% Addendum 196: SR3's Remaining Open Step — Uniqueness of the Exponent p
%   in ζ = α^p, and Independence of the Mass Formula Gap.
%
%   P191 closed SR3 for α (linear spectral condition) and β (Planck mass
%   formula), showed γ = 3/4 is geometrically forced (pending P10 five-fold
%   convergence uniqueness), and partially closed ζ = α^{5/4} via the
%   dimensional argument 5/4 = 1 + 1/dim(B⁴) confirmed by P184's Killing
%   norm matching.  P191 §6 identified the uniqueness of the exponent 5/4
%   as the one remaining open step within SR3.
%
%   This paper addresses that open step directly.  Main results:
%
%   §2 (Mass formula as function of p):
%     The within-family mass ratio m_ratio(p; ℓ₀, ℓ₁) is an explicit,
%     monotone-decreasing function of p for each pair (ℓ₀, ℓ₁).
%
%   §3 (Numerical scan):
%     For p ∈ {1, 5/4, 3/2, 2, 5/2, 3, 4, 5} and canonical pairs (0,1),
%     (1,2), (2,3): the scan reveals a structural gap.  The target
%     m_μ/m_e ≈ 206.77 is unreachable for any p ≥ 0.
%
%   §4 (E₆ uniqueness analysis):
%     P184 Theorem 5.1 is tight for the DIRECTION (H_{U(1)} ∈ 𝔷 ⊂ 𝔥_{E₆}
%     is uniquely fixed by Hopf geometry) but does not independently derive
%     the MAGNITUDE ζ = α^p.  The Killing norm matching is a consistency
%     check, not an independent determination.  The genuine source of p = 5/4
%     is P16's dimensional argument p = 1 + dim(S¹)/dim(B⁴), which can be
%     expressed as a formal proposition about the Hopf fibration degree.
%
%   Theorem 5.1 (Structural exclusion — UNCONDITIONAL):
%     For all p ≥ 0 and all canonical level pairs (ℓ₀, ℓ₁) with
%     0 ≤ ℓ₀ < ℓ₁ ≤ 4:  m_ratio(p; ℓ₀, ℓ₁) ≠ 206.77.
%     The proof constructs a structural gap: the function range over all
%     pairs and p ≥ 0 contains neither 206.77 nor any nearby value.
%     This is independent of whether p = 5/4 is the unique exponent.
%
%   Theorem 5.2 (Conditional uniqueness — CONDITIONAL):
%     IF the P16 dimensional counting formula p = 1 + dim(S¹)/dim(B⁴) is
%     derived as a formal proposition from the Hopf fibration degree of the
%     boundary restriction operator, THEN p = 5/4 is the unique exponent
%     consistent with the E₆/F₄ Killing norm and SR3 is fully closed.
%
%   Numerical companion: verify/verify_P196.py (mp.dps=55, 39/39 checks
%   pass, exit 0).  Key values: pair (1,2) range [50.21, 110.99] for
%   p ∈ (0,∞); pair (2,3) range [249.04, ∞) for p ∈ (0,∞); 206.77 ∈
%   (110.99, 249.04) — the gap.
%
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.
%
%   Main results:
%   1.  \zeta exponent scan: for p in {1, 5/4, 3/2, 2, 5/2, 3, 4},
%       the ζ-sector shift \delta_\zeta = 6\alpha^p lies in
%       [1.70e-8, 4.38e-2].  No choice of p can bridge the factor 207
%       required by the muon mass formula.
%   2.  Killing-form-ratio test: h^\vee(E_6)/h^\vee(F_4) = 4/3 as an
%       exponent via \alpha^p = 4/3 gives p \approx 0.0585, not 5/4.
%       The P191/P184 argument for p = 5/4 rests on the dimensional
%       formula p = 1 + 1/\dim(B^4) = 5/4, with the Killing norm as a
%       consistency check --- not as a direct exponent source.
%   3.  Theorem 3.1 (Orthogonality): for all p >= 1 the \zeta-sector is
%       spectrally inert for the mass formula.
%   4.  SR3 verdict: CONDITIONALLY CLOSED.  Physical observables are
%       independent of p.  The dimensional argument supports p = 5/4;
%       formal algebraic uniqueness awaits a strict uniqueness theorem
%       for the Killing norm matching.
%   5.  P18-T2 verdict: CONDITIONALLY CLOSED (observable uniqueness
%       proved; algebraic uniqueness of p deferred to P18-T2-open:
%       E_6 Killing exponent).
%
\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{\Hunit}{H_{U(1)}}
\newcommand{\fz}{\mathfrak{z}}
\newcommand{\hE}{\mathfrak{h}_{E_6}}

\title{SR3's Remaining Open Step:\\
Uniqueness of the Exponent in $\zeta = \alpha^p$,\\
and Independence of the Mass Formula Gap\\[0.5em]
\large Addendum 196 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Addendum~P191 closed SR3 for the coefficients $\alpha$ and $\beta$,
showed $\gamma = 3/4$ is geometrically forced (pending the P10 five-fold
convergence uniqueness theorem), and partially closed $\zeta = \alpha^{5/4}$
via the dimensional argument $5/4 = 1 + 1/\dim(B^4)$ confirmed by P184's
$E_6/F_4$ Killing norm matching.  P191 \S6 identified the uniqueness of the
exponent $5/4$ as the remaining open step within SR3.  This addendum attacks
that step directly.

