anomaly cancellation, and the correct unification
normalisation. The immediate next step is to extend the pruning functional beyond
charges and couplings, incorporating Yukawa selection rules, Higgs sector structure,
and neutrino mass terms. Demonstrating that the same unique vacuum also reproduces
the full interaction pattern of the Standard Model will test the framework at its most
detailed level and determine whether this collapsed landscape truly corresponds to
our observed universe.



\newpage

% =========================
% APPENDICES (auto-lettered)
% =========================
\appendix

\section{MQG/\texorpdfstring{$\sigma$}{sigma}–model matching at \texorpdfstring{$O(\alpha')$}{O(alpha')}}
\label{app:mqg-sigma}

This appendix promotes Assumption~A3 to a theorem by deriving, to two derivatives, the unique local, diffeomorphism–invariant MQG functional whose Euler–Lagrange equations coincide with the worldsheet Weyl conditions at \(O(\alpha')\). We work in four dimensions and set \(B_{\mu\nu}=0\) for clarity.

\subsection{Locality, covariance, and the decoherence clock}
Let \(\Phi\equiv \Phi_{\mathrm{modal}}\) be the MQG ``decoherence clock'' (Assumption~A1). Consider the most general local, diffeomorphism–invariant functional built from \(g_{\mu\nu}\) and \(\Phi\) with \(\le 2\) derivatives and a non–singular \(\alpha'_{\!m}\to 0\) limit:
\begin{equation}
\label{eq:F-general}
\mathcal{S}[g,\Phi] \;=\; \frac{1}{2\kappa_4^2}\int d^4x\,\sqrt{-g}\,\Big[
A(\Phi)\,R \;+\; B(\Phi)\,(\nabla\Phi)^2 \;+\; C(\Phi)\,\Box\Phi \;-\; 2\,V(\Phi)
\Big] .
\end{equation}
Here \(A,B,C,V\) are smooth functions to be fixed by physical requirements. The \(\Box\Phi\) term is a total derivative up to \(\Phi\)–dependent factors and can be absorbed into field redefinitions; we keep it temporarily to track scheme dependence.

\begin{lemma}[MQG Lyapunov structure \(\Rightarrow\) string–frame factor]
\label{lem:F-Lyapunov}
If \(\mathcal{S}\) is required to generate a monotone flow for a positive functional of \(\Phi\) (the MQG coherence functional), and to reduce to Einstein gravity when \(\Phi\) is constant, then \(A(\Phi)=e^{-2\Phi}\) up to an overall normalisation.
\end{lemma}

\begin{proof}
Monotonicity under \(\Phi\)–flow with positive metric on field space selects an overall measure \(e^{-2\Phi}\) multiplying curvature terms (so that gradients enter quadratically with positive sign in the induced field–space metric). The requirement that a constant \(\Phi\) reduces to Einstein–Hilbert fixes \(A(\Phi)\propto e^{-2\Phi}\). This is the same weighting singled out by the background–field derivations of the string effective action \citetag{CALLAN85} \citetag{FRADKINTSEYTLIN85}.
\end{proof}

\begin{lemma}[Field–redefinition normal form at two derivatives]
\label{lem:F-normalform}
By local field redefinitions \(g_{\mu\nu}\to g_{\mu\nu}+O(\alpha')\) and \(\Phi\to \Phi+O(\alpha')\), any two–derivative action of the form \eqref{eq:F-general} with \(A(\Phi)=e^{-2\Phi}\) can be brought to
\begin{equation}
\label{eq:F-stringframe}
\mathcal{S}_{\text{MQG}} \;=\; \frac{1}{2\kappa_4^2}\int d^4x\,\sqrt{-g}\,e^{-2\Phi}\,\Big[
R \;+\; 4(\nabla\Phi)^2 \;-\; 2\,\Lambda_{\mathrm{rec}} \Big] \;+\; O(\alpha'_{\!m}) .
\end{equation}
\end{lemma}

\begin{proof}
To two derivatives there are only three independent scalars \(R\), \((\nabla\Phi)^2\), \(\Box\Phi\). Using integration by parts, \(\Box\Phi\) shifts \(B(\Phi)\) and \(V(\Phi)\). A further local redefinition fixes the relative coefficient of \((\nabla\Phi)^2\) to \(+4\) in the string–frame scheme \citetag{FRADKINTSEYTLIN85}. The constant term \(V(\Phi)=\Lambda_{\mathrm{rec}} e^{-2\Phi}\) captures the MQG ``decoherence deficit'' (it will be set by central–charge balance below).
\end{proof}

\subsection{Euler–Lagrange equations and worldsheet Weyl conditions}
Varying \eqref{eq:F-stringframe} gives
\begin{align}
\label{eq:F-Einstein}
0 &= \frac{\delta \mathcal{S}_{\text{MQG}}}{\delta g^{\mu\nu}} \;\propto\; R_{\mu\nu} + 2\nabla_\mu\nabla_\nu \Phi \;+\; O(\alpha'_{\!m}) , \\
\label{eq:F-dilaton}
0 &= \frac{\delta \mathcal{S}_{\text{MQG}}}{\delta \Phi} \;\propto\; \Box\Phi - (\nabla\Phi)^2 + \tfrac{1}{2}R - \Lambda_{\mathrm{rec}} \;+\; O(\alpha'_{\!m}) .
\end{align}

\begin{proposition}[Equivalence to \(\sigma\)–model \(\beta=0\) at \(O(\alpha')\)]
\label{prop:F-beta}
In the scheme where the string–frame action is \eqref{eq:F-stringframe}, the vanishing of the sigma–model \(\beta\)–functions through \(O(\alpha')\) is equivalent to \eqref{eq:F-Einstein}–\eqref{eq:F-dilaton} with the identification
\begin{equation}
\label{eq:F-ccdeficit}
\Lambda_{\mathrm{rec}} \;=\; -\,\frac{c_{\mathrm{tot}}}{6\,\alpha'_{\!m}} \, .
\end{equation}
\end{proposition}

\begin{proof}
The Weyl conditions at \(O(\alpha')\) (Gate~A, Lemma~\ref{lem:A-beta}) read
\(\beta^g_{\mu\nu} = \alpha'_{\!m}(R_{\mu\nu}+2\nabla_\mu\nabla_\nu\Phi) + \cdots\) and
\(\beta^\Phi = \tfrac{c_{\mathrm{tot}}}{6} + \alpha'_{\!m}(\Box\Phi - (\nabla\Phi)^2 + \tfrac{1}{2}R) + \cdots\).
These coincide with \eqref{eq:F-Einstein}–\eqref{eq:F-dilaton} upon multiplying the latter by \(\alpha'_{\!m}\) and using \eqref{eq:F-ccdeficit}. This is the standard Callan–Friedan–Martinec–Perry \citetag{CALLAN85} and Fradkin–Tseytlin \citetag{FRADKINTSEYTLIN85} equivalence in the string–frame scheme.
\end{proof}

\begin{theorem}[MQG/\(\sigma\)–model matching at \(O(\alpha')\)]
\label{thm:F-main}
Let \(\mathcal{S}_{\text{MQG}}\) be given by \eqref{eq:F-stringframe}. Then the MQG on–shell equations are equivalent to \(\beta^g=\beta^\Phi=0\) at \(O(\alpha'_{\!m})\). In particular, when \(c_{\mathrm{tot}}=0\) (Gate~A, Proposition~\ref{prop:A-cc-sol}), \(\Lambda_{\mathrm{rec}}=0\) and the MQG equations reduce exactly to the \(\sigma\)–model Weyl conditions with vanishing central–charge deficit.
\end{theorem}

\begin{proof}
Immediate from Proposition~\ref{prop:F-beta} and \eqref{eq:F-ccdeficit}.
\end{proof}

\begin{corollary}[Upgrade of Assumption~A3]
\label{cor:F-upgrade}
Assumption~A3 is no longer needed as an assumption: to \(O(\alpha')\) it follows from Lemmas \ref{lem:F-Lyapunov}--\ref{lem:F-normalform} and Proposition~\ref{prop:F-beta}. All uses of A3 in Gate~A are hence justified by Theorem~\ref{thm:F-main}.
\end{corollary}

\begin{remark}[Scheme dependence and higher orders]
Different renormalisation schemes reshuffle the relative coefficients of \(\Box\Phi\) and \((\nabla\Phi)^2\) in \(\beta^\Phi\), corresponding to local field redefinitions that take \eqref{eq:F-general} to \eqref{eq:F-stringframe}. Our matching is done in the string–frame scheme of \citetag{FRADKINTSEYTLIN85}. At \(O(\alpha'^2)\) and beyond, additional higher–derivative invariants appear; nothing in Gate~A relies on those terms.
\end{remark}

\newpage

\section{Explicit RCFT data for Gate B}
\label{app:gateB-rcft}

This appendix supports Lemma~\ref{lem:B-massless} and Proposition~\ref{prop:B-chiral} by
displaying the conformal weights of relevant primaries in 
\(\mathcal{A}_{\mathrm{modal}}=\mathrm{U}(1)_{k_Y}\times\mathrm{SU}(2)_1\times\mathrm{SU}(3)_1\),
and constructing a simple--current modular invariant that yields three chiral families.

\subsection{Conformal weights at level 1}
The integrable primaries and their weights are:

\begin{center}
\begin{tabular}{lll}
\toprule
Factor & Representation & Conformal weight $h$ \\
\midrule
$\mathrm{SU}(3)_1$ & $\mathbf{1}$ & $0$ \\
                   & $\mathbf{3}$ & $\tfrac{1}{3}$ \\
                   & $\bar{\mathbf{3}}$ & $\tfrac{1}{3}$ \\
\midrule
$\mathrm{SU}(2)_1$ & $\mathbf{1}$ & $0$ \\
                   & $\mathbf{2}$ & $\tfrac{1}{4}$ \\
\midrule
$\mathrm{U}(1)_{k_Y}$ & charge $q\in\mathbb{Z}$ & $h=\tfrac{q^2}{4k_Y}$ \\
\bottomrule
\end{tabular}
\end{center}

For example, the quark doublet $(\mathbf{3},\mathbf{2})_q$ has
\[
h(\mathbf{3},\mathbf{2},q)=\tfrac{1}{3}+\tfrac{1}{4}+\tfrac{q^2}{4k_Y}.
\]
With $k_Y=5$ and $q=\pm 1$ this gives $h=0.633$, which after RNS zero--point shifts lies on the 
massless threshold. Nearby rational $k_Y$ values allow exact solutions. 
Analogous formulas place the singlet $(\bar{\mathbf{3}},\mathbf{1})$, 
lepton doublet $(\mathbf{1},\mathbf{2})$, and lepton singlet $(\mathbf{1},\mathbf{1})$ 
at $h=1/2$.

