% 296_Addendum_IndexMatching.tex
% The open hearts, compressed further. (i) The selection rule
% reformulated EXACTLY: family n draws the coefficient of x^n in rho —
% index matching between level number and monomial degree; the lepton
% families are the Taylor coefficients of the density at the
% unobservable center. (ii) Third mechanism exclusion: the bare
% monomial probe fails (scale ~35x off, n-dependence non-uniform).
% (iii) The ten: 3pi/20 = pi*d_edge/(d_bnd*d_bulk) exactly — the
% constant is built from the layers' first-moment denominators;
% structured flag, not derivation. Both OIs remain open in final
% compressed form. Verifier: verify_P296.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[margin=1in]{geometry}
\usepackage{hyperref}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}

\title{Addendum 296: Index Matching\\
\large The selection rule in final form; a structured flag\\
for the integer ten}
\author{TOE Corpus}
\date{2026-06-11}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B (exact reformulation $+$ third exclusion $+$
structured flag; both OIs open, compressed) \quad|\quad
\textbf{Addresses:} OI-287-1, OI-282-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P296.py} --- all PASS

\begin{abstract}
\emph{(i) The selection rule, reformulated exactly.} The lepton law's
coefficients $(c_e, c_\mu, c_\tau) = (2, 3, 16)$ are the coefficients
of $x^1, x^2, x^3$ in $\rho(x)/\pi^{\deg}$: family $n$ draws precisely
the degree-$n$ monomial of the density. The level-to-layer assignment
of A287 is therefore \emph{index matching}, level number equal to
monomial degree, and since the monomial coefficients are the Taylor
data of $\rho$ at $x = 0$, the rule reads: \textbf{the charged-lepton
families are the Taylor coefficients of the density at the
unobservable center}, the point where observation vanishes (A281).
OI-287-1's final form: derive why level $n$ reads degree $n$.
\emph{(ii) Third mechanism exclusion.} The bare Taylor-probe coupling
(perturbation $\kappa\,c_n\pi^n x^n$ acting diagonally on level $n$)
fails: it underestimates the required log-eigenvalue shifts by
factors $32$ and $42$ at $n = 2, 3$, and the two factors differ by
$28\%$, so no overall rescaling repairs it. With A292 (universal
first-order) and A295 (dressing, wrong $\kappa$-order), three
mechanism classes are now closed; what survives must couple level $n$
to degree $n$ at first order in $\kappa$ with the specific strength
$\kappa\,\Omega/\lambda_1$.
\emph{(iii) The ten, structurally flagged.} The hierarchy constant
decomposes as
\[
\frac{3\pi}{20} \;=\; \pi\cdot\frac{d_{\rm edge}}{d_{\rm bnd}\,
d_{\rm blk}}\,,
\qquad (d_{\rm edge}, d_{\rm bnd}, d_{\rm blk}) = (3, 4, 5)\,,
\]
where $d_\ell = k_\ell + 2$ are exactly the denominators the three
layers contribute to the first moment $\mu_1 = \sum_\ell
c_\ell\pi^{k_\ell}/(k_\ell{+}2)$. Equivalently $10 = d_{\rm bnd}
d_{\rm blk}/2$. The constant of the twenty-two-decade formula is
assembled from the same layer data as everything else in the
framework, with the edge's denominator in the numerator and the
interior denominators below; the structure is exact arithmetic and
is recorded as a flag per the promotion convention, since no
derivation produces it. (The Pythagorean triple $(3,4,5)$ formed by
the three denominators is noted without weight.) OI-282-1's final
form: derive the moment-denominator combination, or the integer ten,
from the flow; either suffices.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{Index Matching}

\begin{theorem}[The selection rule is degree matching]
Write $\rho(x) = \sum_{k=1}^{3} c_k \pi^k x^k$ with
$(c_1, c_2, c_3) = (2, 3, 16)$. The lepton law's coefficients satisfy
$c_n = c_{k=n}$: the anchor (electron, $n=1$) sits at degree one, the
muon ($n=2$) draws the degree-two coefficient, the tau ($n=3$) the
degree-three coefficient. Exact by inspection; the content is the
reformulation, which replaces the two-part assignment (level
$\to$ layer, layer $\to$ coefficient) by a single rule (level $n$
$\to$ degree $n$) and relocates the data to the Taylor expansion at
the origin: $c_k = \rho^{(k)}(0)/(\pi^k\, k!)$. The origin is the
center, where observation terminates; the families are, in this
reading, the structure of the world at the point no measurement
reaches, made visible as masses.
\end{theorem}

\begin{proposition}[Third exclusion]
The minimal mechanism suggested by the reformulation, a diagonal
coupling of level $n$ to its own monomial at strength
$\kappa c_n \pi^n$, gives
$\delta\ln\lambda_n = -\kappa c_n\pi^n\langle x^n\rangle_n/\lambda_n
= -0.00039$ ($n{=}2$), $-0.0026$ ($n{=}3$), against required
$-0.0126$ and $-0.1084$: short by factors $32$ and $42$, which
moreover disagree with each other by $28\%$. Excluded both on scale
and on shape. Three mechanism classes are now closed (A292, A295,
this); the surviving requirement is stated in the abstract.
\end{proposition}

% ────────────────────────────────────────────────────────────────────────────
\section{The Ten}

\begin{proposition}[Moment-denominator decomposition]
$3\pi/20 = \pi\cdot 3/(4\cdot5)$ exactly, with $(3,4,5)$ the
first-moment denominators of the edge, boundary, and bulk terms
respectively. Hence the hierarchy formula reads
$\ln(M_{\rm Pl}/m_e) = \pi\,\frac{d_e}{d_b d_B}\,\mu_1(1+\mu_1\alpha^2)$:
every constant in it is layer data. Recorded as a structured flag;
a derivation must still produce the combination. The previous
candidates (metric components, $2\times5$, $\binom52$) remain listed
in A285; this decomposition supersedes them as the sharpest, being
built from objects the formula already contains.
\end{proposition}

\paragraph{Status.} OI-287-1 and OI-282-1 remain open. Both are now
in single-sentence form: \emph{why does level $n$ read degree $n$},
and \emph{why does the flow weigh $\mu_1$ by
$\pi d_e/(d_b d_B)$}. The package publishes with these as its stated
open hearts; nothing in this addendum changes a published number.

\paragraph{Verifier.} \texttt{verify\_P296.py}: the coefficient
identification; the Taylor reading; the probe-mechanism exclusion
numerics (re-run); the decomposition arithmetic and its uniqueness
among small denominator combinations; the (3,4,5) note. All PASS.

\end{document}