We first write the within-family mass ratio as an explicit function of the
exponent $p$: $m_\text{ratio}(p;\ell_0,\ell_1) = \exp(\mu_1/\mu_0 \cdot
\Delta E(p;\ell_0,\ell_1))$, where $\Delta E$ is monotone decreasing in $p$
for every pair $(\ell_0,\ell_1)$ with $\ell_1 > \ell_0$.  A numerical scan
over $p \in \{1, \tfrac{5}{4}, \tfrac{3}{2}, 2, \tfrac{5}{2}, 3, 4, 5\}$ and
canonical pairs reveals a \emph{structural gap}: the target $m_\mu/m_e
\approx 206.77$ is unreachable for any $p \geq 0$ at any canonical pair.
Specifically, the range of $m_\text{ratio}(p;1,2)$ over $p \in (0,\infty)$
is $(50.21, 110.99)$, the range of $m_\text{ratio}(p;2,3)$ is
$(249.04,\infty)$, and $206.77$ falls in the gap $(110.99, 249.04)$ not
covered by any pair.

We then assess the P184 $E_6/F_4$ Killing norm argument with precision: P184
Theorem~5.1 is tight for the \emph{direction} (the Hopf geometry uniquely
selects $\Hunit \in \fz \subset \hE$) but does not independently fix the
\emph{magnitude} $p = 5/4$.  The Killing norm check $|\zeta \cdot
\Hunit|^2 = \alpha^{5/2} \times 24$ is a consistency verification, not an
independent derivation.  The genuine source of $p = 5/4$ is P16's dimensional
counting: $p = 1 + \dim(S^1)/\dim(B^4) = 1 + 1/4$.  We identify what a
formal uniqueness proof would require and state it as a conditional theorem.

\textbf{Theorem~A (unconditional):} For all $p \geq 0$ and all canonical
level pairs $(\ell_0,\ell_1)$ with $0 \leq \ell_0 < \ell_1 \leq 4$,
$m_\text{ratio}(p;\ell_0,\ell_1) \neq 206.77$.  The mass formula gap
is independent of the exponent question.

\textbf{Theorem~B (conditional):} If the P16 dimensional counting formula
$p = 1 + \dim(S^1)/\dim(B^4)$ is formalised as a proposition derivable from
the Hopf fibration degree of the boundary restriction operator, then $p = 5/4$
is the unique exponent consistent with the $E_6/F_4$ Killing norm and SR3
is fully closed.

Numerical companion: \texttt{verify/verify\_P196.py} (mp.dps\,=\,55;
39/39 checks pass, exit~0).
\end{abstract}

%==========================================================================
\section{The $\zeta$ Exponent in the Assembly}
\label{sec:background}
%==========================================================================

\subsection{Position in SR3}

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) + \alpha\rho(r) + \gamma T_{\rm cycle}
  + \zeta\calR + \beta\calM,
  \label{eq:ohat}
\end{equation}
with four coefficients
\begin{equation}
  \alpha = \frac{1}{\ALPHAINV} \approx \frac{1}{137.036},\qquad
  \gamma = \frac{3}{4},\qquad
  \zeta  = \alpha^{5/4},\qquad
  \beta  = \frac{3\pi}{20}.
  \label{eq:coeffs}
\end{equation}
The assembly-uniqueness question (P18-T2) decomposes into three sub-results.
SR2 is closed (P186: R8 forces additive composition).  SR3 addresses whether
$(\alpha,\gamma,\zeta,\beta)$ are the only spectrum-matching coefficients.
P191 showed SR3 is partially closed: $\alpha$ and $\beta$ are uniquely forced
by linear spectral conditions (closed); $\gamma = 3/4$ is geometrically forced
pending P10's five-fold convergence uniqueness (partially closed); $\zeta =
\alpha^{5/4}$ is derived from the dimensional formula and P184's Killing norm
confirmation, but the uniqueness of the exponent $5/4$ was recorded as an open
step.

\begin{assertion}[P191 open step for $\zeta$]
\label{ass:open-step}
P191 Open Problem~6(ii) records: \emph{``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.''}  This is the
remaining open step within SR3.  The present paper addresses it.
\end{assertion}

\subsection{What is established about $\zeta = \alpha^{5/4}$}

Three independent sources converge on $\zeta = \alpha^{5/4}$.

\begin{assertion}[P18 coefficient determination]
P18 Proposition~4.2 states: the renormalization-generator coefficient is
$\zeta = \kappa = \alpha^{5/4}$ where the oscillation scale $\kappa$ is
derived in Paper~16 from the dimensional analysis $5/4 = 1 + 3/(4\times 3)$.
\end{assertion}

\begin{assertion}[P184 formal bridge]
P184 Theorem~5.1 proves: $\zeta\calR|_{S^3,\mathrm{fibre}} = \zeta \cdot
\Hunit$, where $\Hunit$ is the Hopf right-$U(1)$ generator in the
$F_4$-centralising complement $\fz \subset \hE$.  The direction is uniquely
fixed by Hopf geometry; the coefficient $\zeta = \alpha^{5/4}$ is imported
from P18.  P184 Assertion~7.3 states explicitly: \emph{``The coefficient
$\zeta = \alpha^{5/4}$ is a positive scalar.  It rescales the generator
$\Hunit$ but does not change which direction in $\fz$ is selected.''}
\end{assertion}

\begin{assertion}[Dimensional formula]
\label{ass:dim-formula}
The exponent $5/4$ has the geometric decomposition:
\begin{equation}
  \frac{5}{4}
  \;=\; 1 + \frac{1}{\dim(B^4)}
  \;=\; 1 + \frac{\dim(S^1)}{\dim(B^4)}
  \;=\; \underbrace{1}_{\text{fibre coupling}}
  + \underbrace{\tfrac{1}{4}}_{\text{bulk reciprocal}},
  \label{eq:exponent-dim}
\end{equation}
encoding the fibre dimension $\dim(S^1) = 1$ and the reciprocal bulk
dimension $1/\dim(B^4) = 1/4$.  P18 records the equivalent form
$5/4 = 1 + 3/(4\times 3)$, where $3 = \dim(S^3)$ (boundary dimension).
Both are topological invariants of the Hopf triple
$S^1 \hookrightarrow S^3 \to S^2$.
\end{assertion}

The key gap identified by P191: \emph{are there other values of $p$ that are
equally consistent with the $E_6/F_4$ structure?}  Section~\ref{sec:e6}
shows that P184's argument pins the direction but not the magnitude, and the
dimensional formula is the genuine source of $p = 5/4$.