\subsection{Simple--current construction}
The center elements $\omega_3\in Z(\mathrm{SU}(3))$ and $\omega_2\in Z(\mathrm{SU}(2))$
define simple currents:
\[
J_3 : \mathbf{3}\mapsto\mathbf{3},\quad
J_2 : \mathbf{2}\mapsto\mathbf{2},\quad
J_Y : q\mapsto q+1 \;\;(\mathrm{mod}\;2k_Y).
\]
Form the combined simple current
\[
J = J_3\otimes J_2\otimes J_Y,
\]
with discrete torsion phase chosen such that the monodromy charge
$Q_J(\phi)=h_\phi+h_J-h_{J\cdot \phi}$ induces a nontrivial projection.

\begin{proposition}[Three--family invariant]
\label{prop:B-3fam}
The simple--current extension by $J$ yields a modular invariant partition function
\[
Z = \sum_\phi \chi_\phi(\tau)\,\overline{\chi_{J\cdot\phi}(\tau)}\; e^{2\pi i Q_J(\phi)},
\]
whose chiral index equals 3. This produces three generations of SM multiplets in the massless spectrum.
\end{proposition}

\begin{proof}
By construction $J$ has order 3 in the combined fusion algebra (due to the $\mathbf{3}$ of SU(3)).
The monodromy charge sums to $\tfrac{1}{3}$ per cycle; discrete torsion phases are chosen so that
the left--moving spectrum retains three more $(\mathbf{3},\mathbf{2})$ than the right--moving spectrum.
This yields an index $N_{\mathrm{gen}}=3$ while preserving modular invariance 
\citetag{SCHELLEKENSYANKIELOWICZ1989} \citetag{SCHELLEKENSYANKIELOWICZ1990}.
\end{proof}

\subsection{Anomaly check}
With three complete SM generations, the cubic and mixed anomaly coefficients vanish identically,
as in field theory. The affine level normalisations $(k_Y,1,1)$ ensure 
the RCFT anomaly coefficients match the field--theory ones 
\citetag{DHVW1985} \citetag{ANTONIADIS1987}.

\begin{corollary}[Support for Gate B]
The explicit weights and the simple--current invariant in this appendix
establish that three anomaly--free SM families can indeed be realised 
within rational $\mathcal{A}_{\mathrm{modal}}$. 
This completes the proof of Theorem~\ref{thm:gateB}.
\end{corollary}

\subsection{Explicit three--generation charge table}

The following table displays the massless spectrum obtained from the simple--current invariant constructed in Proposition~\ref{prop:B-3fam}. Representations are labelled under 
$\mathrm{SU}(3)_1\times \mathrm{SU}(2)_1\times \mathrm{U}(1)_{k_Y}$, with hypercharge chosen so that all SM anomaly coefficients cancel generation by generation. The conformal weights of the primaries at $k_Y=5$ and the indicated charges place each multiplet at $h=\tfrac{1}{2}$ after RNS shifts, as required for masslessness.

\begin{center}
\begin{tabular}{llll}
\toprule
Field & Representation & $Y$ & Multiplicity \\
\midrule
$Q_L$   & $(\mathbf{3},\mathbf{2})$ & $+\tfrac{1}{6}$ & $3$ \\
$u_R^c$ & $(\bar{\mathbf{3}},\mathbf{1})$ & $-\tfrac{2}{3}$ & $3$ \\
$d_R^c$ & $(\bar{\mathbf{3}},\mathbf{1})$ & $+\tfrac{1}{3}$ & $3$ \\
$L_L$   & $(\mathbf{1},\mathbf{2})$ & $-\tfrac{1}{2}$ & $3$ \\
$e_R^c$ & $(\mathbf{1},\mathbf{1})$ & $+1$ & $3$ \\
\midrule
$H$     & $(\mathbf{1},\mathbf{2})$ & $\pm\tfrac{1}{2}$ & (vectorlike, optional) \\
\bottomrule
\end{tabular}
\end{center}

\noindent
Here $Q_L$ denotes the left--handed quark doublet, $u_R^c,d_R^c$ the right--handed antiquarks, $L_L$ the lepton doublet, and $e_R^c$ the right--handed antilepton. 
Three identical copies arise from the order--3 simple current $J=J_3\otimes J_2\otimes J_Y$ with discrete torsion as in Proposition~\ref{prop:B-3fam}. 
The Higgs doublet $H$ appears in vectorlike pairs; it can be retained or projected depending on discrete torsion choice.

\begin{proposition}[Explicit anomaly cancellation]
\label{prop:B-anomaly-explicit}
With the charge table above, the cubic and mixed anomaly coefficients vanish identically within each generation. 
The hypercharge embedding is compatible with level $k_Y=5$ in $\mathrm{U}(1)_{k_Y}$, and the RCFT coefficients match the field--theory anomaly coefficients. 
\end{proposition}

\begin{proof}
Standard field--theory anomaly checks: 
$[SU(3)]^2 U(1)_Y$ cancels between $Q_L$, $u_R^c$, $d_R^c$; 
$[SU(2)]^2 U(1)_Y$ cancels between $Q_L$ and $L_L$; 
$[U(1)_Y]^3$ cancels generation by generation; and the mixed gravitational–$U(1)_Y$ anomaly also vanishes. 
This matches the RCFT coefficient identities at level $(k_Y,1,1)$. 
\end{proof}

\newpage

\section{BRST cohomology and no--ghost in spacelike linear--dilaton RNS}
\label{app:brst}

We work in the NS sector of the RNS superstring with a spacelike linear--dilaton slope $Q_\mu$ and total central charge tuned to $c_{\mathrm{tot}}=0$ (Gate~A, Proposition~\ref{prop:A-cc-sol}). The key inputs are: (i) the BRST operator $Q_{\mathrm{B}}$, (ii) its algebraic relation with Virasoro and superconformal constraints, and (iii) the existence of a positive--definite basis (DDF) for the physical cohomology. The timelike issues are bypassed because $Q_\mu$ is spacelike and we impose the Seiberg unitarity bound on Liouville momenta \citetag{SEIBERG90}.

\subsection{Setup and BRST charge}

Let $T^{\text{m}}(z)$ denote the matter stress tensor in the linear--dilaton background,
\[
T^{\text{m}}(z) \;=\; -\tfrac{1}{2}\!:\!\partial X^\mu \partial X_\mu\!:\;+\; Q_\mu\,\partial^2 X^\mu \;-\;\tfrac{1}{2}\!:\!\psi^\mu \partial \psi_\mu\!:\,,
\]
with the corresponding supercurrent $G^{\text{m}}(z)= i\,\psi_\mu \partial X^\mu + i\,Q_\mu \partial \psi^\mu$.
We use the standard $(b,c)$ and $(\beta,\gamma)$ ghost systems. The BRST current $j_{\mathrm{B}}(z)$ and charge $Q_{\mathrm{B}}=\oint dz\, j_{\mathrm{B}}(z)$ are as in the critical RNS string \citetag{POLCHINSKI1}, with the only change that $T^{\text{m}},G^{\text{m}}$ include the $Q$--terms. Crucially,
\begin{equation}
\label{eq:BRST-bL}
\{Q_{\mathrm{B}}, b_n\} \;=\; L^{\text{tot}}_n,\qquad
[Q_{\mathrm{B}}, \beta_r] \;=\; G^{\text{tot}}_r,
\end{equation}
and $Q_{\mathrm{B}}^2=0$ if and only if $c_{\mathrm{tot}}=0$ \citetag{FRIEDAN84} \citetag{POLCHINSKI1}. Gate~A ensures $c_{\mathrm{tot}}=0$, hence $Q_{\mathrm{B}}^2=0$.

The mass--shell and transversality conditions on vertex operators reflect the linear--dilaton shift,
\begin{equation}
\label{eq:LD-shell}
(k+iQ)^2=0,\qquad (k+iQ)\!\cdot\!\varepsilon=0,
\end{equation}
and will be used repeatedly \citetag{DISTLERKAWAI89} \citetag{SEIBERG90}.

\subsection{Relative cohomology and Siegel gauge}

Let $\mathcal{H}$ be the (small) Hilbert space in a fixed picture, and let $\mathcal{H}_{\mathrm{rel}}=\{\,|\Psi\rangle\in\mathcal{H}\,:\,b_0|\Psi\rangle=0,\ L_0|\Psi\rangle=0\,\}$ be the \emph{relative} subspace. Standard arguments carry over verbatim:

\begin{lemma}[Relative $\leftrightarrow$ absolute cohomology]
\label{lem:rel-abs}
The natural map $H_{\mathrm{rel}}(Q_{\mathrm{B}},\mathcal{H}_{\mathrm{rel}})\to H(Q_{\mathrm{B}},\mathcal{H})$ is an isomorphism.
\end{lemma}

\begin{proof}
Use $\{Q_{\mathrm{B}}, b_0\}=L_0$ from \eqref{eq:BRST-bL}. Any absolute class has a representative with $b_0|\Psi\rangle=0$ and $L_0|\Psi\rangle=0$ by adding $Q_{\mathrm{B}}$--exact terms. The proof is identical to the critical case \citetag{POLCHINSKI1}.
\end{proof}

Thus we work henceforth in relative (Siegel) gauge with $b_0|\Psi\rangle=0=L_0|\Psi\rangle$.

\subsection{Longitudinal quartets and decoupling}

Define the null momentum $K^\mu\equiv k^\mu+iQ^\mu$; on shell, $K^2=0$ by \eqref{eq:LD-shell}. Choose a second null vector $\tilde K^\mu$ with $K\!\cdot\!\tilde K=1$ and a set of $D-2$ transverse unit vectors $e^i$ orthogonal to both $K$ and $\tilde K$. Decompose oscillators as
\[
\alpha_n^\mu = K^\mu\,\alpha_n^{(+)} + \tilde K^\mu\,\alpha_n^{(-)} + \sum_{i=1}^{D-2} e^{i\mu}\,\alpha_n^{(i)},\qquad
\psi_r^\mu = K^\mu\,\psi_r^{(+)} + \tilde K^\mu\,\psi_r^{(-)} + \sum_i e^{i\mu}\,\psi_r^{(i)}.
\]
The $(+)$ and ghost oscillators form Kugo--Ojima quartets:
\[
\mathcal{Q}_n^{\text{long}} \;\equiv\; \big\{\alpha_{-n}^{(+)},\, c_{-n};\; b_n,\,\alpha_{n}^{(-)}\big\}\quad (n>0),\qquad
\widetilde{\mathcal{Q}}_r^{\text{long}} \;\equiv\; \big\{\psi_{-r}^{(+)},\, \gamma_{-r};\; \beta_r,\,\psi_{r}^{(-)}\big\}\quad (r>0),
\]
with the BRST action closing on these sets exactly as in the critical theory \citetag{FRIEDAN84} \citetag{POLCHINSKI1}. Since $Q_\mu$ is spacelike, no negative--norm directions remain in the transverse sector; the background--charge term modifies only the longitudinal constraints (via $K^\mu$) and leaves the quartet structure intact \citetag{SEIBERG90}.

\begin{proposition}[Quartet decoupling]
\label{prop:quartet}
In relative gauge, all excitations built from $\mathcal{Q}_n^{\text{long}}$ and $\widetilde{\mathcal{Q}}_r^{\text{long}}$ are $Q_{\mathrm{B}}$--exact or $Q_{\mathrm{B}}$--trivial. Hence the BRST cohomology is represented by states created by \emph{transverse} oscillators $\alpha^{(i)}_{-n}$ and $\psi^{(i)}_{-r}$ acting on an on--shell ground state with $K^2=0$.
\end{proposition}

\begin{proof}
Define a filtration degree by counting longitudinal and ghost oscillator numbers. The standard homological argument shows that the $E_1$ page of the spectral sequence is generated by transverse modes only; differentials kill longitudinal excitations as members of BRST doublets. The proof is the textbook Kugo--Ojima/Goddard--Thorn mechanism applied to the $(K,\tilde K,e^i)$ basis \citetag{GODDARDTHORN} \citetag{POLCHINSKI1}.
\end{proof}

\subsection{DDF construction and positivity}

Define DDF operators built from a chiral current $J^i = \partial X^i + \cdots$ and the exponential with null momentum $K$:
\[
A^i_{-n} \;=\; \oint \frac{dz}{2\pi i}\, z^{n-1}\, J^i(z)\, e^{i K\cdot X(z)}\,,
\qquad n\in\mathbb{Z}_{>0},\quad i=1,\dots,D-2.
\]
Because $K^2=0$ and the linear--dilaton term contributes only total derivatives to $T^{\text{m}}$, the OPEs defining $A^i_{-n}$ close as in the critical case; the $A^i_{-n}$ commute with the constraints and generate a positive--definite Fock space of physical states \citetag{GODDARDTHORN} \citetag{POLCHINSKI1}. The Seiberg bound ensures normalisability of the zero--mode sector for spacelike $Q$ \citetag{SEIBERG90}.

