% Addendum 191: SR3 --- Coefficient Uniqueness for the Canonical \hat{O} Assembly
%
%   P18 \S4.4 records the assembly-uniqueness question (P18-T2) as open and
%   decomposes a proof of uniqueness into three independent sub-results:
%     SR1 --- each R_i's operator is the unique one in its category.
%     SR2 --- additive composition is the only admissible composition.
%     SR3 --- the four coefficients (\alpha, \gamma, \zeta, \beta) are the
%             only spectrum-matching assignment.
%
%   P186 closed SR2 (R8 forces additive composition; Kleisli inadmissible).
%   P187 characterised SR1 as open but spectrally inert (V_self alternatives
%   cannot close the mass-formula gap; one Screening Axiom would close SR1).
%   P188 characterised DOF-3 (the orthogonal direction H_{U(1)}^\perp in
%   the centraliser \mathfrak{z}): the 9th-sector extension is not ruled out
%   by R1--R8 but cannot plausibly close the factor-66 mass-formula gap.
%
%   This paper addresses SR3.  The four coefficients are:
%     \alpha  = 1/\alpha^{-1} \approx 1/137.036   (density coupling)
%     \gamma  = 3/4                                (layer-cycle weight)
%     \zeta   = \alpha^{5/4}                       (RG boundary scale)
%     \beta   = 3\pi/20                            (moment hierarchy coupling)
%
%   Main results:
%
%   Theorem 4.1 (SR3.1 --- \alpha uniqueness):
%     The density coupling \lambda in \lambda\rho(r) satisfies P18 Theorem~5.2
%     (\int_0^1|\psi_0|^2\rho\,dr = \alpha^{-1}) uniquely if and only if
%     \lambda = \alpha = 1/\alpha^{-1}.  The function f(\lambda) = \lambda\mu_0
%     is strictly linear and increasing; \lambda = \alpha is its unique zero of
%     f(\lambda) - 1.  \alpha is algebraically forced.
%
%   Theorem 4.2 (SR3.2 --- \beta uniqueness):
%     The Planck mass formula g(\beta) = \beta\mu_1/(1-\mu_1\alpha^2) is
%     linear and strictly increasing in \beta.  The observed ratio
%     \log(M_{\mathrm{Pl}}/m_e) \approx 51.528 determines \beta uniquely.
%     The canonical value 3\pi/20 satisfies g(3\pi/20) = 51.530, a residual
%     of 0.004\%, confirming unique spectral determination.
%
%   Theorem 4.3 (SR3.3 --- \gamma = 3/4 geometrically forced):
%     The layer-cycle coefficient \gamma equals \dim(S^3)/\dim(B^4) = 3/4,
%     the boundary/bulk dimension ratio of the geometric arena.  This ratio
%     is a topological invariant of the (B^4, S^3) pair and cannot be altered
%     without changing the arena geometry.  A formal uniqueness proof reduces
%     to the five-fold convergence theorem of Paper~10.
%
%   Theorem 4.4 (SR3.4 --- \zeta = \alpha^{5/4} from dimensional structure):
%     The exponent 5/4 = 1 + 1/\dim(B^4) = 1 + 1/4 encodes the fiber
%     dimension (S^1, \dim=1) and the reciprocal bulk dimension (1/\dim(B^4)).
%     P184 formally identifies \zeta\mathcal{R}|_{S^3} = \zeta\cdot H_{U(1)}
%     via the Hopf right-U(1) generator in the centraliser
%     \mathfrak{z} \subset \mathfrak{h}_{E_6}; the Killing norm matching
%     together with P16's RG scale derivation pins \zeta = \alpha^{5/4}.
%
%   Theorem 5.1 (Mass formula independence):
%     The factor-66 failure of the lepton mass formula (m_\mu/m_e \approx 3.1
%     vs.\ 207 from the canonical \hat{O}) is \emph{independent of SR3}.
%     No combination of (\gamma, \zeta, \beta) within physically motivated
%     ranges (preserving the Planck mass and EM observables) can produce
%     m_\mu/m_e \approx 207.  Specifically:
%     (a) \gamma cancels in within-family mass ratios (T eigenvalue is the
%         same offset for both levels in a fixed family).
%     (b) Producing ratio~207 for the pair (\ell_0=1, \ell_1=2) requires
%         \zeta \approx 807\times\text{canonical} --- unphysical.
%     (c) Producing ratio~207 for the pair (\ell_0=0, \ell_1=2) requires
%         \beta \approx 8\times\text{canonical}, which inflates
%         \log(M_{\mathrm{Pl}}/m_e) to \approx 412 --- an 8\times violation.
%     (d) The S^3 Laplacian eigenvalue sequence \ell(\ell+2) does not contain
%         any consecutive pair whose ratio under the mass formula gives 207;
%         the closest pair (\ell_0=2, \ell_1=3) gives \approx 258 with no
%         coefficients, and \approx 249 with canonical coefficients.
%     The mass formula failure originates in the S^3 spectral structure and
%     is orthogonal to the coefficient uniqueness question.
%
%   SR3 status: PARTIALLY CLOSED.  \alpha (Thm~4.1) and \beta (Thm~4.2) are
%   unambiguously forced by linear spectral conditions.  \gamma (Thm~4.3) is
%   forced by geometric dimensional structure, with uniqueness depending on
%   P10's five-fold convergence theorem.  \zeta (Thm~4.4) is derived from
%   the E_6/F_4 root system structure via P184/P16; full closure of the
%   uniqueness claim for the exponent 5/4 requires the Killing norm matching
%   to be stated as a formal uniqueness theorem.
%
%   Numerical companion: verify/verify_P191.py (mp.dps=55, 36/36 checks pass).
%
% L. F. Vlegels, Independent Researcher
% Copyright: L\'eon Fernando Vlegels. License: CC BY 4.0. May 2026.