%==========================================================================
\section{Mass Formula as a Function of $p$}
\label{sec:mass-formula}
%==========================================================================

\subsection{The canonical within-family mass ratio}

From P18~\S5.4 (Mass Spectrum Conjecture) and P191~\S2.3, the within-family
mass ratio between levels $\ell_0$ and $\ell_1$ (same family index $k$) is
\begin{equation}
  m_\text{ratio}(p;\ell_0,\ell_1)
  \;=\; \exp\!\left(\frac{\mu_1}{\mu_0}\cdot\Delta E(p;\ell_0,\ell_1)\right),
  \label{eq:mass-ratio}
\end{equation}
where the energy gap is
\begin{equation}
  \Delta E(p;\ell_0,\ell_1)
  \;=\;
  \underbrace{\bigl[\ell_1(\ell_1+2) - \ell_0(\ell_0+2)\bigr]}_{\Delta E_{S^3}}
  + \underbrace{\alpha^p\cdot(\ell_1-\ell_0)}_{\zeta\text{ contribution}}
  + \underbrace{\beta\,\frac{\mu_{\ell_1}-\mu_{\ell_0}}{\mu_0}}_{\beta\text{ contribution}}.
  \label{eq:deltaE}
\end{equation}
The $\gamma T_{\rm cycle}$ contribution cancels in within-family ratios (P191~\S2.3,
Remark~3.4).

\begin{remark}[Constants]
$\alpha^{-1} = \mu_0 = 4\pi^3 + \pi^2 + \pi \approx 137.036$,
$\mu_1 = 16\pi^3/5 + 3\pi^2/4 + 2\pi/3 \approx 108.717$,
$\mu_1/\mu_0 \approx 0.7933$, $\beta = 3\pi/20 \approx 0.47124$.
The moments $\mu_n = 16\pi^3/(n+4) + 3\pi^2/(n+3) + 2\pi/(n+2)$ are
strictly decreasing (verified: Check~2e of \texttt{verify\_P196.py}).
\end{remark}

\subsection{Properties of $m_\text{ratio}(p;\ell_0,\ell_1)$}

\begin{proposition}[Monotonicity in $p$]
\label{prop:mono}
For every pair $(\ell_0,\ell_1)$ with $\ell_1 > \ell_0$, the function
$p \mapsto m_\text{ratio}(p;\ell_0,\ell_1)$ is strictly decreasing on
$(0,\infty)$.
\end{proposition}

\begin{proof}
Since $\alpha < 1$, the function $p \mapsto \alpha^p$ is strictly decreasing
and positive for all $p \in \RR$.  The term $\alpha^p(\ell_1-\ell_0) > 0$
is therefore strictly decreasing in $p$.  The remaining terms in
$\Delta E$ do not depend on $p$.  Hence $\Delta E$ is strictly decreasing in
$p$, and $m_\text{ratio} = \exp((\mu_1/\mu_0)\cdot\Delta E)$ is strictly
decreasing (Checks~5a--5c of \texttt{verify\_P196.py}).
\end{proof}

\begin{proposition}[Asymptotic limits]
\label{prop:limits}
For each pair $(\ell_0,\ell_1)$ with $\ell_1 > \ell_0$, define
$\Delta E_\infty = \Delta E_{S^3} + \beta(\mu_{\ell_1}-\mu_{\ell_0})/\mu_0$
(the limit as $p \to \infty$, $\zeta \to 0$) and
$\Delta E_0 = \Delta E_\infty + (\ell_1-\ell_0)$ (the limit as $p \to 0^+$,
$\zeta \to 1$).  Then:
\begin{equation}
  \lim_{p\to\infty} m_\text{ratio}(p;\ell_0,\ell_1)
  = \exp\!\left(\tfrac{\mu_1}{\mu_0}\cdot\Delta E_\infty\right)
  =: R_\infty(\ell_0,\ell_1),
  \label{eq:limit-inf}
\end{equation}
and $m_\text{ratio}$ approaches (but does not reach)
$R_0(\ell_0,\ell_1) := \exp((\mu_1/\mu_0)\cdot\Delta E_0)$
as $p \to 0^+$.  For $p \in (0,\infty)$:
\begin{equation}
  m_\text{ratio}(p;\ell_0,\ell_1) \;\in\;
  \bigl(R_\infty(\ell_0,\ell_1),\;R_0(\ell_0,\ell_1)\bigr).
  \label{eq:range}
\end{equation}
\end{proposition}

\begin{proof}
Immediate from Proposition~\ref{prop:mono} and the limits
$\lim_{p\to\infty}\alpha^p = 0$, $\lim_{p\to 0^+}\alpha^p = 1$.
\end{proof}

