% Addendum 194: Mass Identification Rule Alternatives (Route B of P18-Conj1)
% Seven rules tested; six closed; heat-kernel near-miss λ₅^{3/2}≈207.06 (0.14%).
%
% VERIFY: 18/18 checks
%  (1) λ_5=5×7=35  (2) λ_24=24×26=624  (3) α⁻¹≈137.036  (4) T•≈430.512
%  (5) ω₀≈7.752   (6) R*≈206.768  (7) ρ_ℓ=1−(ℓ+1)^{−2} (algebraic)
%  (8) sup ρ_ℓ/ρ_1→4/3  (9) 624/3=208, gap 0.596%  (10) 35^{3/2}≈207.063
%  (11) gap (207.063−206.768)/206.768=0.143%  (12) vol(S³)=2π², K(t)~√π/4·t^{-3/2}
%  (13) 3α⁻¹/2=205.554, gap 0.587%  (14) ζ(s) converges iff s>3/2
%  (15) window l_min=2 (λ_2=8>7.752), l_max=19 (λ_19=399≤430.5)
%  (16) λ_19=399, λ_20=440>T•  (17) λ_1=3<ω₀  (18) (R*)^{2/3}≈34.97≈λ_5=35
%
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

\documentclass[11pt,a4paper]{article}
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\usepackage{booktabs}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue]{hyperref}

\newtheorem{proposition}{Proposition}[section]
\theoremstyle{remark}
\newtheorem{remark}[proposition]{Remark}

\newcommand{\Dop}{\Delta_{S^3}}
\newcommand{\ainv}{\alpha^{-1}}
\newcommand{\Tbullet}{T_{\!\bullet}}

\title{Addendum 194: Mass Identification Rule Alternatives}
\author{L\'eon Fernando Vlegels}
\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
Paper~18 maps $m_e\leftrightarrow\lambda_1=3$ and requires $\lambda_\ell\approx620.3$
for the muon; P191 proved no integer solution exists (nearest miss $\lambda_{24}=624$,
gap $0.6\%$). We test seven alternative identification rules against the $S^3$ spectrum
and TOE constants. Six rules are definitively closed. One near-miss survives: the fifth
eigenvalue $\lambda_5=35$ satisfies $\lambda_5^{3/2}=35\sqrt{35}\approx207.063$,
within $0.143\%$ of $m_\mu/m_e\approx206.768$. P18-Conj1 is left open with a
single spectrally motivated candidate requiring $t_\mu/t_e=\lambda_5$.
\end{abstract}

%------------------------------------------------------------------
\section{Setup}

$S^3$ spectrum: $\lambda_\ell=\ell(\ell+2)$, $d_\ell=(\ell+1)^2$. Key values:
$\lambda_1{=}3$, $\lambda_2{=}8$, $\lambda_5{=}35$, $\lambda_{19}{=}399$,
$\lambda_{24}{=}624$. TOE constants (frozen in \texttt{kernel/math/quat\_s3.py}):
\begin{equation}
  \ainv=4\pi^3{+}\pi^2{+}\pi\approx137.036,\quad
  \Tbullet=\pi\ainv\approx430.51,\quad
  \omega_0=\tfrac{\pi^3}{4}\approx7.752.
\end{equation}
Target: $R^*=m_\mu/m_e=105.6584/0.51100\approx206.768$.

%------------------------------------------------------------------
\section{Rule~1 — Density-Weighted Eigenvalues}

\begin{proposition}
$\rho_\ell=\lambda_\ell/d_\ell=\ell(\ell+2)/(\ell+1)^2=1-(\ell+1)^{-2}$,
so $\rho_\ell\in(0,1)$ strictly. The supremum of any ratio is
$\sup_{\ell>\ell'}\rho_\ell/\rho_{\ell'}<1/\rho_1=4/3\approx1.333$.
\end{proposition}

Representative values: $\rho_1=3/4$, $\rho_2=8/9$, $\rho_{24}=624/625$.
All density-weighted ratios are bounded by $4/3$; target $R^*\approx206.77$
is structurally inaccessible. \textbf{CLOSED.}

%------------------------------------------------------------------
\section{Rule~2 — Heat Kernel at Natural Times}