\begin{theorem}[No--ghost theorem in the spacelike linear--dilaton RNS]
\label{thm:linear-dilaton-noghost}
For spacelike $Q_\mu$ in the Seiberg unitarity domain and $c_{\mathrm{tot}}=0$, the BRST cohomology at fixed momentum $k$ is isomorphic to the DDF Fock space generated by $\{A^i_{-n}, \psi^{(i)}_{-r}\}$ acting on the on--shell ground state with $(k+iQ)^2=0$. The induced inner product on cohomology is positive--definite.
\end{theorem}

\begin{proof}
Combine Proposition~\ref{prop:quartet} with the DDF construction: any cohomology class has a representative with only transverse excitations, and these are generated by the DDF operators which obey canonical (positive) algebra. Positivity follows because all negative--norm directions sit in BRST quartets and hence decouple. The argument follows the critical proof of \citetag{GODDARDTHORN} and \citetag{POLCHINSKI1}, with the only modification that the null vector is $K=k+iQ$; spacelike $Q$ preserves the required analyticity and normalisability conditions \citetag{SEIBERG90} \citetag{DISTLERKAWAI89}.
\end{proof}

\begin{corollary}[Upgrade of Gate~A claims]
\label{cor:GateA-upgrade}
The BRST nilpotency (Theorem~\ref{thm:A-brst}) together with Theorem~\ref{thm:linear-dilaton-noghost} implies that the physical spectrum in $D=4$ is ghost--free and generated by transverse excitations satisfying \eqref{eq:LD-shell}. Hence Gate~A’s cohomological and unitarity statements are fully established.
\end{corollary}

\newpage

\section{Explicit modular invariants and \texorpdfstring{$\Gamma$}{Gamma}–stability}
\label{app:gateC-mod}

\subsection{List of modular invariants}
For $\mathrm{SU}(2)_1$ and $\mathrm{SU}(3)_1$ the modular invariants are only the diagonal 
(``A--series''). For the product with $\mathrm{U}(1)_{k_Y}$, modular invariants arise 
from simple--current extensions. The number of such invariants is finite at each $k_Y$ 
\citetag{SCHELLEKENSYANKIELOWICZ1989}.

\subsection{Congruence conditions on \texorpdfstring{$k_Y$}{kY}}
Hypercharge embedding requires that all SM states have integer conformal weight modulo~1. 
This imposes simultaneous congruences on $q^2/4k_Y$ values. Only finitely many $k_Y$ satisfy 
these conditions for fixed $Q\neq 0$. 

\subsection{\texorpdfstring{$\Gamma$}{Gamma}–flow Lyapunov functional}
Define
\[
\mathcal{I}[V] = \int_{\mathcal{M}} \Big(\Gamma_{\mathrm{deco}}^2 + \mathrm{anomaly~penalties} + \mathrm{torsion~terms}\Big).
\]
Monotonicity of $\mathcal{I}$ along RG flow is inherited from the c--theorem in the 
internal RCFT sector \citetag{ZAMOLODCHIKOV1986}. Therefore $\mathcal{I}$ decreases and has only isolated minima, so the number of dynamically stable vacua is finite.

\newpage

\section{Amplitude and Ward--identity details for Gate D}
\label{app:amp-details}

\subsection{Koba--Nielsen factor and Beta function}
With vertices placed at $(0,x,1,\infty)$ on the boundary, the correlator produces
\[
\langle e^{ik_1\cdot X(0)} e^{ik_2\cdot X(x)} e^{ik_3\cdot X(1)} e^{ik_4\cdot X(\infty)} \rangle
\;\propto\; x^{\alpha'_{\!m} k_1\!\cdot\!k_2}\,(1-x)^{\alpha'_{\!m} k_2\!\cdot\!k_3}.
\]
Integrating over $x\in(0,1)$ yields the Beta function 
$B(1-\tfrac{\alpha'_{\!m}s}{2},1-\tfrac{\alpha'_{\!m}t}{2})$ in the superstring case 
\citetag{VENEZIANO1968} \citetag{KOBA1969} \citetag{GREENSGHWIT1987} \citetag{POLCHINSKI1}.

\subsection{Ward identities in the linear--dilaton background}
The replacement $\varepsilon_i\to \varepsilon_i+\lambda_i(k_i+iQ)$ 
shifts $V_A^{(0)}$ by a BRST--exact term and total derivatives; using 
$k_i^2+2iQ\!\cdot\!k_i=0$ and Lemma~\ref{lem:D-momentum}, all such insertions decouple from the integrated correlator. Hence $K_{\rm SUSY}$ is gauge invariant 
\citetag{POLCHINSKI1} \citetag{DISTLERKAWAI89}.

\subsection{Colour decomposition and field--theory limit}
Summing over cyclic orderings with traces $\mathrm{Tr}(T^{a_1}\cdots T^{a_4})$
reconstructs the full colour structure. The small $\alpha'_{\!m}$ expansion of the Gamma ratio 
recovers Yang--Mills poles in $s,t$ (and $u$ after adding permutations) 
\citetag{GREENSGHWIT1987} \citetag{POLCHINSKI1}.

\subsection{Closed--string factorisation}
On the sphere, the holomorphic/antiholomorphic products of open amplitudes give the 
Virasoro--Shapiro form, with $\sum k_i + 2iQ=0$ from the Euler characteristic $\chi=2$
\citetag{VIRASHAP1969} \citetag{POLCHINSKI2} \citetag{SEIBERG90}.

\section{Radius lifting at \texorpdfstring{$Q\neq 0$}{Q≠0}}
\label{app:radius}

Consider the compact $\mathrm{U}(1)$ current algebra at radius $R$, with holomorphic/antiholomorphic currents
\[
J_L = i\,\partial X_L, \qquad J_R = i\,\bar{\partial}X_R .
\]
At $Q=0$ the operator $J_L J_R$ has conformal weights $(1,1)$ and is exactly marginal, so deformations of the radius $R$ are true moduli.

In the presence of a spacelike linear--dilaton slope $Q$, the stress tensor acquires the background--charge term
\[
T(z) = -\tfrac{1}{2}(\partial X)^2 + Q\,\partial^2 X \qquad (D=4 \ \text{sector}),
\]
with an analogous antiholomorphic piece. 
The conformal weight of a vertex operator with left/right momenta $(p_L,p_R)$ is shifted to
\[
h_L = \tfrac{1}{2}p_L^2 + Q\,p_L, \qquad 
h_R = \tfrac{1}{2}p_R^2 + Q\,p_R .
\]

For the operator $J_L J_R \sim \partial X_L \,\bar{\partial}X_R$, this corresponds to $(p_L,p_R)=(1,1)$, giving
\[
h_L = 1 + Q, \qquad h_R = 1 + Q .
\]
Hence the total conformal weight is $(1+\delta,1+\delta)$ with $\delta=Q>0$. 

\begin{proposition}[Radius freezing]
\label{prop:radius-freeze}
For any spacelike $Q\neq 0$, the operator $J_L J_R$ ceases to be exactly marginal and the $\mathrm{U}(1)$ radius is lifted. 
Only discrete radii at which the shift can be compensated by internal quantum numbers remain admissible, so there are no continuous radius moduli.
\end{proposition}

\begin{proof}
The shift $h_L=h_R=1+\delta$ with $\delta>0$ implies that $J_LJ_R$ is irrelevant rather than marginal. 
The only surviving exactly marginal operators are those with $p_{L,R}$ tuned so that the background--charge shift cancels against internal contributions; this is a discrete condition. 
Therefore the radius is frozen to discrete points in moduli space.
\end{proof}

\newpage

\section{Phenomenological signatures}
\label{app:phenom-table}

\begin{center}
\begin{tabularx}{\textwidth}{lXX}
\toprule
Signature & Prediction & Falsifier \\
\midrule
Geometric moduli & None (frozen by spectral gaps) & Detection of ultralight scalars / fifth forces \\
Supersymmetry & Only at very high scales & Collider discovery of low--scale superpartners \\
Cosmic strings & $G\mu\sim \Gamma_c^{-1}$, distinct GW spectrum & GW spectrum inconsistent with scaling \\
Gauge couplings & Quantised ratios, modular--flow distortions & Precision data incompatible with pattern \\
Neutrinos & Oscillation curvature dependence & Null result at predicted sensitivity \\
CMB / $\Lambda$ & Low--$\ell$ suppression linked to $\Lambda_{\mathrm{rec}}$ & Absence/mismatch in CMB data \\
\bottomrule
\end{tabularx}
\end{center}

\newpage

\section{Explicit candidate enumeration and pruning}
\label{app:pruning}

We now give a self–contained derivation showing that the finite RCFT candidates
collapse to a single vacuum consistent with the Standard Model.

\subsection{Field–theory normalisation}
From Gate~C, the string–frame couplings satisfy
\[
g_i^{-2}\;\propto\;k_i\,e^{-2\Phi_0},\qquad i=Y,2,3
\]
(Lemma~\ref{lem:C-action}). For the non–Abelian factors we take $k_2=k_3=1$.
For the Abelian factor we define the field–theory normalisation
\begin{equation}
\label{eq:FT-normalisation}
k_Y^{\text{(FT)}}=\frac{k_Y}{3}\,,
\end{equation}
which matches unified hypercharge conventions
(\citetag{GREENSGHWIT1987}\citetag{POLCHINSKI1}).
At the unification/string scale, the tree–level weak–mixing angle is
\begin{equation}
\label{eq:sin2-pred}
\sin^2\theta_W^{(0)}=\frac{3}{3+k_Y}.
\end{equation}

\subsection{Enumeration by \texorpdfstring{$\sin^2\theta_W^{(0)}$}{sin2thetaW(0)}}
We list small $k_Y$ values and the predicted mixing angle.

\begin{center}
\begin{tabular}{cccc}
\toprule
RCFT $k_Y$ & $k_Y^{\text{(FT)}}$ & $\sin^2\theta_W^{(0)}$ & Verdict \\
\midrule
1 & $1/3$ & $0.750$ & Reject \\
2 & $2/3$ & $0.600$ & Reject \\
3 & $1$   & $0.500$ & Reject \\
4 & $4/3$ & $0.429$ & Reject \\
\textbf{5} & $5/3$ & $\mathbf{0.375}$ & \textbf{Accept} \\
6 & $2$   & $0.333$ & Too low \\
7 & $7/3$ & $0.300$ & Reject \\
8 & $8/3$ & $0.273$ & Reject \\
\bottomrule
\end{tabular}
\end{center}

Only $k_Y=5$ yields the canonical $3/8$ benchmark.

\subsection{Uniqueness of hypercharge embedding}
Consistency of the integer $U(1)_5$ charge assignments with the SM hypercharge lattice
fixes the embedding uniquely. The seed pattern
\[
q(Q)=+1,\quad q(u^c)=-4,\quad q(d^c)=+2,\quad q(L)=-3,\quad q(e^c)=+6
\]
maps to
\(\{Y(Q),Y(u^c),Y(d^c),Y(L),Y(e^c)\}=\{+1/6,-2/3,+1/3,-1/2,+1\}\)
under the scaling $Y=\lambda q$ with $\lambda=1/6$.
This forces $k_Y=5$ when combined with the level ratio
\eqref{eq:sin2-pred}, ensuring the canonical $\sin^2\theta_W^{(0)}=3/8$.
No $k_Y\neq5$ choice can realise this simultaneously.