\subsection{Target for $m_\mu/m_e$}

The experimental ratio is $m_\mu/m_e \approx 206.77$ (CODATA 2018:
$m_\mu/m_e = 206.7682830$).  The required energy gap for this target is
\begin{equation}
  \Delta E_* \;=\; \frac{\ln(206.77)}{\mu_1/\mu_0}
  \;=\; \frac{5.332}{0.7933}
  \;\approx\; 6.722.
  \label{eq:target-dE}
\end{equation}

%==========================================================================
\section{Numerical Scan: $p$ Versus Mass Ratio}
\label{sec:scan}
%==========================================================================

\subsection{Scan table}

Table~\ref{tab:scan} shows $m_\text{ratio}(p;\ell_0,\ell_1)$ for three
representative pairs and eight values of $p$, computed at mp.dps\,=\,55.

\begin{table}[h]
\centering
\caption{Mass ratios $m_\text{ratio}(p;\ell_0,\ell_1)$ vs.\ exponent $p$.
Target: $206.77$.  The canonical value is $p = 5/4$ (bold row).
Pair $(1,2)$ is the ``nearest below'' pair; pair $(2,3)$ is the ``nearest
above'' pair.  The target falls in the structural gap between them.}
\label{tab:scan}
\begin{tabular}{@{}rrrrr@{}}
\toprule
$p$ & $\zeta = \alpha^p$ & ratio$(0,1)$ & ratio$(1,2)$ & ratio$(2,3)$ \\\midrule
$1$             & $7.297\times10^{-3}$ & $10.060$ & $50.497$ & $250.49$ \\
$\mathbf{5/4}$ & $\mathbf{2.133\times10^{-3}}$ & $\mathbf{10.019}$ & $\mathbf{50.291}$ & $\mathbf{249.46}$ \\
$3/2$           & $6.234\times10^{-4}$ & $10.007$ & $50.231$ & $249.17$ \\
$2$             & $5.325\times10^{-5}$ & $10.002$ & $50.208$ & $249.05$ \\
$5/2$           & $4.549\times10^{-6}$ & $10.002$ & $50.206$ & $249.04$ \\
$3$             & $3.886\times10^{-7}$ & $10.002$ & $50.206$ & $249.04$ \\
$4$             & $2.836\times10^{-9}$ & $10.002$ & $50.206$ & $249.04$ \\
$5$             & $2.069\times10^{-11}$ & $10.002$ & $50.206$ & $249.04$ \\
\midrule
\multicolumn{2}{l}{\emph{target}} & \multicolumn{3}{r}{$\longleftarrow 206.77 \longrightarrow$} \\
\bottomrule
\end{tabular}
\end{table}

\begin{remark}[Rapid convergence of $\zeta$ to zero]
\label{rem:rapid}
For $p \geq 2$, the contribution $\zeta = \alpha^p < 5.3\times10^{-5}$ is
negligible: the table entries stabilise to four-digit agreement with the
$p \to \infty$ limits.  The observable range of $m_\text{ratio}$ as $p$
varies over $(0,\infty)$ is determined almost entirely by the $S^3$
eigenvalue structure, not by the precise value of $p$.
\end{remark}

\subsection{Structural gap}

Table~\ref{tab:scan} reveals the central finding of this paper: no value of
$p \in \{1, \tfrac{5}{4}, \tfrac{3}{2}, 2, \tfrac{5}{2}, 3, 4, 5\}$, and
indeed no $p \geq 0$, produces $m_\text{ratio}(p;\ell_0,\ell_1) \approx 206.77$
for any of the three canonical pairs.  The column for pair $(2,3)$ is
consistently above $249$; the column for pair $(1,2)$ is consistently near
$50$.

\begin{proposition}[Range bounds]
\label{prop:ranges}
For $p \in (0,\infty)$:
\begin{enumerate}[label=(\alph*)]
  \item $m_\text{ratio}(p;1,2) \in (50.21,\; 111.0)$
    \quad (infimum at $p \to \infty$, supremum as $p \to 0^+$).
  \item $m_\text{ratio}(p;2,3) \in (249.0,\; \infty)$
    \quad (infimum at $p \to \infty$, no finite supremum).
  \item $m_\text{ratio}(p;0,1) \in (10.00,\; 22.1)$.
  \item $m_\text{ratio}(p;0,2) \in (502,\; \infty)$.
\end{enumerate}
In particular, $206.77 \in (111.0, 249.0)$ falls in the \emph{structural
gap} between the supremum of range~(a) and the infimum of range~(b).
No canonical pair covers this gap for any $p \geq 0$.
\end{proposition}

\begin{proof}
Direct computation from Proposition~\ref{prop:limits} using the values in
Table~\ref{tab:limits}.  Verified to 55-digit precision in
Checks~6a--6f of \texttt{verify\_P196.py}.
\end{proof}

\begin{table}[h]
\centering
\caption{Asymptotic limits $R_\infty$ and $R_0$ for canonical pairs.}
\label{tab:limits}
\begin{tabular}{@{}lrrrr@{}}
\toprule
Pair $(\ell_0,\ell_1)$ & $\Delta E_\infty$ & $\Delta E_0$ & $R_\infty$ ($p\to\infty$) & $R_0$ ($p\to 0^+$) \\\midrule
$(0,1)$ & $2.9026$ & $3.9026$ & $10.002$ & $22.11$ \\
$(1,2)$ & $4.9362$ & $5.9362$ & $50.21$  & $111.0$ \\
$(2,3)$ & $6.9549$ & $7.9549$ & $249.0$  & $550.6$ \\
$(0,2)$ & $7.8388$ & $9.8388$ & $502$    & $2449$  \\
\bottomrule
\end{tabular}
\end{table}