The heat trace on $S^3$ and its small-$t$ asymptotic
(unit sphere, $\mathrm{vol}=2\pi^2$, curvature coefficient $a_1=1$):
\begin{equation}
  K(t)=\!\sum_{\ell\ge0}(\ell{+}1)^2 e^{-\lambda_\ell t},\qquad
  K(t)\sim\frac{\sqrt\pi}{4}\,t^{-3/2}(1+t+O(t^2)),\quad t\to0^+.
  \label{eq:hk}
\end{equation}
With electron/muon times $t_e$ and $t_\mu=r\,t_e$, the leading-order ratio is
\begin{equation}
  \frac{K(t_e)}{K(t_\mu)}\approx r^{3/2},\qquad
  r^{3/2}=R^* \;\Longrightarrow\; r=(R^*)^{2/3}\approx34.97.
\end{equation}

\begin{proposition}
\label{prop:hk}
$\lambda_5^{3/2}=35\sqrt{35}\approx207.063$, gap
$(207.063-206.768)/206.768=0.143\%$. No other $S^3$ eigenvalue
$\lambda_\ell^{3/2}$ lies within $1\%$ of $R^*$
($\lambda_4^{3/2}\approx117.6$, gap $43\%$; $\lambda_6^{3/2}\approx332.6$, gap $61\%$).
\end{proposition}

The identification hypothesis is $t_\mu/t_e=\lambda_5=35$.
$(R^*)^{2/3}\approx34.97$: the fifth eigenvalue is the unique integer-valued
spectral value satisfying this.

\begin{remark}[Sub-leading correction]
At $t_e=1/\Tbullet\approx0.00232$ the first-order correction factor
$(1+t_e)/(1+35t_e)\approx0.927$ gives $207.06\times0.927\approx191.9$
(gap $\approx7.2\%$). The leading-order near-miss is a ratio property that holds
independently of the absolute scale $t_e$; the sub-leading behaviour
requires a geometric pinning of $t_e$.
\end{remark}

\textbf{PROMISING.} Near-miss $\lambda_5^{3/2}\approx207.06$, gap $0.14\%$.

%------------------------------------------------------------------
\section{Rule~3 — Spectral Zeta Ratios}

$\zeta(s)=\sum_{\ell\ge1}(\ell+1)^2[\ell(\ell+2)]^{-s}$ converges iff $s>3/2$
(terms $\sim\ell^{2-2s}$). Pole at $s=3/2$ with residue $(2\sqrt\pi)^{-1}$.
At $s=2$: partial sum to $\ell=200$ gives $\zeta(2)\approx0.880$; all convergent
values are $O(1)$. Residue ratios at the pole are trivially~$1$.
Individual-term tuning to hit $R^*$ introduces the same unjustified free parameter
as Rule~2. \textbf{CLOSED.}

%------------------------------------------------------------------
\section{Rule~4 — GON Lapse}

The GON lapse $m=|\sin2\beta|$ gives $m_\mu/m_e=\sin2\beta_\mu/\sin2\beta_e$
only if $\sin2\beta_e\le1/R^*\approx4.84\times10^{-3}$.
The smallest TOE-natural angle $\pi/(\ainv)^2\approx1.67\times10^{-4}$~rad
gives $\sin(2\beta_e)\approx3.35\times10^{-4}$, requiring $\beta_\mu\approx\pi/90.8$;
the denominator $90.8\approx\ainv/1.51$ has no canonical TOE derivation.
For canonical angles $\beta\in\{\pi/6,\pi/4,\pi/3\}$ the maximum lapse ratio
is $2/\sqrt3\approx1.155$, far below $R^*$. \textbf{CLOSED.}

%------------------------------------------------------------------
\section{Rule~5 — Moment Hierarchy}

The $n$-th regulated moment $\mu_n(t)=(-\partial_t)^n K(t)$ satisfies
$\mu_1(t)/\mu_0(t)\sim3/(2t)$ as $t\to0^+$.
Ratios of $\mu_0$ at two times reduce to $(t_\mu/t_e)^{3/2}$ — exactly Rule~2.
No new spectral information is accessed. \textbf{CLOSED (reduces to \S3).}

%------------------------------------------------------------------
\section{Rule~6 — Windowed Spectrum}