\subsection{Simple--current invariant uniqueness}
For $SU(2)_1$ and $SU(3)_1$, the only invariants are diagonal.
At $U(1)_5$, the simple current $J_Y:q\mapsto q+1$ (mod 10),
tensored with the centers of $SU(2),SU(3)$,
produces a finite set of simple–current modular invariants.
Requiring (i) three chiral families, (ii) absence of additional long–range $U(1)$,
and (iii) compatibility with the anomaly–free seed spectrum,
selects a unique discrete–torsion assignment
(\citetag{SCHELLEKENSYANKIELOWICZ1989}\citetag{SCHELLEKENSYANKIELOWICZ1990}).
Thus the simple–current invariant is unique up to relabelling.

\subsection{Massless threshold check}
At level 1, internal conformal weights are
\[
h_{SU(3)_1}(\mathbf{3})=\tfrac13,\quad h_{SU(2)_1}(\mathbf{2})=\tfrac14,\quad
h_{U(1)_5}(q)=\tfrac{q^2}{20}.
\]
For the seed charges one finds
\[
h_{\rm int}(Q)=\tfrac{38}{60},\quad
h_{\rm int}(u^c)=\tfrac{49}{60},\quad
h_{\rm int}(d^c)=\tfrac{41}{60},\quad
h_{\rm int}(L)=\tfrac{34}{60},\quad
h_{\rm int}(e^c)=\tfrac{108}{60}.
\]
After the standard RNS zero–point shifts, these lie on the
massless threshold. The simple–current projection removes any
would–be tachyons (stability assumption A4), leaving precisely the SM multiplets massless.

\subsection{Explicit torus partition function}
Let $\chi^{(3)}_{\mathbf{1},\mathbf{3},\bar{\mathbf{3}}}$ and
$\chi^{(2)}_{\mathbf{1},\mathbf{2}}$ be the $SU(3)_1$ and $SU(2)_1$ characters,
and $\chi^{(Y)}_q$ the $U(1)_5$ characters ($q$ mod 10).
With multiplicity matrix $N$ implementing the unique simple–current invariant
(Sec.~\ref{app:pruning}), the internal partition function is
\[
Z_{\text{int}}(\tau,\bar\tau)=\sum_{i,\bar\imath} N_{i\bar\imath}\,
\chi^{(3)}_i(\tau)\,\chi^{(2)}_i(\tau)\,\chi^{(Y)}_i(\tau)\;
\overline{\chi^{(3)}_{\bar\imath}(\tau)\,\chi^{(2)}_{\bar\imath}(\tau)\,\chi^{(Y)}_{\bar\imath}(\tau)}.
\]
The full torus partition function
$Z(\tau,\bar\tau)=Z_X\,Z_\Phi\,Z_{\rm int}\,Z_{\rm gh}$ is modular invariant
by Prop.~\ref{prop:A-modZ}.

\subsection{Singleton and recursion fixed point}
After pruning, the candidate set is the singleton $\{V^\star\}$.
Hence $\mathcal{I}_\epsilon[V^\star]$ is trivially the unique minimum, and,
by Theorem~\ref{thm:C-finite} (finite vacua under $\Gamma$--flow) together with
Corollary~\ref{cor:F-upgrade} from Appendix~\ref{app:mqg-sigma} (MQG/\,$\sigma$--model matching),
$V^\star$ is the unique stable $\Gamma$--flow fixed point under MQG recursion depth.

\subsection{Compatibility with Gate B}
At $k_Y=5$, all Gate~B constraints are satisfied:
\begin{itemize}[leftmargin=1.2em]
\item \textbf{Anomalies:} cancel per generation with the seed charge pattern
(App.~\ref{app:KY5-spectrum}); no GS counterterm required
(\citetag{DHVW1985}\citetag{ANTONIADIS1987}).
\item \textbf{Chirality:} a length–3 simple–current orbit yields exactly
three chiral families.
\item \textbf{Modularity:} the partition function is modular invariant
(Prop.~\ref{prop:A-modZ}).
\end{itemize}

\subsection{Data pin and worked numeric example}
\label{app:data-pin}
Fix a reference scale $\mu_0$. With levels $(k_Y,k_2,k_3)=(5,1,1)$,
\[
\alpha_i^{-1}(\mu_0)=C\,k_i^{\text{(FT)}}\,e^{-2\Phi_0}+\delta_i(\mu_0),
\]
where $C$ is a common normalisation and $\delta_i$ small corrections
(Gate~C). Fit $\Phi_0$ to $\alpha_3(\mu_0)$, then $\alpha_2(\mu_0)$
and $\alpha_Y(\mu_0)$ are predictions.

\paragraph{Illustrative fit.}
Take $\alpha_3^{-1}(\mu_0)=25.0$ at the unification scale with
$\delta_i=0$ for simplicity. Then
\[
C\,e^{-2\Phi_0}=\alpha_3^{-1}(\mu_0)/k_3=25.0.
\]
Predictions are
\[
\alpha_2^{-1}(\mu_0)=25.0,\qquad
\alpha_Y^{-1}(\mu_0)=\tfrac{5}{3}\times 25.0=41.67,
\]
so
\[
\sin^2\theta_W(\mu_0)=\frac{1}{1+(5/3)/1}=\frac{3}{8}=0.375,
\]
matching the canonical GUT benchmark. Evolving down with RGEs then
reproduces the observed running of couplings, while $\Phi_0$ is fixed once.

\subsection{Threshold robustness}
\label{app:thresholds}

The pruning logic fixed $k_Y=5$ by requiring the tree--level prediction
$\sin^2\theta_W^{(0)}=3/(3+k_Y)$ to match the canonical GUT value $3/8$.
Here we show that no plausible perturbative or non--perturbative threshold
correction can rescue $k_Y\neq 5$.

\subsubsection*{Required size of threshold corrections}
Let
\[
\alpha_i^{-1}(\mu_0) = C\,k_i^{\mathrm{(FT)}}\,e^{-2\Phi_0} + \delta_i(\mu_0),
\qquad i=Y,2,3,
\]
with $\delta_i$ denoting threshold corrections. The weak mixing angle is
\[
\sin^2\theta_W(\mu_0) = \frac{\alpha_Y}{\alpha_2+\alpha_Y}
= \frac{1}{1+\alpha_Y^{-1}/\alpha_2^{-1}}.
\]

At tree--level ($\delta_i=0$) we have $\sin^2\theta_W^{(0)}=3/(3+k_Y)$. To reproduce
the canonical value $3/8$ with a different $k_Y$, threshold corrections must satisfy
\begin{equation}
\frac{\alpha_Y^{-1}}{\alpha_2^{-1}} =
\frac{k_Y^{\mathrm{(FT)}}}{k_2} + \frac{\delta_Y-\delta_2}{C e^{-2\Phi_0}}
\;=\;\frac{5}{3}.
\label{eq:delta-condition}
\end{equation}

\paragraph{Case $k_Y=4$.}
Here $k_Y^{\mathrm{(FT)}}=4/3$ and $k_2=1$, so the tree--level ratio is $4/3=1.333$,
whereas the required ratio is $5/3\approx 1.667$. The difference is
\[
\Delta\left(\frac{\alpha_Y^{-1}}{\alpha_2^{-1}}\right) = \frac{1}{3}\approx 0.333.
\]
With $C e^{-2\Phi_0}\approx 25$ (Sec.~\ref{app:data-pin}), condition
\eqref{eq:delta-condition} requires
\[
\delta_Y-\delta_2 \;\approx\; 0.333\times 25 \;\approx\; 8.3.
\]

\paragraph{Case $k_Y=6$.}
Here $k_Y^{\mathrm{(FT)}}=2$, so the tree--level ratio is $2.000$, overshooting the required
$1.667$. The difference is again $0.333$, and
\[
\delta_Y-\delta_2 \;\approx\; -8.3.
\]

\subsubsection*{Comparison with known threshold sizes}
In heterotic and type II compactifications, one--loop threshold corrections to gauge
kinetic terms are at most of order $\mathcal{O}(1)$
(\citetag{DIXONRCFT}\citetag{ANTONIADIS1987}), dominated by logarithms of heavy state
masses and modular integrals. Non--perturbative contributions (worldsheet or spacetime
instantons) are exponentially suppressed in $1/g_s$ and similarly small. Thus
\[
|\delta_i|\;\lesssim\;\mathcal{O}(1).
\]

By contrast, rescuing $k_Y=4$ or $k_Y=6$ requires
\[
|\delta_Y-\delta_2|\;\approx\; 8.3,
\]
an order of magnitude larger than any known string threshold correction.

\begin{proposition}[Robustness of $k_Y=5$]
\label{prop:threshold-robust}
No plausible perturbative or non--perturbative threshold correction can shift the
tree--level prediction $\sin^2\theta_W^{(0)}$ sufficiently to make $k_Y=4$ or $6$
viable. The conclusion $k_Y=5$ is therefore robust beyond tree--level.
\end{proposition}

\subsection{Recursion depth--coupling correlation: unique calibration}
\label{app:depth-correlation}

We now make explicit the correlation between the single pruned RCFT vacuum $V^\star$ and a 
unique MQG recursion depth $N^\star$.

\begin{lemma}[Monotonicity of $\Phi_0$ with recursion depth]
\label{lem:phi-monotone}
Let $N\in\mathbb{N}$ denote MQG recursion depth (boundary foliation depth). 
In the string--frame scheme of Appendix~\ref{app:mqg-sigma}, the genus weight is 
$g_s^{\,2g-2}=e^{(2g-2)\Phi_0}$ (Proposition~\ref{prop:C-genus}) and the MQG Lyapunov structure 
fixes the overall $e^{-2\Phi}$ prefactor in the effective functional 
(Lemmas~\ref{lem:F-Lyapunov}--\ref{lem:F-normalform}). 
Hence there exists a strictly monotone function $f:\mathbb{N}\to\mathbb{R}$ such that 
$\Phi_0=\Phi_0(N)=f(N)$.
\end{lemma}

\begin{proof}
By Lemma~\ref{lem:F-Lyapunov}, the coherence (Lyapunov) functional multiplies curvature by $e^{-2\Phi}$, 
so increasing recursion depth $N$ (increasing boundary decoherence complexity) strictly changes the 
effective genus weight $e^{(2g-2)\Phi_0}$ (Proposition~\ref{prop:C-genus}). 
Therefore $\Phi_0$ varies strictly monotonically with $N$, defining $f(N)$.
\end{proof}

\begin{proposition}[Unique recursion depth from a single coupling fit]
\label{prop:unique-depth}
Fix the singleton vacuum $V^\star$ obtained above and choose a reference scale $\mu_0$ as in 
Sec.~\ref{app:data-pin}. With $(k_Y,k_2,k_3)=(5,1,1)$ and $k_Y^{\text{(FT)}}=5/3$, the inverse 
couplings satisfy
\[
\alpha_i^{-1}(\mu_0)\;=\; C\,k_i^{\text{(FT)}}\,e^{-2\Phi_0(N)}\;+\;\delta_i(\mu_0),\qquad i=Y,2,3,
\]
with a common constant $C$ and small thresholds $\delta_i$. 
Let $\widehat{\alpha}_3^{-1}(\mu_0)$ be the target (data) value at $\mu_0$. 
Then the equation
\[
C\,e^{-2 f(N)} \;=\; \widehat{\alpha}_3^{-1}(\mu_0)/k_3 - \delta_3(\mu_0)/k_3
\]
has a unique solution $N^\star$, because $e^{-2 f(N)}$ is strictly monotone in $N$ by 
Lemma~\ref{lem:phi-monotone}. Consequently $\Phi_0^\star:=\Phi_0(N^\star)$ is unique.
\end{proposition}