\subsection{Values of $p^*$ for which $m_\text{ratio} = 206.77$}

For each pair, we solve $m_\text{ratio}(p;\ell_0,\ell_1) = 206.77$ for $p$.
From equation~\eqref{eq:deltaE}: the required $\zeta^* = (\Delta E_* -
\Delta E_\infty)/(\ell_1-\ell_0)$ and $p^* = \log\zeta^*/\log\alpha$.

\begin{table}[h]
\centering
\caption{Values of $p^*$ satisfying $m_\text{ratio}(p^*;\ell_0,\ell_1)=206.77$.
All values are unphysical: either $p^* < 0$ (so $\zeta^* > 1$, violating
the perturbativity requirement $\zeta \ll 1$) or $\zeta^* < 0$ (impossible).}
\label{tab:pstar}
\begin{tabular}{@{}lrrrr@{}}
\toprule
Pair $(\ell_0,\ell_1)$ & $\Delta E_\infty$ & $\zeta^* = \Delta E_* - \Delta E_\infty$ & $p^*$ & Physical? \\\midrule
$(0,1)$ & $2.9026$ & $3.819$ & $-0.272$ & No ($\zeta^*>1$, $p^*<0$) \\
$(1,2)$ & $4.9362$ & $1.786$ & $-0.118$ & No ($\zeta^*>1$, $p^*<0$) \\
$(2,3)$ & $6.9549$ & $-0.233$ & --- & No ($\zeta^*<0$, impossible) \\
$(0,2)$ & $7.8388$ & $-1.117/2$ & --- & No ($\zeta^*<0$, impossible) \\
\bottomrule
\end{tabular}
\end{table}

\noindent For pairs $(0,1)$ and $(1,2)$: the solution exists but requires
$p^* < 0$, i.e.\ $\zeta^* = \alpha^{p^*} > 1$, which is unphysical
($\zeta \approx 2.1\times 10^{-3}$ in the physical assembly and must satisfy
$\zeta \ll 1$).
For pairs $(2,3)$ and $(0,2)$: the $S^3$ eigenvalue gap alone already
exceeds the target $\Delta E_* = 6.722$, so no real solution exists.
(Verified in Checks~7a--7h of \texttt{verify\_P196.py}.)

%==========================================================================
\section{$E_6$ Uniqueness Analysis: Does P184 Close It?}
\label{sec:e6}
%==========================================================================

\subsection{What P184 proves}

P184 Theorem~5.1 (the formal bridge theorem) establishes:
\begin{equation}
  \zeta\calR\big|_{S^3,\,\mathrm{fibre}}
  \;=\; \zeta \cdot \Hunit,
  \qquad \Hunit \in \fz \subset \hE.
  \label{eq:bridge}
\end{equation}
The proof chains: (i)~the holomorphic Euler decomposition of
$\calR = r\partial/\partial r + \Delta_\psi$ on $\CC^2 \supset B^4$ separates
the radial component (normal to $S^3$) from the Hopf fibre component
$\partial/\partial\psi$; (ii)~P176 Theorem~4.4 identifies $\partial/\partial\psi
= \Hunit$, the $F_4$-centralising Cartan generator with Killing norm squared
$|\Hunit|^2 = 24$ (P182: $24 = 2|\Phi_{G_2}|$).

\begin{assertion}[P184 direction uniqueness --- tight]
\label{ass:direction-tight}
The choice of $\Hunit$ \emph{as the direction} in $\fz$ is uniquely forced:
\begin{enumerate}[label=(\roman*)]
  \item The centraliser $\fz \subset \hE$ is 2-dimensional
    (P176 Proposition~4.1: $\mathrm{rank}(E_6) - \mathrm{rank}(F_4) = 2$).
  \item The Hopf fibration on $S^3$ is unique up to orientation.  The Hopf
    right-$U(1)$ action $(z_1,z_2) \mapsto (e^{i\theta}z_1, e^{i\theta}z_2)$
    commutes with the $F_4$ left-action on $\mathrm{Sp}(1) \cong S^3$, so its
    generator lies in $\fz$.
  \item The orthogonal direction $\Hunit^\perp = \partial/\partial\phi$ (the
    $S^2$ azimuthal rotation, P188) vanishes at the Hopf poles $\chi=0,\pi$
    (P189 Theorem~4.1: Hopf Regularity Axiom), excluding it from the assembly.
    $\Hunit = \partial/\partial\psi$ is nowhere-vanishing.
\end{enumerate}
Hence the direction $\Hunit \in \fz$ is uniquely forced by the Hopf
fibration geometry, independently of the magnitude $\zeta$.
\end{assertion}

\subsection{What P184 does not prove: the magnitude}

\begin{assertion}[P184 magnitude import]
\label{ass:magnitude-import}
P184 Assertion~7.3 states explicitly: \emph{``The particular value
$\zeta = \alpha^{5/4}$ is fixed by the P18 coefficient-determination theorem
(P18 Proposition~4.2) and encodes the renormalization group scale.''}  P184
does not derive $p = 5/4$ independently from the $E_6/F_4$ Killing norm.
The consistency check is:
\begin{equation}
  |\zeta \cdot \Hunit|^2 \;=\; \zeta^2 \times 24
  \;=\; \alpha^{5/2} \times 24,
  \label{eq:killing-norm}
\end{equation}
which is satisfied when $\zeta = \alpha^{5/4}$.  But equation~\eqref{eq:killing-norm}
does not independently constrain $p$: any value $\zeta = \alpha^p$ gives
$|\zeta\Hunit|^2 = \alpha^{2p} \times 24$, and the constraint merely reflects
that $p = 5/4$ once $\zeta$ is taken as $\alpha^{5/4}$ from P18.
\end{assertion}

\subsection{The Killing norm as consistency check vs.\ independent derivation}

To close the uniqueness gap, the Killing norm matching must impose an
\emph{external} constraint on $p$, not merely verify a substitution.  Three
approaches are available.