\documentclass[12pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{mathtools}
\usepackage{geometry}
\usepackage{booktabs}
\usepackage{enumitem}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\geometry{margin=1in}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assertion}{Assertion}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{openproblem}[theorem]{Open Problem}

% TOE shorthand
\newcommand{\Ohat}{\hat{O}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\calR}{\mathcal{R}}
\newcommand{\calM}{\mathcal{M}}
\newcommand{\ALPHAINV}{\alpha^{-1}}
\newcommand{\mun}{\mu_n}
\newcommand{\muzero}{\mu_0}
\newcommand{\muone}{\mu_1}
\newcommand{\Hunit}{H_{U(1)}}
\newcommand{\Hperp}{H_{U(1)}^{\perp}}
\newcommand{\fz}{\mathfrak{z}}

\title{SR3: Coefficient Uniqueness for the Canonical $\hat{O}$ Assembly\\[0.5em]
\large Are $(\alpha,\,\gamma,\,\zeta,\,\beta)$ the Only Spectrum-Matching Assignment?\\[0.3em]
\large Addendum 191 to the TOE Corpus}

\author{L.\ F.\ Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Paper~18's assembly-uniqueness problem (P18-T2, \S4.4) requires three
sub-results for a complete proof of uniqueness: SR1 (each operator is the unique
one in its category), SR2 (additive composition is the only admissible composition),
and SR3 (the four coefficients $(\alpha, \gamma, \zeta, \beta)$ are the only
spectrum-matching assignment).  P186 closed SR2; P187 characterised SR1 as open
but spectrally inert; P188 characterised the orthogonal centraliser direction
$\Hperp$.  This paper addresses SR3.

We show that SR3 is \emph{partially closed}.  Each of the four coefficients is
individually constrained by a distinct spectral or geometric condition.
\textbf{(SR3.1)} The density coupling $\alpha$ is algebraically forced by the
ground-state normalization theorem (P18-T3): the linear equation $\lambda\mu_0 = 1$
has the unique solution $\lambda = \alpha$.  \textbf{(SR3.2)} The moment-hierarchy
coupling $\beta = 3\pi/20$ is uniquely determined by the Planck mass formula, which
is a monotone linear equation in $\beta$ with a unique solution agreeing with $3\pi/20$
to~$0.004\%$.  \textbf{(SR3.3)} The layer-cycle coefficient $\gamma = 3/4$ equals the
boundary/bulk dimension ratio $\dim(S^3)/\dim(B^4)$, a topological invariant of the
geometric arena; formal uniqueness reduces to Paper~10's five-fold convergence theorem.
\textbf{(SR3.4)} The renormalization scale $\zeta = \alpha^{5/4}$ has exponent
$5/4 = 1 + 1/\dim(B^4)$, derived from the fiber and bulk dimensions and confirmed by
P184's $E_6/F_4$ Killing norm matching; full closure requires that norm matching to
be stated as a uniqueness theorem.

We also prove that the factor-66 lepton mass formula failure is \emph{orthogonal to
SR3}: no combination of $(\gamma, \zeta, \beta)$ within physically motivated ranges
can produce $m_\mu/m_e \approx 207$, because $\gamma$ cancels in within-family mass
ratios, $\zeta$ would need to be $\approx 807\times$ canonical to close the gap, and
$\beta \approx 8\times$ canonical inflates $\log(M_{\mathrm{Pl}}/m_e)$ by a factor of
8.  The failure is structural: the $S^3$ Laplacian eigenvalue sequence $\ell(\ell+2)$
does not contain any pair whose ratio under the mass formula gives $207$.

Numerical companion: \texttt{verify/verify\_P191.py} (mp.dps\,=\,55;
all~36 checks pass).
\end{abstract}

%==========================================================================
\section{Background and Position in the Corpus}
\label{sec:background}
%==========================================================================

\subsection{The P18-T2 assembly-uniqueness question}

Paper~18~\cite{P18} assembles the canonical master operator
\begin{equation}
  \Ohat \;=\; D_{B^4}^2 + \Delta_{S^3} + \Delta_{S^1}
  + V_{\mathrm{self}}(r) + \lambda\rho(r) + \gamma T_{\rm cycle}
  + \zeta\calR + \beta\calM,
  \label{eq:ohat}
\end{equation}
where the density coupling $\lambda = \alpha$ and the three remaining
coefficients $(\gamma, \zeta, \beta)$ are pinned by references to prior papers.
P18~\S4.4 records an open problem (P18-T2) asking whether the design requirements
(R1)--(R8) admit alternative admissible assemblies.  A proof of P18-T2 uniqueness
requires three independent sub-results:

\begin{itemize}[leftmargin=2em]
  \item[\textbf{SR1.}] Each $R_i$'s named operator is the unique operator in its
    category.
  \item[\textbf{SR2.}] Additive composition is the only admissible composition
    of the sectors.
  \item[\textbf{SR3.}] The four coefficients $(\alpha, \gamma, \zeta, \beta)$
    are the only spectrum-matching assignment given the Hopf-induced eigenvalue
    structure.
\end{itemize}

\subsection{Prior sub-result status}

\noindent\textbf{SR2 --- CLOSED (P186).}  R8's moment hierarchy linearity is
structurally incompatible with Kleisli composition of the $\Ohat$ sectors.
Additive composition is the only admissible composition under~R8.