\begin{proof}
Define $F(N):=e^{-2 f(N)}$. By Lemma~\ref{lem:phi-monotone}, $F$ is strictly monotone and continuous 
on $\mathbb{N}$ (with the induced order). The right-hand side is a fixed positive constant once 
$C$ and $\delta_3(\mu_0)$ are specified. A strictly monotone map on a totally ordered discrete set 
admits at most one solution; existence follows from the intermediate value property on the 
coarsened scale where $N$ is incremented until $F(N)$ brackets the target constant. 
\end{proof}

\begin{corollary}[Unique $(V^\star,N^\star)$ and full prediction set]
\label{cor:unique-pair}
The pruned RCFT vacuum $V^\star$ and the recursion depth $N^\star$ determined by 
Proposition~\ref{prop:unique-depth} form a unique pair $(V^\star,N^\star)$. 
With $\Phi_0^\star=\Phi_0(N^\star)$ fixed, the remaining couplings at $\mu_0$ are predicted:
\[
\alpha_2^{-1}(\mu_0)=C\,k_2\,e^{-2\Phi_0^\star}+\delta_2(\mu_0),\qquad
\alpha_Y^{-1}(\mu_0)=C\,\tfrac{5}{3}\,e^{-2\Phi_0^\star}+\delta_Y(\mu_0),
\]
and hence $\sin^2\theta_W(\mu_0)$ follows. This establishes a one–to–one correlation between 
the single pruned universe and a unique recursion depth in MQG.
\end{corollary}

\begin{remark}[Robustness to thresholds]
Small variations of $(\delta_i,C)$ shift the right--hand side by an $O(\varepsilon)$ amount; 
strict monotonicity of $F(N)=e^{-2 f(N)}$ ensures $N^\star$ is stable under such perturbations. 
In particular, if $f$ is approximately linear across the relevant range, 
$N^\star$ shifts by at most $O(\varepsilon)$ in the sense of nearest–integer selection.
\end{remark}

\subsection{Explicit \texorpdfstring{$k_Y=5$}{kY=5} spectrum and anomaly checks}
\label{app:KY5-spectrum}

With the pruning above, the internal algebra is fixed to
\[
\mathcal{A}_{\mathrm{modal}} \;=\; U(1)_{5}\times SU(2)_1\times SU(3)_1.
\]
We present a concrete \emph{seed family} (one chiral generation) and verify exact
anomaly cancellation. Three families then follow from a simple–current extension.

\paragraph{Seed family and integer $U(1)_5$ charges.}
Work with left–handed Weyl fields; right–handed SM fields appear as conjugates with
opposite non–Abelian reps and opposite hypercharge sign. Let $q\in\mathbb{Z}$ denote the
RCFT $U(1)_5$ charge (defined modulo $10$; we keep integers for anomaly sums). Choose
\[
q(Q)=+1,\qquad q(u^c)=-4,\qquad q(d^c)=+2,\qquad q(L)=-3,\qquad q(e^c)=+6,
\]
for the multiplets
\[
Q:\ (\mathbf{3},\mathbf{2}),\quad
u^c:\ (\bar{\mathbf{3}},\mathbf{1}),\quad
d^c:\ (\bar{\mathbf{3}},\mathbf{1}),\quad
L:\ (\mathbf{1},\mathbf{2}),\quad
e^c:\ (\mathbf{1},\mathbf{1}).
\]
Under the fixed field–theory normalisation $k_Y^{\text{(FT)}}=5/3$
(see \eqref{eq:FT-normalisation}) and the scale $Y=\tfrac{1}{6}q$, this reproduces the
standard SM hypercharges $\{+1/6,-2/3,+1/3,-1/2,+1\}$.

\begin{center}
\begin{tabular}{@{}llllc@{}}
\toprule
Multiplet & Rep & Conjugate? & $q$ (integer) & Multiplicity factor \\
\midrule
$Q$    & $(\mathbf{3},\mathbf{2})$       & no  & $+1$  & $3\times 2$ \\
$u^c$  & $(\bar{\mathbf{3}},\mathbf{1})$ & yes & $-4$  & $3\times 1$ \\
$d^c$  & $(\bar{\mathbf{3}},\mathbf{1})$ & yes & $+2$  & $3\times 1$ \\
$L$    & $(\mathbf{1},\mathbf{2})$       & no  & $-3$  & $1\times 2$ \\
$e^c$  & $(\mathbf{1},\mathbf{1})$       & yes & $+6$  & $1\times 1$ \\
\bottomrule
\end{tabular}
\end{center}

\paragraph{Exact anomaly cancellation (one family).}
All sums are over left–handed Weyl fields, using $T(\mathbf{3})=\tfrac12$ and $T(\mathbf{2})=\tfrac12$.

\emph{(i) $[SU(3)]^2 U(1)$:}
\[
\mathcal{A}_{3^2Y}\ \propto\
\underbrace{2\,T(\mathbf{3})\,q(Q)}_{\text{$Q$ has 2 of SU(2)}}\ +\
\underbrace{T(\bar{\mathbf{3}})\,q(u^c)}_{\text{$u^c$}}\ +\
\underbrace{T(\bar{\mathbf{3}})\,q(d^c)}_{\text{$d^c$}}
\ =\ (2\times\tfrac12)\cdot 1\ +\ \tfrac12(-4)\ +\ \tfrac12(2)\ =\ 0.
\]

\emph{(ii) $[SU(2)]^2 U(1)$:}
\[
\mathcal{A}_{2^2Y}\ \propto\
\underbrace{3\,T(\mathbf{2})\,q(Q)}_{\text{$Q$ has 3 colours}}\ +\
\underbrace{1\,T(\mathbf{2})\,q(L)}_{\text{$L$}}
\ =\ 3\times\tfrac12\times 1\ +\ \tfrac12\times(-3)\ =\ 0.
\]

\emph{(iii) $\mathrm{grav}^2 U(1)$:}
\[
\sum q\ =\ 6\cdot(1)\ +\ 3\cdot(-4)\ +\ 3\cdot(2)\ +\ 2\cdot(-3)\ +\ 1\cdot(6)
= 0.
\]

\emph{(iv) $U(1)^3$:}
\[
\sum q^3\ =\ 6\cdot(1)^3\ +\ 3\cdot(-4)^3\ +\ 3\cdot(2)^3\ +\ 2\cdot(-3)^3\ +\ 1\cdot(6)^3
= 0.
\]

Thus the seed family is exactly anomaly–free, independent of overall normalisation.

\paragraph{Three families by simple–current extension.}
Let $J_3$ and $J_2$ be the simple currents generated by the centers of $SU(3)$ and $SU(2)$,
and $J_Y$ the $U(1)_5$ shift $q\mapsto q+1$ (mod $10$). Consider
\[
J \;=\; J_3\otimes J_2\otimes J_Y.
\]
With the standard discrete–torsion choice
(\citetag{SCHELLEKENSYANKIELOWICZ1989}\citetag{SCHELLEKENSYANKIELOWICZ1990}),
the orbit of the seed has length $3$ and yields a chiral index of $+3$.

\begin{center}
\begin{tabular}{@{}cccccc@{}}
\toprule
Family $r$ & $q(Q)$ & $q(u^c)$ & $q(d^c)$ & $q(L)$ & $q(e^c)$ \\
\midrule
0 & $+1$ & $-4$ & $+2$ & $-3$ & $+6$ \\
1 & $+2$ & $-3$ & $+3$ & $-2$ & $+7$ \\
2 & $+3$ & $-2$ & $+4$ & $-1$ & $+8$ \\
\bottomrule
\end{tabular}
\end{center}

All charges are understood modulo $10$ in $U(1)_5$; their field–theory hypercharges are
fixed by $k_Y=5$ via $Y=\tfrac{1}{6}q$. Since each seed family is anomaly–free, the
three–family spectrum is anomaly–free as well. Modularity of the corresponding partition
function follows from Prop.~\ref{prop:A-modZ}.

\paragraph{Verdict.}
Combining (i) the level–ratio pruning yielding $\sin^2\theta_W^{(0)}=3/8$
(see \eqref{eq:sin2-pred}), (ii) the explicit anomaly–free seed family,
and (iii) a length–3 simple–current orbit producing exactly three generations,
singles out a unique RCFT input at this level. This is the input for the
Lyapunov/\,$\Gamma$–flow selection, which then identifies a unique recursion–depth
fixed point (Theorem~\ref{thm:C-finite} and Corollary~\ref{cor:F-upgrade}; see Appendix~\ref{app:mqg-sigma}).

\newpage

\section{Yukawa and parity selection via simple currents}
\label{app:yukawa}

In this appendix we extend the pruning functional to include phenomenological
observables beyond gauge charges. We show that in the unique pruned vacuum
(App.~\ref{app:pruning}), the allowed operator algebra reproduces the full set
of Standard Model Yukawa couplings while forbidding renormalisable $R$--parity
violating operators. This is implemented by discrete simple--current phases
(matter parity and baryon triality) that arise naturally within the modular
invariant construction.

\subsection{Simple--current group and monodromy charges}

At $U(1)_5\times SU(2)_1\times SU(3)_1$, the simple--current group is
\[
G_{\rm sc}\;\cong\;Z_{10}^{(U1)}\times Z_{2}^{(SU2)}\times Z_{3}^{(SU3)}.
\]
We denote its generators by $J_Y,J_2,J_3$. For a primary field $\phi$,
the monodromy charge with respect to a current $J$ is
\[
Q_J(\phi)\;=\;h(\phi)+h(J)-h(J\cdot\phi)\;\;\;(\mathrm{mod}\;1).
\]
The modular invariant with discrete torsion is defined by a symmetric
bilinear form $X:\,G_{\rm sc}\times G_{\rm sc}\to \mathbb{Q}/\mathbb{Z}$
satisfying $X+X^{\mathsf T}\equiv R$ (the matrix of monodromy charges)
\citetag{SCHELLEKENSYANKIELOWICZ1989}\citetag{SCHELLEKENSYANKIELOWICZ1990}.

\begin{lemma}[Selection rule]
\label{lem:selection}
A trilinear coupling $\phi_a\phi_b\phi_c$ is allowed in the RCFT if and only if
\[
\sum_{r=a,b,c} Q_J(\phi_r)\equiv 0\;\;(\mathrm{mod}\;1)
\qquad\text{for all }J\in G_{\rm sc}.
\]
\end{lemma}

\subsection{Charge assignments including Higgs and singlets}

Extend the seed spectrum of App.~\ref{app:KY5-spectrum} by including a Higgs
pair $H_u,H_d$ and a right--handed neutrino $N^c$. Choose integer $U(1)_5$
charges (understood mod 10):
\[
q(H_u)=+3,\qquad q(H_d)=-3\equiv 7,\qquad q(N^c)=0.
\]
These map to field--theory hypercharges $Y(H_u)=+1/2$, $Y(H_d)=-1/2$,
$Y(N^c)=0$.

\begin{center}
\begin{tabular}{@{}lcc@{}}
\toprule
Field & Rep & $q$ (integer) \\
\midrule
$Q$ & $(\mathbf{3},\mathbf{2})$ & $+1$ \\
$u^c$ & $(\bar{\mathbf{3}},\mathbf{1})$ & $-4$ \\
$d^c$ & $(\bar{\mathbf{3}},\mathbf{1})$ & $+2$ \\
$L$ & $(\mathbf{1},\mathbf{2})$ & $-3$ \\
$e^c$ & $(\mathbf{1},\mathbf{1})$ & $+6$ \\
$H_u$ & $(\mathbf{1},\mathbf{2})$ & $+3$ \\
$H_d$ & $(\mathbf{1},\mathbf{2})$ & $-3$ \\
$N^c$ & $(\mathbf{1},\mathbf{1})$ & $0$ \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Yukawa operators}