\paragraph{Approach A (Dual Coxeter ratio).}
The dual Coxeter ratio $h^\vee(E_6)/h^\vee(F_4) = 12/9 = 4/3$ is the
natural $E_6/F_4$ algebraic invariant.  Setting $\alpha^p = 4/3$ gives
$p = \ln(4/3)/\ln(\ALPHAINV) \approx 0.0585$, which is \emph{not} $5/4$.
The Coxeter ratio as a direct exponent source is excluded.

\paragraph{Approach B (Dimensional coupling).}
The renormalization generator $\calR$ acts on $B^4$ (dimension 4) and its
boundary restriction lives on the Hopf fibre $S^1$ (dimension 1).  The
natural coupling of $\calR$ to the fine-structure constant $\alpha$ is:
one power of $\alpha$ for the intrinsic scale of $\calR$ (a first-order
operator on $B^4$, contributing exponent $1$); one additional power
$\alpha^{1/\dim(B^4)} = \alpha^{1/4}$ for the boundary restriction
($B^4 \to S^3$ projection reduces the effective dimension by $\dim(B^4) = 4$,
contributing the reciprocal).  The total coupling is $\alpha^{1+1/4} =
\alpha^{5/4}$, giving $p = 5/4$.  This is the argument recorded in P16
and cited in P18 and P191.

\paragraph{Approach C (Hopf fibration degree).}
A formal proposition (the candidate Killing norm uniqueness theorem) would
state: the unique coupling $\zeta = \alpha^p$ such that $\zeta \cdot \Hunit$
carries Hopf fibration degree $\deg(\pi) = 1$ at the $\alpha$-scale is
$\zeta = \alpha^{5/4}$.  This route derives $p = 5/4$ from the Hopf
invariant and the bulk dimension without appealing to P16's dimensional
counting.

\begin{remark}[Status of the three approaches]
Approach~A is excluded (wrong prediction: $p \approx 0.06 \neq 5/4$).
Approach~B is the actual P16 derivation; it can be converted into a formal
proposition by specifying how ``fibre coupling exponent 1'' and ``bulk
reciprocal exponent $1/4$'' are computed from the Hopf geometry.
Approach~C is the algebraic route through the Hopf invariant; if made precise,
it would provide an independent algebraic derivation of $p = 5/4$ within the
$E_6/F_4$ framework.
\end{remark}

\subsection{Summary: what P184 closes and what remains}

\begin{table}[h]
\centering
\caption{Decomposition of the $\zeta$ uniqueness claim.}
\label{tab:decomp}
\begin{tabular}{@{}lll@{}}
\toprule
Component & Status & Evidence \\\midrule
Direction: $\Hunit \in \fz$ & \textbf{Closed} (P184) & Hopf geometry, P176 Thm~4.4 \\
Magnitude: $p = 5/4$ & \textbf{Partially closed} & P16 (dimensional), P18 \\
Formal uniqueness of $p$ & \textbf{Open step} & Requires formal proposition \\
\bottomrule
\end{tabular}
\end{table}

The gap is precise: P184 proves the direction uniquely; P16 derives the
magnitude via dimensional counting.  Full formal closure requires converting
P16's dimensional argument into a proposition derivable from the Hopf
fibration geometry alone, without ad hoc dimensional counting.  This is a
well-defined task (Approach~C above) requiring no new corpus papers but
only formalising the Hopf fibration degree argument.

%==========================================================================
\section{Theorem: SR3 Status for $\zeta$}
\label{sec:theorem}
%==========================================================================

\begin{theorem}[Structural exclusion --- unconditional]
\label{thm:exclusion}
For all $p \geq 0$ and all canonical level pairs $(\ell_0,\ell_1)$ with
$0 \leq \ell_0 < \ell_1 \leq 4$:
\begin{equation}
  m_\text{ratio}(p;\ell_0,\ell_1) \;\neq\; 206.77.
  \label{eq:exclusion}
\end{equation}
Equivalently, the mass formula gap is \emph{independent of the exponent $p$}
and cannot be closed by varying $p$ within the physical range $p > 0$.
\end{theorem}

\begin{proof}
We partition the canonical level pairs by the sign of
$\Delta E_\infty - \Delta E_*$, where $\Delta E_* = 6.722$
(equation~\eqref{eq:target-dE}).

\textbf{Below pairs} ($\Delta E_\infty < \Delta E_*$): pairs $(0,1)$ and
$(1,2)$.

The range of $m_\text{ratio}(p;\ell_0,\ell_1)$ over $p \in (0,\infty)$
is $(R_\infty, R_0)$ (Proposition~\ref{prop:ranges}).  We verify
$R_0 < 206.77$ (the supremum, approached as $p \to 0^+$, is below target):
\begin{align}
  R_0(1,2) &= \exp\!\bigl(0.7933 \times (4.936 + 1)\bigr)
  = \exp(4.706) \approx 111.0 < 206.77, \label{eq:R0-12} \\
  R_0(0,1) &= \exp\!\bigl(0.7933 \times (2.903 + 1)\bigr)
  = \exp(3.096) \approx 22.1 < 206.77. \label{eq:R0-01}
\end{align}
Hence $m_\text{ratio}(p;\ell_0,\ell_1) < 206.77$ for all $p \in (0,\infty)$
for these pairs.

\textbf{Above pairs} ($\Delta E_\infty \geq \Delta E_*$): pairs $(2,3)$,
$(0,2)$, and all higher pairs.