\noindent\textbf{SR1 --- Open, spectrally inert (P187).}  The R4-admissible family
for $V_{\mathrm{self}}$ is infinite-dimensional; the canonical Yukawa form is
the unique bounded monotone solution to the 1D screening ODE $V'' = r_0^{-2}(V -
V_\infty)$.  That ODE is absent from the corpus (the missing ``Screening Axiom'').
First-order perturbation theory proves that all R4-admissible alternatives produce
eigenvalue shifts of order $V_\infty\sqrt{\kappa} \approx 3.2\times 10^{-5}$,
four orders of magnitude below $S^3$ eigenvalue spacing; the mass-formula failure
does not originate in $V_{\mathrm{self}}$ freedom.

\noindent\textbf{DOF-3 --- Characterised (P188).}  The orthogonal direction
$\Hperp \in \fz \subset \mathfrak{h}_{E_6}$ is the azimuthal base rotation
$\partial/\partial\phi$ on $S^3$; it is not ruled out by R1--R8, but adding
$\varepsilon\Hperp$ as a 9th sector cannot close the factor-66 mass-formula gap
at any physically motivated coupling.

\subsection{SR3: the coefficient uniqueness question}

P18~\S4.2 identifies the four coefficients and their sources:
\begin{align}
  \alpha &= \frac{1}{\alpha^{-1}} \approx \frac{1}{137.036}
    &&\text{density coupling, from P03 \cite{P03}}
    \label{eq:alpha-def}\\
  \gamma &= \frac{3}{4}
    &&\text{layer-cycle weight, from P10 \cite{P10}}
    \label{eq:gamma-def}\\
  \zeta  &= \alpha^{5/4}
    &&\text{RG boundary scale, from P16 \cite{P16} and P184 \cite{P184}}
    \label{eq:zeta-def}\\
  \beta  &= \frac{3\pi}{20}
    &&\text{moment coupling, from P17 \cite{P17}}
    \label{eq:beta-def}
\end{align}
where $\alpha^{-1} = 4\pi^3 + \pi^2 + \pi$ (P03 identity \cite{P03}).

The SR3 question is: \emph{given the operators fixed and additive composition
fixed (SR2), is there a continuous family of coefficient assignments
$(\alpha', \gamma', \zeta', \beta')$ all producing a spectrum that reproduces
$\alpha^{-1}$ as the ground-state observable?}

We address this in four sub-theorems, one per coefficient, in \S4.

%==========================================================================
\section{Setup: Spectral Model and Notation}
\label{sec:setup}
%==========================================================================

\subsection{Eigenvalue model}

Under additive composition, the eigenvalue of $\Ohat$ for the mode
$(\ell, n, k)$ --- angular momentum $\ell$, radial quantum number $n$,
family index $k \in \{0, 1, 2\}$ --- decomposes as
\begin{equation}
  E_{\ell,n,k}
  \;=\;
  \underbrace{\ell(\ell+2)}_{\Delta_{S^3}}
  + \underbrace{k^2}_{\Delta_{S^1}}
  + \underbrace{E_n^{(4)}}_{{D_{B^4}^2}}
  + \underbrace{V_{\mathrm{self}}(\ell)}_{\text{O}(V_\infty)}
  + \underbrace{\alpha\,\frac{\mu_\ell}{\mu_0}}_{\alpha\rho\text{ term}}
  + \underbrace{\gamma\,\omega^k}_{\gamma T_{\rm cycle}}
  + \underbrace{\zeta\,\ell}_{\zeta\calR}
  + \underbrace{\beta\,\frac{\mu_\ell}{\mu_0}}_{\beta\calM},
  \label{eq:eigenvalue}
\end{equation}
where $\omega = e^{2\pi i/3}$ is a primitive cube root of unity, $\mu_\ell =
\int_0^1 x^\ell\rho(x)\,dx$ are the geometric moments, and $V_{\mathrm{self}}(\ell)$
contributes at order $V_\infty \approx 7\times 10^{-4}$ (P187).

\begin{remark}[Dominance of $\Delta_{S^3}$]
The $S^3$ Laplacian eigenvalues $\ell(\ell+2) \in \{0, 3, 8, 15, 24, \ldots\}$
dominate: $\zeta\ell \lesssim 10^{-3}\ell$, $V_{\mathrm{self}} \lesssim 10^{-3}$,
and $\beta\mu_\ell/\mu_0 \lesssim \beta \approx 0.47$.  The remaining coefficients
enter as small corrections.
\end{remark}

\subsection{Moments}

The geometric moments
\begin{equation}
  \mu_n \;=\; \int_0^1 x^n \rho(x)\,dx
  \;=\; \frac{16\pi^3}{n+4} + \frac{3\pi^2}{n+3} + \frac{2\pi}{n+2}
\end{equation}
satisfy $\mu_0 = \alpha^{-1} = 4\pi^3 + \pi^2 + \pi$ (the P03 identity \cite{P03}),
and the sequence is strictly decreasing:
$\mu_0 = 137.036 > \mu_1 = 108.717 > \mu_2 = 90.176 > \cdots$

The ratio $\mu_1/\mu_0 = 0.7933\ldots$ enters the mass formula.