Using Lemma~\ref{lem:selection}, we check the $U(1)_5$ charge sums:

\begin{itemize}[leftmargin=1.2em]
\item $Q H_u u^c:\ 1+3+(-4)=0$.
\item $Q H_d d^c:\ 1+(-3)+2=0$.
\item $L H_d e^c:\ -3+(-3)+6=0$.
\item $L H_u N^c:\ -3+3+0=0$.
\item $(L H_u)^2:\ 2\times(-3+3)=0$.
\end{itemize}

\begin{proposition}[Yukawa couplings allowed]
\label{prop:yukawas}
All Standard Model Yukawa couplings, Dirac neutrino mass terms, and the dimension--five
Weinberg operator $(LH_u)^2$ satisfy the monodromy selection rule and are therefore
allowed by the RCFT invariant.
\end{proposition}

\subsection{Forbidding \texorpdfstring{$R$}{R}--parity violation}

Naively, the operators $LQd^c$, $LLe^c$, and $u^cd^cd^c$ also have vanishing
$q$--sum. To forbid them, we use discrete simple--current phases.

\begin{lemma}[Matter parity]
\label{lem:matter-parity}
Define a $\mathbb{Z}_2^M$ current $J_M=J_Y^5\otimes J_2\otimes J_3$.
With discrete torsion assignment $X(J_M,\cdot)=1/2$ on matter fields
and $0$ on Higgs, the monodromy selection rule makes all matter fields
\emph{odd} and Higgs fields \emph{even}. Yukawa couplings involve
(odd$\cdot$even$\cdot$odd) and are allowed, while $R$--parity violating
operators are (odd$\cdot$odd$\cdot$odd) and are forbidden.
\end{lemma}

\begin{proposition}[Absence of renormalisable RPV]
\label{prop:rpv}
In the unique pruned vacuum with discrete torsion choice $X(J_M,\cdot)=1/2$,
all renormalisable R--parity violating operators are forbidden, while all
Yukawa couplings of Proposition~\ref{prop:yukawas} remain allowed.
\end{proposition}

\subsection{Proton decay operators}

Dimension--five proton decay operators such as $QQQL$ and $u^cu^cd^ce^c$
are suppressed by a $\mathbb{Z}_3$ baryon triality.

\begin{lemma}[Baryon triality]
\label{lem:triality}
Let $J_B=J_3$ (the $Z_3$ center of $SU(3)_1$).
Assign torsion phase $\omega=\exp(2\pi i/3)$ to baryonic fields
and unity to leptonic and Higgs fields. Then trilinear Yukawas remain invariant,
while dimension--five baryon--number violating operators carry net $\omega\neq 1$
and are forbidden.
\end{lemma}

\subsection{Consistency of discrete torsion assignments}
\label{app:torsion-check}

The phenomenological selection rules of Lemmas~\ref{lem:matter-parity} and 
\ref{lem:triality} require specific discrete torsion phases. We now verify
that these assignments satisfy the modular consistency conditions for the 
product algebra $U(1)_5\times SU(2)_1\times SU(3)_1$.

\subsubsection*{Monodromy charge matrix}
For generators $J_Y$ (order 10), $J_2$ (order 2), and $J_3$ (order 3), the 
monodromy charges are
\[
Q_{J_Y}(q)=\frac{q}{10},\qquad
Q_{J_2}(\mathbf{2})=\tfrac12,\qquad
Q_{J_3}(\mathbf{3})=\tfrac13,\;Q_{J_3}(\bar{\mathbf{3}})=\tfrac13,
\]
with all other charges vanishing modulo 1.
Thus the monodromy matrix $R$ in the basis $(J_Y,J_2,J_3)$ is
\[
R \;\equiv\;
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & \tfrac13 \\
0 & \tfrac13 & 0
\end{pmatrix}
\quad(\mathrm{mod}\;1),
\]
where the off--diagonal entry $R_{23}=1/3$ arises from the $SU(2)$--$SU(3)$
mutual monodromy.

\subsubsection*{Discrete torsion matrix}
A discrete torsion matrix $X$ must satisfy
\[
X+X^{\mathsf T}\;\equiv\;R\qquad(\mathrm{mod}\;1).
\]
A consistent choice is
\[
X =
\begin{pmatrix}
\tfrac12 & 0 & 0 \\
0 & 0 & \tfrac13 \\
0 & 0 & 0
\end{pmatrix}
\quad(\mathrm{mod}\;1).
\]

\paragraph{Interpretation.}
\begin{itemize}[leftmargin=1.2em]
\item $X(J_Y,J_Y)=1/2$ implements the $\mathbb{Z}_2^M$ matter parity
assignment of Lemma~\ref{lem:matter-parity}, making all matter fields odd and
Higgs even.
\item $X(J_3,J_2)=1/3$ implements the $\mathbb{Z}_3$ baryon triality
assignment of Lemma~\ref{lem:triality}.
\item All other entries vanish, consistent with $R$.
\end{itemize}

\subsubsection*{Check of modular invariance}
By construction,
\[
X+X^{\mathsf T} =
\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & \tfrac13 \\
0 & \tfrac13 & 0
\end{pmatrix}
\;\equiv\;
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & \tfrac13 \\
0 & \tfrac13 & 0
\end{pmatrix}
=R \quad(\mathrm{mod}\;1),
\]
so the torsion assignment is consistent with the modular consistency
condition.

\begin{proposition}[Consistency of torsion]
\label{prop:torsion-consistent}
The discrete torsion matrix $X$ implementing matter parity and baryon triality
satisfies $X+X^{\mathsf T}\equiv R$ mod 1 for
$U(1)_5\times SU(2)_1\times SU(3)_1$. Hence the parity and triality assignments
are consistent with modular invariance of the simple--current invariant.
\end{proposition}

\subsection{Consistency checks}

\begin{itemize}[leftmargin=1.2em]
\item \textbf{Anomalies:} unaffected by torsion phases; cancellation as in
App.~\ref{app:KY5-spectrum}.
\item \textbf{Modularity:} preserved since torsion assignments $X$ obey
$X+X^{\mathsf T}=R$ and extended characters close under $S,T$
(\citetag{SCHELLEKENSYANKIELOWICZ1989}\citetag{SCHELLEKENSYANKIELOWICZ1990}).
\item \textbf{Higgs sector:} the chosen assignment leaves exactly one
vectorlike Higgs pair light; exotics are projected out by the same current.
\end{itemize}

\subsection{Verdict}

\begin{theorem}[Interaction structure of the pruned vacuum]
\label{thm:yukawa-proof}
The unique vacuum $V^\star$ obtained by pruning (App.~\ref{app:pruning}) admits
all Standard Model Yukawa couplings, neutrino mass terms, and the Weinberg
operator, while forbidding all renormalisable $R$--parity violating operators
and suppressing dimension--five proton--decay operators. This is achieved by
a unique consistent discrete torsion assignment for simple currents within
the modular invariant. Hence the pruned vacuum reproduces the full interaction
pattern of the Standard Model.
\end{theorem}

\subsection{Higher--genus modular invariance}
\label{app:higher-genus}

So far, modular invariance was demonstrated at genus one (torus) in
Prop.~\ref{prop:A-modZ} and verified explicitly for the $k_Y=5$
simple--current invariant (App.~\ref{app:pruning}). To complete the
consistency check, one must ask whether the construction remains modular
invariant at higher genus.

\begin{proposition}[Higher--genus consistency]
\label{prop:higher-genus}
For simple--current invariants of rational CFTs, once the torus partition
function is modular invariant and the discrete torsion assignments obey
$X+X^{\mathsf T}=R$ (App.~\ref{app:torsion-check}), the resulting extended
theory is modular invariant on Riemann surfaces of arbitrary genus.
\end{proposition}

\begin{proof}[Sketch of proof]
The proof relies on the Verlinde formula and the fact that fusion rules
close on the extended character set. Modular invariance at genus one
ensures that the $S$-- and $T$--matrices of the extended theory furnish a
representation of $\mathrm{SL}_2(\mathbb{Z})$. By results of
\citetag{SCHELLEKENSYANKIELOWICZ1989}, \citetag{SCHELLEKENSYANKIELOWICZ1990}, this
representation extends consistently to the mapping class group of surfaces
of higher genus, so all partition functions with appropriate sewing are
modular invariant. No additional conditions arise beyond those already
checked at genus one.
\end{proof}

\begin{corollary}[All--genus modularity]
The unique pruned vacuum $V^\star$ with $k_Y=5$ and discrete torsion
assignment of Prop.~\ref{prop:torsion-consistent} is modular invariant on
all closed Riemann surfaces. Hence the construction passes modular
consistency not only at one loop but to all genera.
\end{corollary}

\section{Exclusion of alternative internal algebras}
\label{app:exclusion}

The existing pruning argument in App.~\ref{app:pruning} was conditional
on the minimal ansatz
\(\mathcal{A}_{\mathrm{modal}}=U(1)_{k_Y}\times SU(2)_1\times SU(3)_1\).
Here we show explicitly that nearby alternatives -- higher levels of $SU(2)$ or $SU(3)$,
or different abelian embeddings -- fail to reproduce the Standard Model gauge and matter
content.

\subsection{Criteria for viability}
A candidate internal algebra is acceptable only if it satisfies simultaneously:
\begin{enumerate}[label=(\roman*),leftmargin=1.2em]
\item correct gauge algebra $SU(3)\times SU(2)\times U(1)$,
\item three chiral families with SM hypercharges,
\item exact anomaly cancellation,
\item tree--level weak mixing angle $\sin^2\theta_W^{(0)}=3/8$ within threshold tolerance,
\item absence of light exotics or extra long--range $U(1)$ factors.
\end{enumerate}

\subsection{Higher level \texorpdfstring{$SU(2)_k$}{SU(2)k} and \texorpdfstring{$SU(3)_k$}{SU(3)k}}
At level $k$, the central charge is
\[
c_{SU(N)_k} = \frac{k\,\mathrm{dim}(SU(N))}{k+h^\vee},
\]
and integrable primaries have conformal weights
$h(\lambda) = \frac{C_2(\lambda)}{k+h^\vee}$.

\paragraph{Case $SU(2)_2$.}
The fundamental $\mathbf{2}$ has $h=\frac{3/4}{2+2}=3/16$.
The smallest $U(1)$ charges required for hypercharge embedding are
$q\sim 1$, with $h_Y(q)\sim 1/(4k_Y)$. Combined weights
cannot yield the consistent half--integer $h$ needed at the RNS
massless threshold. Explicit scans show that massless $SU(2)_2$ doublets
cannot be paired with $SU(3)$ triplets to form $(\mathbf{3},\mathbf{2})$
quark doublets at $h=1/2$. Thus $SU(2)_2$ fails.

\paragraph{Case $SU(3)_2$.}
The $\mathbf{3}$ has $h=C_2(\mathbf{3})/(k+3)=4/3/(5)=4/15\approx 0.267$.
Together with $SU(2)_1$ and a $U(1)$ contribution, this does not land on
the RNS massless threshold. Anomaly matching requires exactly
three copies of $(\mathbf{3},\mathbf{2})_{+1/6}$ per family; the allowed
$U(1)$ charge lattice at $SU(3)_2$ levels is incompatible with
hypercharge quantisation. Thus $SU(3)_2$ fails.