For these pairs $R_\infty \geq 249.0 > 206.77$ and $m_\text{ratio}$ is
decreasing in $p$, so $m_\text{ratio}(p;\ell_0,\ell_1) \geq R_\infty > 206.77$
for all $p \in (0,\infty)$.

\textbf{At $p = 0$}: the boundary value $\zeta = \alpha^0 = 1$ is unphysical,
but for completeness: below pairs remain below $206.77$
(equations~\eqref{eq:R0-12}--\eqref{eq:R0-01}); above pairs are even larger.

\textbf{Gap location}: The range of $m_\text{ratio}$ over all canonical pairs
and all $p \geq 0$ contains neither $206.77$ nor any value in the interval
$(111.0, 249.0)$.  The target $206.77 \in (111.0, 249.0)$ is structurally
excluded.

Numerical verification: Check~8a of \texttt{verify\_P196.py} performs an
exhaustive scan over all pairs $0 \leq \ell_0 < \ell_1 \leq 4$ and
$p \in \{0, 0.25, 0.50, \ldots, 10.0\}$ (41 $p$-values) and confirms no
pair comes within $1.0$ of the target.
\end{proof}

\begin{corollary}[Mass formula gap is orthogonal to $\zeta$ exponent question]
\label{cor:orthogonal}
The gap between the computed ratio $\approx 50$ (pair $(1,2)$, canonical $p$)
and the experimental ratio $\approx 206.77$ cannot be closed by any choice
of $p$.  Resolving the $\zeta$ exponent uniqueness question (closing SR3 for
$\zeta$) would not repair the mass formula gap.  The two questions are
orthogonal.
\end{corollary}

\begin{theorem}[Conditional uniqueness of $p = 5/4$]
\label{thm:uniqueness}
Suppose the following formal proposition holds:

\begin{proposition}[Hopf degree coupling]
\label{prop:hopf-degree}
The unique coupling $\zeta = \alpha^p$ such that $\zeta\calR$, when
restricted to the Hopf fibre $S^1 \subset S^3 = \partial B^4$, carries
coupling exponent equal to $\dim(S^1) + 1/\dim(B^4) = 1 + 1/4 = 5/4$
is $p = 5/4$.
\end{proposition}

Then $p = 5/4$ is the unique exponent consistent with the $E_6/F_4$ Killing
norm and the Hopf fibration data, and SR3 is fully closed for $\zeta$.
\end{theorem}

\begin{proof}
If Proposition~\ref{prop:hopf-degree} holds, then $p = 5/4$ is the unique
solution of the Hopf degree constraint.  Combined with the P184 uniqueness of
the direction (Assertion~\ref{ass:direction-tight}), the complete operator
$\zeta\calR = \alpha^{5/4}\calR$ is uniquely determined.  The P18-T3 and P18-T4
spectral conditions fix $\alpha$ and $\beta$ (P191 Theorems~4.1--4.2); the
geometric dimension ratio fixes $\gamma$ (P191 Theorem~4.3).  All four
coefficients being uniquely determined closes SR3.
\end{proof}

\begin{remark}[Status of Proposition~\ref{prop:hopf-degree}]
Proposition~\ref{prop:hopf-degree} is not yet proved in the corpus.  It is
the formalised version of P16's dimensional counting argument
($p = 1 + \dim(S^1)/\dim(B^4)$, which P18 records as the source of
$\zeta = \alpha^{5/4}$).  The proposition requires specifying: (a)~the
``coupling exponent'' of a differential operator on $B^4$ with respect to
$\alpha$, in the sense of the $E_6/F_4$-graded Sobolev scale on $B^4$;
(b)~showing that $\dim(S^1) + 1/\dim(B^4)$ is the unique exponent consistent
with the Hopf fibration degree $\deg(\pi)=1$.  This is a well-defined
mathematical task requiring no new corpus papers.
\end{remark}

\subsection{SR3 status update}

\begin{table}[h]
\centering
\caption{SR3 coefficient uniqueness status after P196.}
\label{tab:status}
\begin{tabular}{@{}lllll@{}}
\toprule
Coeff. & Value & Prior status (P191) & P196 contribution & Updated status \\\midrule
$\alpha$ & $1/\alpha^{-1}$ & Closed & Confirmed & \textbf{Closed} \\
$\beta$  & $3\pi/20$        & Closed & Confirmed & \textbf{Closed} \\
$\gamma$ & $3/4$            & Partially closed & Not addressed & Partially closed \\
$\zeta$  & $\alpha^{5/4}$   & Partially closed & Direction closed; & Partially closed \\
         &                  &                  & magnitude gap characterised & \\
\bottomrule
\end{tabular}
\end{table}

The key advance of P196 relative to P191 for $\zeta$: P196 confirms that P184
\emph{tightly closes} the direction sub-question (no gap there, Assertion~\ref{ass:direction-tight}),
characterises the magnitude gap precisely (requires formalising the Hopf degree coupling,
Proposition~\ref{prop:hopf-degree}), and proves unconditionally that the mass
formula gap is independent of $p$ (Theorem~\ref{thm:exclusion}).

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

The remaining open step in SR3 is the formal uniqueness of the exponent
$p = 5/4$ in $\zeta = \alpha^p$.  This paper has:

\begin{enumerate}[label=(\roman*)]
  \item Written the mass formula as an explicit function of $p$
    (equations~\eqref{eq:mass-ratio}--\eqref{eq:deltaE}) and proved it is
    monotone decreasing in $p$ for every canonical level pair
    (Proposition~\ref{prop:mono}).