Window $W=[\omega_0,\Tbullet]\approx[7.752,\,430.5]$ contains levels
$\ell=2,\ldots,19$ ($\lambda_2=8>\omega_0$; $\lambda_{19}=399\le\Tbullet$;
$\lambda_{20}=440>\Tbullet$; $\lambda_1=3<\omega_0$). Eighteen levels total.
Multiplicity-weighted sum: $S_W=\sum_{\ell=2}^{19}d_\ell\lambda_\ell=719\,784$.
Ratio $S_W/(d_1\lambda_1)\approx59\,982$: three orders of magnitude above $R^*$.
By the $S^3$ Weyl law ($N(\lambda)\sim\lambda^{3/2}$) window sum ratios grow
polynomially and cannot produce $O(200)$ values between adjacent windows.
\textbf{CLOSED.}

%------------------------------------------------------------------
\section{Rule~7 — Casimir-Type Combinations}

\noindent Searching for $\alpha_1\lambda_{\ell_1}+\alpha_2\lambda_{\ell_2}
= R^*(\alpha_1\lambda_{\ell_3}+\alpha_2\lambda_{\ell_4})$ with TOE-natural $\alpha_i$:
\begin{align}
  \lambda_{24}/\lambda_1 &= 624/3 = 208.000, &
  \text{gap} &= (208.000-206.768)/206.768 = 0.596\%. \\
  \tfrac{3}{2}\ainv &= \tfrac{3}{2}{\times}137.036 = 205.554, &
  \text{gap} &= (206.768-205.554)/206.768 = 0.587\%.
\end{align}
Both near-misses are at $\sim0.6\%$, inferior to the $0.14\%$ of \S3.
The constant $\omega_0\approx7.752$ is not a $S^3$ eigenvalue, so
$\omega_0$-weighted combinations produce no exact spectral expression.
\textbf{CLOSED (near-misses at $0.6\%$).}

%------------------------------------------------------------------
\section{Summary Table}

\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{@{}lllr@{}}
\toprule
\textbf{Rule} & \textbf{Result} & \textbf{Best value} & \textbf{Gap} \\
\midrule
\S2 Density-weighted   & CLOSED    & $\sup\le4/3$              & $>100$ \\
\S3 Heat kernel        & PROMISING & $\lambda_5^{3/2}=207.063$ & $0.143\%$ \\
\S4 Spectral zeta      & CLOSED    & $\zeta$-ratios $O(1)$     & $>100$ \\
\S5 GON lapse          & CLOSED    & No natural $\beta$-pair   & $>100$ \\
\S6 Moment hierarchy   & CLOSED    & Reduces to \S3            & --- \\
\S7 Windowed spectrum  & CLOSED    & Scale mismatch $O(10^4)$  & $>100$ \\
\S8 Casimir combns.    & CLOSED    & $\lambda_{24}/\lambda_1=208$ & $0.596\%$ \\
\bottomrule
\end{tabular}
\end{center}

\medskip
\noindent\textbf{Key numerical identity.}
$(R^*)^{2/3}=206.768^{2/3}\approx34.97\approx\lambda_5=35$; gap $0.09\%$.
By Proposition~\ref{prop:hk}, $\lambda_5$ is the unique $S^3$ eigenvalue
whose $\tfrac{3}{2}$-power matches $R^*$ to better than $1\%$.

%------------------------------------------------------------------
\section{Conclusion}

Six rules are definitively closed: density-weighted ratios are bounded by $4/3$,
spectral-zeta global values are $O(1)$, GON lapse values require
unphysical pole proximity, windowed sums have $O(10^4)$ scale mismatch, moment
hierarchy reduces to heat kernel, and Casimir near-misses fall at $0.6\%$.

\textbf{Open with single candidate.}
The heat-kernel asymptotic scaling $K(t_e)/K(t_\mu)\approx(t_\mu/t_e)^{3/2}$
matches $R^*$ when $t_\mu/t_e=\lambda_5=35$, yielding $35^{3/2}=207.06$
(gap $0.14\%$). This is the closest approach found across all seven routes.
The candidate requires a geometric argument for why the muon heat-kernel time
exceeds the electron time by $\lambda_5$; no such argument is currently available.

\textbf{Constraint on P18-Conj1.}
Any valid Route-B identification rule must produce the factor $\lambda_5^{3/2}$
or account for the $0.14\%$ residual. The problem reduces to:
\emph{derive $t_\mu/t_e=\lambda_5$ from the TOE geometry.}

\medskip
\noindent\textit{Next:} Addendum~195 investigates whether renormalisation-group
running from the $\hat{\mathcal{O}}$ scale to the physical scale accounts for
the $0.14\%$ residual gap.

\end{document}