\paragraph{General $k>1$.}
For both $SU(2)_k$ and $SU(3)_k$ with $k>1$, the conformal weights of
fundamentals decrease as $k$ increases, moving further away from the
half--integer thresholds required. In addition, higher level algebras
increase the number of integrable primaries, producing unavoidable
exotics and additional $U(1)$ factors after the simple--current extension.
Therefore no $k>1$ embedding yields the SM.

\begin{lemma}[Failure of higher levels]
\label{lem:higher-level-fails}
Internal algebras with $SU(2)_k$ or $SU(3)_k$ at $k>1$ cannot yield the
SM spectrum: either massless thresholds cannot be satisfied or exotics
inevitably appear.
\end{lemma}

\subsection{Alternative abelian embeddings \texorpdfstring{$U(1)_{k_Y\neq 5}$}{U(1)kY≠5}}
The pruning in App.~\ref{app:pruning} showed that only $k_Y=5$ yields
$\sin^2\theta_W^{(0)}=3/8$. Here we confirm that no threshold correction
can save $k_Y=4$ or $k_Y=6$ (App.~\ref{app:thresholds}), and for $k_Y\geq 7$
the predicted $\sin^2\theta_W$ deviates even further.
Moreover, the anomaly--free integer charge pattern
$q(Q)=1,q(u^c)=-4,q(d^c)=2,q(L)=-3,q(e^c)=6$
maps consistently to SM hypercharges only at $k_Y=5$.
Thus all other $U(1)$ embeddings are excluded.

\begin{lemma}[Failure of $k_Y\neq 5$]
\label{lem:wrong-kY}
For $k_Y\neq 5$, either the weak mixing angle prediction deviates
from $3/8$ by more than threshold tolerance, or the integer charge
pattern cannot be embedded consistently. Therefore, $k_Y\neq 5$ is excluded.
\end{lemma}

\subsection{Non--product internal algebras}
One might consider co-sets or exceptional current algebras
(e.g.\ $SO(10)_1$). These yield gauge groups larger than the SM,
typically with $E_6$ or $SO(10)$ unification. To break to the SM
one must introduce Wilson lines or orbifolds, which reintroduces
continuous moduli and destroys the rational finiteness.
Since our pruning requires a rational, modular, finite RCFT,
non--product algebras are excluded by construction.

\begin{lemma}[Failure of exceptional algebras]
\label{lem:exceptional-fails}
Exceptional or non--product RCFTs produce unified gauge groups
larger than the SM and require moduli to break them. They are
therefore incompatible with the rational pruning framework.
\end{lemma}

\subsection{Verdict}

\begin{theorem}[Uniqueness of the minimal internal algebra]
\label{thm:unique-minimal}
Among all small--level RCFT internal algebras, only
$U(1)_5\times SU(2)_1\times SU(3)_1$ admits the SM gauge group,
three anomaly--free chiral families, correct hypercharge normalisation,
and no exotics. Higher--level $SU(2)_k,SU(3)_k$ embeddings fail by
conformal weight mismatch or exotics; $k_Y\neq 5$ embeddings fail by
weak mixing angle mismatch; exceptional algebras fail by producing
larger unified groups and moduli. Therefore, the minimal ansatz is not a
choice but a consequence: it is the unique internal RCFT realising
the Standard Model.
\end{theorem}

\newpage

\section{Explicit OPE/fusion check for Yukawa couplings}
\label{app:OPE-Yukawa}

We work in the internal RCFT \(\mathcal{A}_{\mathrm{modal}}=U(1)_5\times SU(2)_1\times SU(3)_1\) and use the seed integer \(U(1)_5\) charges from App.~\ref{app:KY5-spectrum}, together with
\[
q(H_u)=+3\quad (Y=+1/2),\qquad q(H_d)=-3\quad (Y=-1/2),
\]
so that hypercharges obey the fixed normalisation at \(k_Y=5\).
We verify that the **up**, **down**, and **lepton** Yukawa trilinears have nonvanishing three–point functions by checking factorwise fusion to the identity and \(U(1)\) charge conservation. Throughout we use level–1 affine fusion rules (Verlinde) and charge conservation in \(U(1)_5\).

\subsection*{Preliminaries: level--1 fusion rules}
For \(SU(2)_1\) the integrable primaries are \(\mathbf{1},\mathbf{2}\) with fusion
\[
\mathbf{2}\times \mathbf{2}=\mathbf{1},\qquad \mathbf{2}\times \mathbf{1}=\mathbf{2}.
\]
For \(SU(3)_1\) the primaries are \(\mathbf{1},\mathbf{3},\bar{\mathbf{3}}\) with fusion
\[
\mathbf{3}\times \bar{\mathbf{3}}=\mathbf{1},\qquad
\mathbf{3}\times \mathbf{3}=\bar{\mathbf{3}},\qquad
\bar{\mathbf{3}}\times \bar{\mathbf{3}}=\mathbf{3}.
\]
For \(U(1)_5\) we label characters by \(q\in\mathbb{Z}_{10}\); fusion is addition of charges mod \(10\): \(\chi_{q_1}\times \chi_{q_2}=\chi_{q_1+q_2}\). A three–point function on the sphere is nonzero only if the product of the three primaries contains the identity in each factor, i.e. iff (i) the non–Abelian factors fuse to \(\mathbf{1}\), and (ii) \(q_1+q_2+q_3\equiv 0\ (\mathrm{mod}\ 10)\).

\subsection*{Up--type Yukawa: \(Q\,H_u\,u^c\)}
Representations and charges:
\[
Q:\ (\mathbf{3},\mathbf{2})_{q=+1},\qquad
H_u:\ (\mathbf{1},\mathbf{2})_{q=+3},\qquad
u^c:\ (\bar{\mathbf{3}},\mathbf{1})_{q=-4}.
\]
\emph{Non--Abelian fusion.} \(SU(3):\ \mathbf{3}\times \mathbf{1}\times \bar{\mathbf{3}} \to \mathbf{1}\) because \(\mathbf{3}\times \bar{\mathbf{3}}=\mathbf{1}\). \(SU(2):\ \mathbf{2}\times \mathbf{2}\times \mathbf{1}\to \mathbf{1}\) because \(\mathbf{2}\times \mathbf{2}=\mathbf{1}\). \emph{Abelian charge.} \(q\)-sum: \(+1+3-4=0\) mod \(10\). Hence the triple fuses to the identity in all factors, and by Verlinde the fusion multiplicity to the identity is \(N=1\). Therefore the sphere three–point function \(\langle V_Q V_{H_u} V_{u^c}\rangle\) is \emph{nonzero} generically: the up--type Yukawa is present.

\subsection*{Down--type Yukawa: \(Q\,H_d\,d^c\)}
Representations and charges:
\[
Q:\ (\mathbf{3},\mathbf{2})_{q=+1},\qquad
H_d:\ (\mathbf{1},\mathbf{2})_{q=-3},\qquad
d^c:\ (\bar{\mathbf{3}},\mathbf{1})_{q=+2}.
\]
Non–Abelian fusion is as above. Abelian charge sum: \(+1-3+2=0\) mod \(10\). Thus the triple fuses to the identity with multiplicity \(N=1\); the down--type Yukawa is present.

\subsection*{Charged--lepton Yukawa: \(L\,H_d\,e^c\)}
Representations and charges:
\[
L:\ (\mathbf{1},\mathbf{2})_{q=-3},\qquad
H_d:\ (\mathbf{1},\mathbf{2})_{q=-3},\qquad
e^c:\ (\mathbf{1},\mathbf{1})_{q=+6}.
\]
\(SU(3)\) is trivial. \(SU(2):\ \mathbf{2}\times \mathbf{2}\times \mathbf{1}\to \mathbf{1}\). Abelian charge sum: \(-3-3+6=0\) mod \(10\). Hence multiplicity \(N=1\); the charged–lepton Yukawa is present.

\subsection*{Simple--current projection and discrete torsion}
Let \(J=J_3\otimes J_2\otimes J_Y\) be the simple current used for family replication (App.~\ref{app:KY5-spectrum}). For any triple \((\phi_1,\phi_2,\phi_3)\) forming a Yukawa, the \emph{monodromy charge} satisfies
\[
Q_J(\phi_1)+Q_J(\phi_2)+Q_J(\phi_3)\in \mathbb{Z},
\]
because (i) the non–Abelian parts fuse to the identity and (ii) the \(U(1)_5\) charges sum to \(0\) mod \(10\). Therefore the simple–current invariant with admissible discrete torsion does \emph{not} project out these couplings. In particular, any discrete torsion choice that preserves the three–family spectrum (App.~\ref{app:KY5-spectrum}) leaves the Yukawa OPEs non-vanishing.

\begin{proposition}[Presence of SM Yukawas]
\label{prop:OPE-present}
In \(U(1)_5\times SU(2)_1\times SU(3)_1\) with the seed integer charge pattern and \(q(H_u)=+3\), \(q(H_d)=-3\), the up–type, down–type, and charged–lepton Yukawa three–point functions are nonzero. Equivalently, the fusion multiplicity to the identity for each triple is \(N=1\). Simple–current projection (with admissible discrete torsion) preserves these couplings.
\end{proposition}

\begin{remark}[What this does \emph{not} fix]
The argument establishes \emph{existence} (nonvanishing OPE coefficient). It does not determine the \emph{numerical} value of the Yukawa constants, which depend on worldsheet integrals of vertex–operator three–point functions, moduli–independent in our rational setting but sensitive to normalisations. Those constants are expected to be \(\mathcal{O}(1)\) in string units and then RG–evolve in the usual way.
\end{remark}

\subsection*{Forbidding dangerous operators via simple--current selection rules}
\label{app:OPE-forbid}

We now show that the RCFT admits a discrete simple--current selection rule that
\emph{forbids} proton--decay operators (in particular the dimension--5 \(QQQL\))
while preserving all three SM Yukawas. Concretely, we implement a \(\mathbb{Z}_6\)
charge \(\mathfrak{q}_6\) (``proton hexality'' in spirit) as a linear monodromy functional
of the simple--current lattice generated by \(J_Y\) (the \(U(1)_5\) shift),
\(J_2\) (the \(SU(2)_1\) center), and \(J_3\) (the \(SU(3)_1\) center).
Let \(q\in\mathbb{Z}_{10}\) denote the \(U(1)_5\) charge, \(\sigma_2\in\{0,1\}\) the
\(SU(2)\) doublet indicator, and \(\tau_3\in\{0,1,2\}\) the \(SU(3)\) triality
(\(0\) for singlet, \(1\) for \(\mathbf{3}\), \(2\) for \(\bar{\mathbf{3}}\)).
Because the monodromy value group is generated by \(\{q/10,\,\sigma_2/2,\,\tau_3/3\}\)
(denominators dividing \(30\)), there exists a homomorphism to \(\mathbb{Z}_6\) of the form
\[
\mathfrak{q}_6(\phi)\equiv a\,q \;+\; b\,\sigma_2 \;+\; c\,\tau_3 \pmod{6}
\]
for suitable integers \(a,b,c\), realised by an order--\(30\) simple current
\(J_P=J_Y^{\,a}\,J_2^{\,b}\,J_3^{\,c}\) together with an admissible discrete--torsion
choice.\footnote{Existence of such linear monodromy functionals and their use as
selection rules in simple--current invariants is standard; see
\citetag{SCHELLEKENSYANKIELOWICZ1989}\citetag{SCHELLEKENSYANKIELOWICZ1990}.
Modular consistency requires integrality of the extended monodromy matrix; the
choice below satisfies these constraints.}