  \item Performed a numerical scan over $p \in \{1, \tfrac{5}{4}, \tfrac{3}{2},
    2, \tfrac{5}{2}, 3, 4, 5\}$ and canonical pairs (Table~\ref{tab:scan}),
    revealing a \emph{structural gap}: the target $m_\mu/m_e \approx 206.77$
    falls in $(111.0, 249.0)$, not covered by any canonical pair for $p \geq 0$.

  \item Proved \textbf{unconditionally} (Theorem~\ref{thm:exclusion}) that
    no exponent $p \geq 0$ produces $m_\text{ratio} \approx 206.77$:
    the mass formula gap is independent of the exponent uniqueness question.
    Corollary~\ref{cor:orthogonal}: the two questions are orthogonal.

  \item Assessed P184's $E_6/F_4$ argument with precision: the
    \emph{direction} ($\Hunit \in \fz$) is tightly closed by Hopf geometry;
    the \emph{magnitude} ($p = 5/4$) is supported by P16's dimensional
    argument but requires formalisation as a Hopf degree coupling proposition
    (the dual Coxeter ratio route gives $p \approx 0.06$, not $5/4$, and is excluded).

  \item Stated \textbf{conditionally} (Theorem~\ref{thm:uniqueness}) that
    if Proposition~\ref{prop:hopf-degree} holds, then $p = 5/4$ is unique
    and SR3 is fully closed for $\zeta$.
\end{enumerate}

\noindent\textbf{What is unconditional.}  Theorem~\ref{thm:exclusion} and
Corollary~\ref{cor:orthogonal} do not depend on any open step.

\noindent\textbf{What is conditional.}  Theorem~\ref{thm:uniqueness} depends
on Proposition~\ref{prop:hopf-degree}, not yet proved in the corpus.  It is
a formalisation of P16's dimensional counting, requiring the notion of
``coupling exponent'' in the $E_6/F_4$-graded Sobolev scale on $B^4$.

\noindent\textbf{SR3 summary.}  Three coefficients are closed or fully pinned:
$\alpha$ (closed), $\beta$ (closed), $\gamma$ (pending P10 five-fold
convergence theorem).  The coefficient $\zeta$ is partially closed: the
direction ($\Hunit$) is tightly closed by P184, and the magnitude ($p = 5/4$)
awaits the Hopf degree coupling proposition.  The two remaining open steps in
SR3 are of comparable tractability — both require formalising an existing
geometric argument, not introducing new machinery.

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

All numerical claims are verified by
\texttt{Lumen/corpus/addenda/verify/verify\_P196.py}
(mp.dps\,=\,55; 39/39 checks pass, exit~0).

Key verified values:
\begin{itemize}
  \item Check~4: canonical ratios at $p=5/4$: pair $(0,1)=10.019$,
    pair $(1,2)=50.291$, pair $(2,3)=249.464$.
  \item Check~6: range bounds: pair $(1,2) \in (50.21, 111.0)$,
    pair $(2,3) \in (249.0, \infty)$, gap $(111.0, 249.0)$ contains $206.77$.
  \item Check~7: $p^*(0,1) = -0.272$, $p^*(1,2) = -0.118$ (both negative);
    pairs $(2,3)$, $(0,2)$, $(1,3)$ have no real $p^*$.
  \item Check~8: exhaustive scan, 41 $p$-values $\times$ 10 pairs, no hit
    within $1.0$ of target.
  \item Check~9: $|\Hunit|^2 = 24$ (P182); $|\zeta\Hunit|^2 = \alpha^{5/2}\times 24$.
  \item Check~10: $p_\text{dim} = 1 + \dim(S^1)/\dim(B^4) = 5/4$ (all three forms).
\end{itemize}

%==========================================================================
\begin{thebibliography}{99}
%==========================================================================

\bibitem{P16}
L.~F.\ Vlegels, \emph{Paper~16: Renormalization Scale and the Oscillation
Parameter}, TOE Corpus, 2024.

\bibitem{P18}
L.~F.\ Vlegels, \emph{Paper~18: The Master Operator $\hat{O}$},
TOE Corpus, 2025.

\bibitem{P176}
L.~F.\ Vlegels, \emph{Addendum~176: Perturb as the $U(1)$ Cartan Generator
in $E_6 \to F_4 \times U(1)$}, TOE Corpus, May 2026.

\bibitem{P182}
L.~F.\ Vlegels, \emph{Addendum~182: Root-Complement Doubling
$24 = 2|\Phi_{G_2}|$}, TOE Corpus, May 2026.

\bibitem{P184}
L.~F.\ Vlegels, \emph{Addendum~184: OP-H Formal Derivation ---
$\mathrm{Perturb} = H_{U(1)}$ via the Hopf Fibre Restriction},
TOE Corpus, May 2026.

\bibitem{P186}
L.~F.\ Vlegels, \emph{Addendum~186: SR2 Closure --- R8 Forces Additive
Composition}, TOE Corpus, May 2026.

\bibitem{P188}
L.~F.\ Vlegels, \emph{Addendum~188: The Orthogonal Direction
$H_{U(1)}^\perp$}, TOE Corpus, May 2026.

\bibitem{P189}
L.~F.\ Vlegels, \emph{Addendum~189: DOF-3 Closed --- Dilatation Exclusion
and Hopf Regularity Axiom}, TOE Corpus, May 2026.

\bibitem{P191}
L.~F.\ Vlegels, \emph{Addendum~191: SR3 --- Coefficient Uniqueness for the
Canonical $\hat{O}$ Assembly}, TOE Corpus, May 2026.

\end{thebibliography}

\end{document}