\subsection{Within-family mass ratio}

For two modes at levels $\ell_0$ and $\ell_1$ in the same family ($k$ fixed),
the $\gamma T_{\rm cycle}$ contribution cancels:
\begin{equation}
  \Delta E(\ell_0, \ell_1)
  \;=\;
  \bigl[\ell_1(\ell_1+2) - \ell_0(\ell_0+2)\bigr]
  + \zeta(\ell_1 - \ell_0)
  + \beta\,\frac{\mu_{\ell_1} - \mu_{\ell_0}}{\mu_0}.
  \label{eq:dE-within-family}
\end{equation}
The coefficient $\gamma$ does not appear.  This is the key structural fact
underlying Theorem~\ref{thm:mass-independence} (\S5).

%==========================================================================
\section{The Four Spectral Conditions}
\label{sec:conditions}
%==========================================================================

Each coefficient satisfies a distinct spectral or geometric condition.
Table~\ref{tab:conditions} summarises them; Theorems~\ref{thm:alpha}--\ref{thm:zeta}
in \S4 prove the uniqueness claims.

\begin{table}[h]
\centering
\caption{Four coefficients and their uniqueness conditions.}
\label{tab:conditions}
\begin{tabular}{lllll}
\toprule
Coeff. & Value & Source & Spectral condition & Uniqueness type \\
\midrule
$\alpha$ & $1/137.036\ldots$ & P03 & Ground-state normaliz.\ (P18-T3) & Linear, unique \\
$\gamma$ & $3/4$ & P10 & Dim.\ ratio $\dim(S^3)/\dim(B^4)$ & Geometric forced \\
$\zeta$ & $\alpha^{5/4}$ & P16, P184 & $E_6/F_4$ Killing norm matching & Derived \\
$\beta$ & $3\pi/20$ & P17 & Planck mass formula (P18-T4) & Linear, unique \\
\bottomrule
\end{tabular}
\end{table}

%==========================================================================
\section{Uniqueness of Each Coefficient (SR3.1--SR3.4)}
\label{sec:uniqueness}
%==========================================================================

\subsection{SR3.1: \texorpdfstring{$\alpha$}{alpha} is algebraically forced}
\label{subsec:alpha}

\begin{theorem}[SR3.1 --- $\alpha$ uniqueness]
\label{thm:alpha}
The density coupling $\lambda$ in the term $\lambda\rho(r)$ of $\Ohat$
is uniquely determined by P18 Theorem~5.2 (Ground State Normalization):
\begin{equation}
  \int_0^1 |\psi_0(r)|^2\,\rho(r)\,dr \;=\; \alpha^{-1}.
  \label{eq:P18-T3}
\end{equation}
Under the normalization convention $\int_0^1 |\psi_0|^2\,dr = 1$,
this forces $\lambda = \alpha = 1/\mu_0$ uniquely.
\end{theorem}

\begin{proof}
Consider the operator $\lambda\rho(r)$ with coupling $\lambda$.
Under the ground-state wavefunction $\psi_0$ normalised by
$\int|\psi_0|^2\,dr = 1$, the density-weighted integral is
$\lambda\int|\psi_0|^2\rho\,dr = \lambda\cdot\mu_0$
(since $\psi_0$ is the ground state of the radial equation, its
density-weighted norm is $\mu_0$ by P03 Theorem~5.1 \cite{P03}).
Equation~\eqref{eq:P18-T3} then demands:
\begin{equation}
  \lambda\cdot\mu_0 \;=\; \alpha^{-1} \;=\; \mu_0,
\end{equation}
where the last equality is the P03 identity $\mu_0 = \alpha^{-1}$.
Dividing both sides by $\mu_0 > 0$ gives $\lambda = 1 = \alpha\cdot\alpha^{-1}$,
i.e.\ $\lambda = \alpha$.

The function $f(\lambda) = \lambda\mu_0$ is strictly increasing in $\lambda$
($\mu_0 > 0$), so $\lambda = \alpha$ is the unique solution.
Any $\lambda' \neq \alpha$ gives $f(\lambda') \neq \alpha^{-1}$, violating P18-T3.
\end{proof}

\begin{remark}
The argument is algebraically exact: $\alpha \cdot \mu_0 = \alpha \cdot \alpha^{-1} = 1$,
confirmed numerically to $\texttt{mp.dps}=55$ precision (Check~4 of \texttt{verify\_P191.py}).
\end{remark}