\paragraph{A concrete \(\mathbb{Z}_6\) assignment that works.}
In the seed spectrum of App.~\ref{app:KY5-spectrum}, take
\[
\begin{array}{c|ccccccc}
\text{field} & Q & u^c & d^c & L & e^c & H_u & H_d \\
\hline
(q,\sigma_2,\tau_3) &
(1,1,1) & (-4,0,2) & (2,0,2) & (-3,1,0) & (6,0,0) & (3,1,0) & (-3,1,0)
\end{array}
\]
and define the \(\mathbb{Z}_6\) charges by
\begin{equation}
\label{eq:P6-assign}
\mathfrak{q}_6(Q)=1,\quad \mathfrak{q}_6(u^c)=5,\quad \mathfrak{q}_6(d^c)=5,\quad
\mathfrak{q}_6(L)=2,\quad \mathfrak{q}_6(e^c)=4,\quad \mathfrak{q}_6(H_u)=\mathfrak{q}_6(H_d)=0 \pmod{6}.
\end{equation}
These are realised by a linear monodromy functional \(\mathfrak{q}_6=a\,q+b\,\sigma_2+c\,\tau_3\)
with integers \((a,b,c)\) chosen so that \eqref{eq:P6-assign} holds on the seven generators
above; admissible choices exist because the seven constraints live in the \(\mathbb{Z}_{30}\)
module spanned by \(\{q,\,5\sigma_2,\,10\tau_3\}\) and reduce consistently mod \(6\).
The corresponding current \(J_P\) has order dividing \(30\), and its \(\mathbb{Z}_6\)
subgroup generated by \(J_P^{\,5}\) produces \(\mathfrak{q}_6\) as a selection rule.

\begin{proposition}[Yukawas allowed, proton decay forbidden]
\label{prop:P6-forbid}
With the \(\mathbb{Z}_6\) charges \eqref{eq:P6-assign}:
\begin{align*}
\text{Yukawas:}\quad &
\mathfrak{q}_6(Q)+\mathfrak{q}_6(H_u)+\mathfrak{q}_6(u^c)=1+0+5=0 \pmod{6},\\
&
\mathfrak{q}_6(Q)+\mathfrak{q}_6(H_d)+\mathfrak{q}_6(d^c)=1+0+5=0 \pmod{6},\\
&
\mathfrak{q}_6(L)+\mathfrak{q}_6(H_d)+\mathfrak{q}_6(e^c)=2+0+4=0 \pmod{6};
\\[4pt]
\text{dim--4 RPV:}\quad &
\mathfrak{q}_6(L)+\mathfrak{q}_6(L)+\mathfrak{q}_6(e^c)=2+2+4=2 \not\equiv 0,\\
&
\mathfrak{q}_6(L)+\mathfrak{q}_6(Q)+\mathfrak{q}_6(d^c)=2+1+5=2 \not\equiv 0,\\
&
\mathfrak{q}_6(u^c)+\mathfrak{q}_6(d^c)+\mathfrak{q}_6(d^c)=5+5+5=3 \not\equiv 0;
\\[4pt]
\text{dim--5 proton decay:}\quad &
\mathfrak{q}_6(Q)+\mathfrak{q}_6(Q)+\mathfrak{q}_6(Q)+\mathfrak{q}_6(L)=1+1+1+2=5 \not\equiv 0.
\end{align*}
Hence all three SM Yukawas are \emph{allowed}, while \(LLE^c\), \(LQD^c\), \(U^cD^cD^c\),
and \(QQQL\) are \emph{forbidden} at the worldsheet level (their three/four--point functions
vanish by the simple--current selection rule).
\end{proposition}

\begin{proof}
The \(\mathbb{Z}_6\) selection rule in a simple--current invariant states that a correlator
\(\langle\prod_i V_{\phi_i}\rangle\) can be nonzero only if
\(\sum_i Q_{J_P}(\phi_i)\in\mathbb{Z}\). Since \(6\,Q_{J_P}\equiv \mathfrak{q}_6 \pmod{6}\),
this is equivalent to \(\sum_i \mathfrak{q}_6(\phi_i)\equiv 0\ (\mathrm{mod}\ 6)\).
The sums above verify the claim.
\end{proof}

\begin{remark}[Compatibility with modular invariance]
The current \(J_P\) is a product of the admissible simple currents of the three factors.
Choosing the discrete--torsion phases so that the extended monodromy matrix is integral
(and symmetric) ensures modularity of the extended partition function
\citetag{SCHELLEKENSYANKIELOWICZ1989}, \citetag{SCHELLEKENSYANKIELOWICZ1990}. The assignment
\eqref{eq:P6-assign} corresponds to a \(\mathbb{Z}_6\) subgroup of the order--\(30\)
current \(J_P\) and is therefore compatible with the Gate~B invariant.
\end{remark}

\paragraph{Further dimension--5 check.}
The baryon-- and lepton--violating operator $U^c U^c D^c E^c$ also violates the selection rule:
\[
\mathfrak{q}_6(u^c)+\mathfrak{q}_6(u^c)+\mathfrak{q}_6(d^c)+\mathfrak{q}_6(e^c)
= 5+5+5+4 = 1 \not\equiv 0 \pmod{6},
\]
hence it is forbidden at the worldsheet level.

\newpage

\section{Formal recursion depth and the dilaton map}
\label{app:recursion-depth}

In this appendix we give a self--contained definition of recursion depth in MQG, show
that the associated Lyapunov structure forces a strictly monotone map
$\Phi_0=f(N)$ from recursion depth $N$ to the string dilaton vev, and
prove uniqueness of calibration once a single coupling is fitted.
This eliminates residual circularity in the MQG$\leftrightarrow$string
dictionary.

\subsection{Definitions and axioms}

\begin{definition}[Recursion depth]
\label{def:recursion}
Let $\mathcal{F}$ be a foliation of a compact 4--manifold into
coherence sheets (codimension--1 leaves). The \emph{recursion depth}
$N(\mathcal{F})\in\mathbb{N}$ is the maximal length of a nested
sequence of nontrivial subfoliations, each refining the previous one
by a boundary decomposition. Equivalently, $N$ is the height of the
directed acyclic graph of nested foliations.
\end{definition}

\begin{definition}[Decoherence curvature]
\label{def:Gamma}
Let $\Gamma$ denote the decoherence curvature scalar defined on each
sheet of $\mathcal{F}$, with $\Gamma\in L^2(\mathcal{F})$ and
$\int_{\mathcal{F}}\Gamma^2<\infty$.
\end{definition}

\begin{definition}[Lyapunov functional]
\label{def:Lyapunov}
The MQG Lyapunov functional is
\[
\mathcal{I}[\mathcal{F}] \;=\; \int_{\mathcal{F}}
\Big(\Gamma^2 + \text{anomaly penalties} + \text{torsion terms}\Big)\,,
\]
monotone under recursion refinements.
\end{definition}

\paragraph{Axioms.}
\begin{enumerate}[label=(R\arabic*),leftmargin=1.2em]
\item \emph{Boundedness.} $|\Gamma|$ is bounded on compact sets; $\mathcal{I}[\mathcal{F}]<\infty$.
\item \emph{Monotonicity.} One recursion step strictly decreases $\mathcal{I}$.
\item \emph{Additivity.} Independent recursion blocks compose additively:
$\mathcal{I}[\mathcal{F}_1\oplus\mathcal{F}_2]=\mathcal{I}[\mathcal{F}_1]+\mathcal{I}[\mathcal{F}_2]$.
\item \emph{Scaling.} Under Weyl rescaling $g_{\mu\nu}\mapsto \lambda g_{\mu\nu}$,
$\mathcal{I}$ rescales by a universal factor $\lambda^2$.
\end{enumerate}

\subsection{Derivation of the string--frame functional}

\begin{theorem}[Unique two--derivative MQG functional]
\label{thm:MQG-func}
From axioms (R1)--(R4), the unique local two--derivative MQG action is
\[
\mathcal{S}_{\mathrm{MQG}}
=\frac{1}{2\kappa_4^2}\int d^4x\sqrt{-g}\,e^{-2\Phi}\Big[
R+4(\nabla\Phi)^2-2\Lambda_{\mathrm{rec}}\Big]\,,
\]
up to local field redefinitions.
\end{theorem}

\begin{proof}[Sketch]
Boundedness and scaling fix an overall exponential factor $e^{-2\Phi}$.
Additivity and monotonicity force the quadratic structure $(R,(\nabla\Phi)^2)$.
The constant term encodes the deficit $\Lambda_{\mathrm{rec}}$. Full proof
follows App.~\ref{app:mqg-sigma}.
\end{proof}

\begin{corollary}[Genus/recursion equivalence]
\label{cor:genus-recursion}
On a genus--$g$ worldsheet, the Euler characteristic term contributes
$g_s^{2g-2}=e^{(2g-2)\Phi_0}$. One recursion step corresponds to a
topological move $\chi\mapsto\chi-2$, so recursion multiplicity and
genus weighting coincide. Thus there exists a function $f:\mathbb{N}\to\mathbb{R}$
such that $\Phi_0(N)=f(N)$.
\end{corollary}

\subsection{Properties of the map \texorpdfstring{$f(N)$}{f(N)}}

\begin{proposition}[Strict monotonicity]
\label{prop:monotone}
By axiom (R2), each recursion step strictly decreases $\mathcal{I}$.
Since $\mathcal{I}\propto e^{-2\Phi}$, this implies $f(N+1)>f(N)$.
\end{proposition}

\begin{proposition}[Injectivity]
\label{prop:injective}
If $f(N_1)=f(N_2)$ with $N_1<N_2$, then the per--step Lyapunov decrease
is zero somewhere, contradicting (R2). Therefore $N_1=N_2$.
\end{proposition}

\begin{proposition}[Unique calibration]
\label{prop:uniqueN}
Fix the pruned vacuum $V^\star$ and choose a reference scale $\mu_0$.
With $(k_Y,k_2,k_3)=(5,1,1)$,
\[
\alpha_i^{-1}(\mu_0)=C\,k_i^{\mathrm{(FT)}}\,e^{-2f(N)}+\delta_i(\mu_0).
\]
Fitting $\alpha_3(\mu_0)$ determines a unique $N^\star$ by injectivity.
All other couplings at $\mu_0$ then follow.
\end{proposition}

\begin{proof}
Injectivity (Prop.~\ref{prop:injective}) guarantees uniqueness of the solution
to $C e^{-2f(N)}=\alpha_3^{-1}(\mu_0)-\delta_3(\mu_0)$.
\end{proof}

\subsection{Threshold robustness}

\begin{proposition}[Stability under small thresholds]
\label{prop:threshold-stable}
If $|\delta_i|\ll C k_i e^{-2f(N)}$, then the inferred $N^\star$ is
stable: small perturbations shift the right--hand side of the calibration
equation by $O(\varepsilon)$, and strict monotonicity ensures the nearest--integer
solution for $N^\star$ is unchanged.
\end{proposition}

\subsection{Independence and falsifiability}

\begin{lemma}[Independence of RCFT choice]
\label{lem:indep}
The definitions of $N,\Gamma,\mathcal{I}$ use only MQG foliation structure
and not the RCFT input. Thus $f(N)$ is an MQG theorem, not a string postulate.
\end{lemma}

\begin{corollary}[Falsifiability hook]
\label{cor:falsify}
Given the unique pair $(V^\star,N^\star)$, the remaining couplings
$\alpha_2,\alpha_Y$ at $\mu_0$ are predictions. Together with MQG--linked
observables (neutrino curvature dependence, CMB/$\Lambda$ suppression),
this provides empirical falsifiers for the recursion--depth dictionary.
\end{corollary}

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\end{description}

\end{document}