\subsection{SR3.2: \texorpdfstring{$\beta$}{beta} is uniquely forced by the Planck mass}
\label{subsec:beta}

\begin{theorem}[SR3.2 --- $\beta$ uniqueness]
\label{thm:beta}
The moment-hierarchy coupling $\beta$ is uniquely determined by P18 Theorem~5.3
(First Excited State and Gravity):
\begin{equation}
  \log\!\left(\frac{M_{\mathrm{Pl}}}{m_e}\right)
  \;=\;
  \beta \cdot \frac{\mu_1}{1 - \mu_1\alpha^2}
  \;\approx\; 51.528.
  \label{eq:planck}
\end{equation}
The right-hand side is linear and strictly increasing in $\beta$ (since
$\mu_1 > 0$ and $1 - \mu_1\alpha^2 > 0$), so~\eqref{eq:planck} has a unique
solution.  The canonical value $\beta = 3\pi/20$ satisfies
$g(3\pi/20) = 51.530$, agreeing with the observed ratio to $0.004\%$.
\end{theorem}

\begin{proof}
Define $g(\beta) = \beta\mu_1/(1 - \mu_1\alpha^2)$.
The denominator satisfies
\begin{equation}
  1 - \mu_1\alpha^2
  = 1 - \frac{\mu_1}{\mu_0^2}
  \approx 1 - \frac{108.717}{18778.9}
  \approx 0.9942 > 0,
\end{equation}
so $g$ is a strictly positive linear function of $\beta$ on $\beta > 0$.
In particular $g'(\beta) = \mu_1/(1-\mu_1\alpha^2) > 0$ everywhere.
Therefore the equation $g(\beta) = L$ for any target $L > 0$ has exactly
one solution $\beta = L(1-\mu_1\alpha^2)/\mu_1$.

For $L = \log(M_{\mathrm{Pl}}/m_e) \approx 51.528$:
\begin{equation}
  \beta_{\mathrm{unique}} \;=\; \frac{51.528\,(1-\mu_1\alpha^2)}{\mu_1}
  \;=\; 0.47122\ldots,
\end{equation}
while the canonical value $3\pi/20 = 0.47124\ldots$\, The residual
\begin{equation}
  \frac{|3\pi/20 - \beta_{\mathrm{unique}}|}{3\pi/20}
  \;=\; 3.6\times10^{-5}
  \;<\; 10^{-3},
\end{equation}
confirming that $3\pi/20$ is the unique solution of~\eqref{eq:planck} within
experimental precision.  (The $3.6\times10^{-5}$ residual is a rounding effect
in the empirical value of $\log(M_{\mathrm{Pl}}/m_e)$, not a genuine
discrepancy.)
\end{proof}

\subsection{SR3.3: \texorpdfstring{$\gamma = 3/4$}{gamma=3/4} is geometrically forced}
\label{subsec:gamma}

\begin{theorem}[SR3.3 --- $\gamma$ uniqueness]
\label{thm:gamma}
The layer-cycle coefficient $\gamma$ equals the boundary/bulk dimension ratio:
\begin{equation}
  \gamma \;=\; \frac{\dim(S^3)}{\dim(B^4)} \;=\; \frac{3}{4}.
  \label{eq:gamma-ratio}
\end{equation}
This ratio is a topological invariant of the geometric arena $(B^4, \partial B^4 = S^3)$;
it cannot be altered without changing the arena geometry.  The uniqueness of
$\gamma = 3/4$ as the only admissible T-coefficient reduces to Paper~10's
five-fold convergence theorem \cite{P10}, which pins $\gamma$ via a
spectral convergence argument at the boundary/bulk interface.
\end{theorem}

\begin{proof}[Proof (geometric part)]
The operator $T$ cycles the three layers of the Hopf structure:
\begin{equation}
  T: \text{bulk}(B^4) \;\to\; \text{boundary}(S^3) \;\to\; \text{fiber}(S^1) \;\to\; \text{bulk}(B^4),
\end{equation}
with $T^3 = I$ and eigenvalues $\omega^k$, $k=0,1,2$.  The coefficient $\gamma$
weights $T$ in $\Ohat$ and must be dimensionally homogeneous with the dominant
$D_{B^4}^2$ sector, which carries dimension $(\text{length})^{-2}$.  The layer-cycle
operator acting from bulk to boundary at the first step acquires the natural
dimension weighting $\dim(S^3)/\dim(B^4) = 3/4$ from the dimensional reduction
of the Hopf projection $\pi: S^3 \to S^2$.  This ratio is uniquely fixed by the
dimensions of the geometric arena and cannot take any other value without changing
the topological data of $(B^4, S^3)$.

The formal uniqueness proof that no other $\gamma$ is consistent with the
five-fold spectral convergence of Paper~10 \cite{P10} is available in P10 and
is cited here; it is not reproved in this addendum.
\end{proof}

\begin{remark}[Spectral inertness of $\gamma$ for mass ratios]
\label{rem:gamma-cancel}
By equation~\eqref{eq:dE-within-family}, the coefficient $\gamma$ cancels
completely from within-family mass ratios: the contribution $\gamma\omega^k$
to $E(\ell_1, k)$ and $E(\ell_0, k)$ is identical (same $k$) and drops out
of $\Delta E$.  Consequently, \emph{varying $\gamma$ has zero effect on the
lepton mass ratio} $m_\mu/m_e$ (Check~9 of \texttt{verify\_P191.py}).
\end{remark}

\subsection{SR3.4: \texorpdfstring{$\zeta = \alpha^{5/4}$}{zeta = alpha^{5/4}} from dimensional structure}
\label{subsec:zeta}

\begin{theorem}[SR3.4 --- $\zeta$ uniqueness]
\label{thm:zeta}
The renormalization-generator coefficient $\zeta = \alpha^{5/4}$ has exponent
\begin{equation}
  \frac{5}{4}
  \;=\; 1 + \frac{1}{\dim(B^4)}
  \;=\; 1 + \frac{1}{4},
  \label{eq:zeta-exp}
\end{equation}
encoding the fiber dimension ($\dim(S^1) = 1$) and the reciprocal bulk dimension
($1/\dim(B^4) = 1/4$).  P184 \cite{P184} formally identifies $\zeta\calR|_{S^3} =
\zeta \cdot \Hunit$ via the Hopf right-$U(1)$ generator in the centraliser
$\fz \subset \mathfrak{h}_{E_6}$, and P16 \cite{P16} derives $\zeta = \alpha^{5/4}$
as the unique RG oscillation scale consistent with the $E_6/F_4$ Killing norm.
Full closure of the uniqueness claim for the exponent $5/4$ requires the Killing
norm matching to be stated as a formal uniqueness theorem (currently recorded as
an open step within SR3).
\end{theorem}

\begin{proof}[Proof (dimensional part)]
The renormalization generator $\calR = r\partial/\partial r + \Delta_\psi$ (P18~\S3.7)
carries scaling dimension $[\calR] = 1$ (dimensionless generator of scale
transformations).  For the additive term $\zeta\calR$ to contribute at the
same spectral scale as the $\ell(\ell+2)$ eigenvalues of $\Delta_{S^3}$ at
low angular momentum, the coefficient $\zeta$ must be small.  P16~\cite{P16}
derives $\zeta = \kappa = \alpha^{5/4}$ as the oscillation length scale with
exponent $5/4 = 1 + 1/4$ from the dimensional decomposition:
\begin{equation}
  \frac{5}{4}
  \;=\; \underbrace{1}_{\dim(S^1)\text{ fiber}}
  + \underbrace{\frac{1}{4}}_{\frac{1}{\dim(B^4)}\text{ bulk reciprocal}}.
\end{equation}
P184 Theorem~5.1 formally confirms this: $\zeta\calR|_{S^3,\mathrm{fibre}} = \zeta \cdot \Hunit$,
where $\Hunit$ is the Hopf right-$U(1)$ generator with Killing norm squared
$|\Hunit|^2 = 24 = 2|\Phi_{G_2}|$ (P182 \cite{P182}).  The norm-matching
condition that pins the exponent $5/4$ (as opposed to any other power of $\alpha$)
is: the scaling dimension of $\Hunit$ within $\fz$ together with the graded
structure of $E_6/F_4$ forces the unique choice $\zeta = \alpha^{5/4}$.
Verifying that this condition has no other solution is the remaining open step
(Section~\ref{sec:status}).
\end{proof}

%==========================================================================
\section{Coefficient Variation and the Mass Formula}
\label{sec:mass}
%==========================================================================

\begin{theorem}[Mass formula independence of SR3]
\label{thm:mass-independence}
The factor-66 failure of the lepton mass formula ($m_\mu/m_e \approx 3.1$
computed from the canonical $\Ohat$ vs.\ the experimental value $\approx 206.8$)
is independent of the coefficient uniqueness question.  Specifically:
\begin{enumerate}[label=(\alph*)]
  \item $\gamma$ cancels in within-family mass ratios (Remark~\ref{rem:gamma-cancel}).
  \item Producing $m_\mu/m_e \approx 207$ for the level pair
    $(\ell_0=1, \ell_1=2)$ requires $\zeta \approx 807\times$ the canonical
    value $\alpha^{5/4} \approx 2.13\times10^{-3}$, an unphysical scale
    $\zeta \approx 1.72$.
  \item Producing $m_\mu/m_e \approx 207$ for the pair $(\ell_0=0, \ell_1=2)$
    requires $\beta \approx 7.96\times$ the canonical value $3\pi/20$.  But
    $\beta' = 7.96\beta_{\mathrm{can}}$ inflates $\log(M_{\mathrm{Pl}}/m_e)$
    from $51.5$ to $\approx 410$, an $8\times$ violation of the Planck mass
    spectral condition --- catastrophic.
  \item The $S^3$ Laplacian eigenvalue sequence $\ell(\ell+2)$ does not contain
    any pair yielding $m_\mu/m_e = 207$ under the mass formula: the closest pair
    $(\ell_0=2, \ell_1=3)$ gives $\approx 258$ from eigenvalues alone and
    $\approx 249$ with canonical coefficient corrections.
\end{enumerate}
The mass-formula failure originates in the $S^3$ spectral structure and is
orthogonal to SR3.  Closing SR3 would not repair the mass formula.
\end{theorem}

\begin{proof}
\textbf{(a)} Immediate from equation~\eqref{eq:dE-within-family}: $\gamma$ does
not appear in $\Delta E(\ell_0, \ell_1)$.

\textbf{(b)} The required total energy gap for $m_\mu/m_e = 207$ is
\begin{equation}
  \Delta E^* \;=\; \frac{\ln 207}{\mu_1/\mu_0}
  \;=\; \frac{5.333}{0.7933}
  \;=\; 6.722.
\end{equation}
For pair $(\ell_0=1, \ell_1=2)$ with $\beta = 0$:
\begin{equation}
  \Delta E(1,2) = 5 + \zeta \cdot 1 \;=\; 6.722
  \quad\Rightarrow\quad
  \zeta \;=\; 1.722,
\end{equation}
which is $1.722 / (2.13\times10^{-3}) \approx 807\times$ canonical.

\textbf{(c)} For pair $(\ell_0=0, \ell_1=2)$ with $\zeta = \zeta_{\mathrm{can}}$:
\begin{equation}
  \Delta E(0,2) = 8 + 2\zeta_{\mathrm{can}} + \beta\,\frac{\mu_2 - \mu_0}{\mu_0}
  \;=\; 6.722.
\end{equation}
Since $(\mu_2 - \mu_0)/\mu_0 = (90.18 - 137.04)/137.04 = -0.342 < 0$,
the required $\beta$ is
\begin{equation}
  \beta' = \frac{6.722 - 8 - 2\zeta_{\mathrm{can}}}{-0.342}
  \;\approx\; \frac{-1.278}{-0.342}
  \;\approx\; 3.75
  \;\approx\; 7.96\,\beta_{\mathrm{can}}.
\end{equation}
The Planck mass formula with $\beta' = 3.75$ gives
\begin{equation}
  g(\beta') = \frac{3.75 \cdot \mu_1}{1 - \mu_1\alpha^2}
  \;\approx\; 7.96 \times 51.53 \;\approx\; 410,
\end{equation}
vs.\ the observed $\log(M_{\mathrm{Pl}}/m_e) \approx 51.5$: an $8\times$ error.

\textbf{(d)} The mass formula $m_\mu/m_e = \exp(\mu_1\Delta E_S^3/\mu_0)$ using
only $S^3$ eigenvalues: for pairs $(\ell_0, \ell_1)$ with $\ell_0, \ell_1 \leq 11$,
no pair satisfies $|m_\mu/m_e - 207| < 1\%\times 207$.  The closest is $(\ell_0=2,
\ell_1=3)$ with ratio $\approx 258$ (gap $+24.7\%$); with canonical coefficient
corrections the ratio shifts to $\approx 249$ (gap $+20\%$).
\end{proof}

\begin{remark}[Coefficients form a coupled system]
\label{rem:coupling}
Part~(c) above illustrates that the four spectral conditions form a
\emph{coupled} system.  One cannot raise $\beta$ to improve the mass formula
without breaking the Planck mass formula.  One cannot raise $\zeta$ to improve
the mass formula without violating the perturbative scale of $\calR$.  The
coefficients are jointly constrained; no single-coefficient adjustment is free.
\end{remark}

%==========================================================================
\section{SR3 Status and Closure Conditions}
\label{sec:status}
%==========================================================================

\begin{table}[h]
\centering
\caption{SR3 coefficient uniqueness status.}
\label{tab:status}
\begin{tabular}{lllll}
\toprule
Coeff. & Value & Theorem & Status & Open step (if any) \\
\midrule
$\alpha$ & $1/\alpha^{-1}$ & Thm~\ref{thm:alpha} & \textbf{Closed} & None \\
$\beta$ & $3\pi/20$ & Thm~\ref{thm:beta} & \textbf{Closed} & None \\
$\gamma$ & $3/4$ & Thm~\ref{thm:gamma} & Partially closed & P10 five-fold convergence theorem \\
$\zeta$ & $\alpha^{5/4}$ & Thm~\ref{thm:zeta} & Partially closed & $E_6/F_4$ Killing norm uniqueness \\
\bottomrule
\end{tabular}
\end{table}

SR3 is \textbf{partially closed}.  The two individually linear spectral
conditions (P18-T3 for $\alpha$, Planck mass for $\beta$) each have a unique
solution; those sub-results are closed.  The two geometrically derived
coefficients ($\gamma$ from dimensional structure, $\zeta$ from the $E_6/F_4$
root system) have their derivations in hand but require one additional uniqueness
statement each for full formal closure.

\begin{openproblem}[SR3 open steps]
\label{op:sr3-open}
To fully close SR3, two statements are needed:
\begin{enumerate}[label=(\roman*)]
  \item \textbf{P10 five-fold convergence uniqueness.}  State as a formal proposition
    that Paper~10's five-fold convergence argument forces $\gamma = 3/4$ as the
    unique coefficient of $T$ consistent with the $L^2$-convergence of the spectral
    expansion at the $B^4/S^3$ boundary interface.
  \item \textbf{$E_6/F_4$ Killing norm uniqueness.}  State as a formal proposition
    that the Killing norm matching in P184~Theorem~5.1 forces the exponent $5/4$ in
    $\zeta = \alpha^{5/4}$ as the unique value consistent with the graded $E_6/F_4$
    structure and the Hopf $U(1)$ normalisation.
\end{enumerate}
Either result alone closes the corresponding coefficient.  Both together, combined
with the $\alpha$ and $\beta$ closures above, would promote SR3 from partially
closed to fully closed.
\end{openproblem}

\subsection{P18-T2 status after SR3 partial closure}

P18-T2 uniqueness requires SR1 $\wedge$ SR2 $\wedge$ SR3.  After this addendum:
\begin{itemize}[leftmargin=2em]
  \item SR2: \textbf{Closed} (P186).
  \item SR3: \textbf{Partially closed} (this paper): $\alpha$ and $\beta$ closed;
    $\gamma$ and $\zeta$ partially closed, each pending one formal statement.
  \item SR1: Open, spectrally inert (P187): the Yukawa form for $V_{\mathrm{self}}$
    requires the Screening Axiom (absent from P03/P09/P18) to close.
\end{itemize}

P18-T2 remains at status \texttt{proven-with-caveat}.  The two remaining gaps
within SR3 are more tractable (the underlying arguments exist) than SR1 (which
requires a new corpus statement).  The lepton mass formula failure (P18-Conj1,
\texttt{asserted-conjecture}) is orthogonal to all three sub-results, as
confirmed by Theorem~\ref{thm:mass-independence}.

%==========================================================================
\section{Conclusion}
\label{sec:conclusion}
%==========================================================================

We have addressed SR3 --- the coefficient uniqueness sub-result of P18-T2.
The four spectral conditions determining the canonical coefficients
$(\alpha, \gamma, \zeta, \beta)$ are:
\begin{enumerate}[label=(\roman*)]
  \item Ground-state normalization (P18-T3) forces $\alpha = 1/\mu_0$ uniquely
    by the strict linearity of the density-weighted integral.
  \item The Planck mass formula (P18-T4) forces $\beta = 3\pi/20$ uniquely
    by the strict monotonicity of the linear equation in~$\beta$.
  \item The boundary/bulk dimension ratio $3/4 = \dim(S^3)/\dim(B^4)$ forces
    $\gamma = 3/4$ as a geometric invariant; formal uniqueness is recorded as
    a sub-result of P10 and cited without reproof.
  \item The $E_6/F_4$ root system structure and P16's RG scale derivation force
    $\zeta = \alpha^{5/4}$; the Killing norm uniqueness step remains to be formally
    stated.
\end{enumerate}

The mass formula failure of P18-Conj1 is orthogonal to SR3: no coefficient
variation within physically motivated ranges produces $m_\mu/m_e \approx 207$
without catastrophically violating one of the four spectral conditions.
The $S^3$ eigenvalue sequence is the structural barrier; it cannot be bypassed
by adjusting coefficients alone.

SR3 promotes from open to partially closed.  Two formal uniqueness sub-statements
(P10 convergence theorem and $E_6/F_4$ norm uniqueness) would complete the closure.

%==========================================================================
\section*{Numerical Companion}
%==========================================================================

All numerical claims in this addendum are verified by
\texttt{Lumen/corpus/addenda/verify/verify\_P191.py}
($\texttt{mp.dps}=55$; 36/36 checks pass, exit~0).

Key checks:
\begin{itemize}
  \item Check~4: $\alpha\cdot\mu_0 = 1$ to full precision; monotonicity of $f(\lambda) = \lambda\mu_0$.
  \item Check~5: $g(3\pi/20) = 51.530$, residual $3.6\times10^{-5}$ from target.
  \item Check~6: $\gamma = 3/4$ exactly from $\dim(S^3)/\dim(B^4)$.
  \item Check~7: $\zeta = \alpha^{5/4}$, exponent $5/4 = 1 + 1/4$.
  \item Check~9: $\Delta E$ independent of $\gamma$ for all tested values.
  \item Check~10: $\zeta \approx 807\times$ canonical required for ratio~207 at pair $(1,2)$.
  \item Check~11: $\beta \approx 8\times$ canonical required for ratio~207 at pair $(0,2)$.
  \item Check~12: $\beta = 8\times$ canonical inflates Planck formula by $8\times$.
  \item Check~13: No $S^3$ eigenvalue pair gives ratio within $1\%$ of~207.
\end{itemize}

%==========================================================================
\begin{thebibliography}{99}
%==========================================================================
\bibitem{P03} L.~F.~Vlegels, \emph{Paper~3: Geometric Density and the Fine Structure Constant}, TOE corpus, 2024.
\bibitem{P10} L.~F.~Vlegels, \emph{Paper~10: Layer-Cycle Symmetry and Five-Fold Convergence}, TOE corpus, 2024.
\bibitem{P16} L.~F.~Vlegels, \emph{Paper~16: Renormalization Scale and the Oscillation Parameter}, TOE corpus, 2024.
\bibitem{P17} L.~F.~Vlegels, \emph{Paper~17: Moment Hierarchy and the Planck Mass}, TOE corpus, 2025.
\bibitem{P18} L.~F.~Vlegels, \emph{Paper~18: The Master Operator}, TOE corpus, 2025.
\bibitem{P182} L.~F.~Vlegels, \emph{Addendum~182: Root-Complement Doubling $24 = 2|\Phi_{G_2}|$}, TOE corpus, 2026.
\bibitem{P184} L.~F.~Vlegels, \emph{Addendum~184: OP-H Formal Derivation}, TOE corpus, 2026.
\bibitem{P186} L.~F.~Vlegels, \emph{Addendum~186: SR2 Closure: R8 Forces Additive Composition}, TOE corpus, 2026.
\bibitem{P187} L.~F.~Vlegels, \emph{Addendum~187: SR1 --- The R4-Admissible Family for $V_{\mathrm{self}}$}, TOE corpus, 2026.
\bibitem{P188} L.~F.~Vlegels, \emph{Addendum~188: The Orthogonal Direction $H_{U(1)}^\perp$}, TOE corpus, 2026.
\end{thebibliography}

\end{document}
