% 326_Addendum_FamilyDegreeBridge.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1 (corrected). A325 located the standing residual as ONE unstated corpus
% bridge with a unique structural reconciler char = (degree - 1) mod 3 under the
% A304-forced forward generator, and flagged three things still in play: the
% fiber-vs-edge identification (P18 names the third layer "fiber"; A287/kernel name
% it "edge"), the electron=k=0=edge anchor, and the projection rule selecting which
% coefficient a layer-spread eigenspace draws. This addendum attempts to make the
% bridge airtight. It RESOLVES the first two flags and the handedness sign as corpus-
% forced -- the fiber IS the edge IS the degree-1 S^1 layer (P05 line 156, line 244,
% section title; P18 line 127/575); the anchor k=0<->edge<->deg1 is forced (P20
% trivial rep + rho(0)=0 + edge=ground); the relative twist is a fixed -1 (both
% actions inherit the one A304 forward rotation). It does NOT close the third: the
% family eigenspaces are Z_3 Fourier combinations spread equally over all three
% layers, so "family k draws a single degree" is a PROJECTION choice on a spread
% state, and the corpus never derives that the projection is the character-resonant
% diagonal (the route A287 flagged; the withdrawn overlap-diagonal of A299-A320 tried
% to compute it and overshot). The bridge therefore reduces to a SINGLE irreducible
% posited premise G1; given G1 the full bridge k=(degree-1) mod 3 is forced with zero
% residual and reproduces (e,mu,tau)<->(2,3,16) end-to-end through the A287 law.
% Verdict PARTIAL (posited pin, sharpened to one premise; two of A325's three open
% items now corpus-forced). The overlap-diagonal stays withdrawn; no published number
% changes. Verifier: verify_P326.py.

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

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

\title{Addendum 326: The Family--Degree Bridge\\
\large One posited premise joins the eigenvalue labelling to the coefficient it draws}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} PARTIAL (the bridge is posited as one irreducible premise; two of
A325's three open items are now corpus-forced) \quad|\quad
\textbf{Addresses:} OI-287-1 (corrected); A325; A287; P20; A304 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P326.py} --- all PASS

\begin{abstract}
A325 located the standing residual of OI-287-1 as a single unstated corpus bridge:
P20 indexes a charged-lepton family by its $T_{\rm cycle}$ eigenvalue exponent $k$,
A287 indexes it by the layer-degree of the coefficient it draws, and the two read
through the grading $\mathrm{char}=\mathrm{degree}\bmod 3$ disagree by a uniform $+1$.
A325 found a unique structural reconciler, $\mathrm{char}=(\mathrm{degree}-1)\bmod 3$,
under the A304-forced forward generator, and flagged three things still in play: the
identification of P18's third layer (``fiber'') with A287's and the kernel's ``edge'';
the electron-as-anchor; and the rule selecting which coefficient a family draws. This
addendum makes the bridge as airtight as the corpus allows. It resolves the first two
flags, and the sign of the twist, as corpus-forced. The fiber \emph{is} the edge: P05
names ``the edge $S^1$ fibers'' carrying the linear term $2\pi x$, the observational
share $2.3\% = \pi/\alpha^{-1}$, so P18's $\text{bulk}\to\text{boundary}\to\text{fiber}$
is the degree cycle $3\to2\to1$ with fiber $=$ edge $=$ degree $1$, not a separate
object. The anchor $k=0\leftrightarrow\text{edge}\leftrightarrow\text{degree }1$ is
forced by the electron being the trivial representation (P20) together with
$\rho(0)=0$, which makes the edge the ground of the present degrees $\{1,2,3\}$. The
relative handedness is a fixed $-1$, not a choosable $\pm1$, because the layer cycle
(degree-decreasing) and the degree grading (character-increasing) both inherit the one
A304-forced forward rotation. What does \emph{not} close is the third flag. The family
eigenspaces $\psi^{(k)}$ are $\mathbb{Z}_3$ Fourier combinations spread equally over
all three layers, so ``family $k$ draws a single degree'' is a \emph{projection} choice
on a spread state; the corpus never derives that the projection is the
character-resonant diagonal rather than another $\mathbb{Z}_3$-equivariant functional.
The bridge therefore reduces to one irreducible premise, $G1$: the mass-law coefficient
a family draws is the character-resonant layer of its eigenspace. \emph{Given} $G1$, the
full bridge $k=(\mathrm{degree}-1)\bmod 3$ is forced with zero residual and reproduces
$(e,\mu,\tau)\leftrightarrow(2,3,16)$ end to end through the A287 law. The verdict is
PARTIAL: a posited pin, but sharpened from A325's bare equality to the single premise
$G1$, with two of A325's three open items now corpus-forced. The overlap-diagonal of
A299 through A320 stays withdrawn; no wavefunction or overlap magnitude is formed; no
published number changes.
\end{abstract}

% ----------------------------------------------------------------------
\section{The Bridge, Stated}

\begin{proposition}[The Family--Degree Bridge]
\label{prop:bridge}
A charged-lepton family's $T_{\rm cycle}$ eigenvalue index $k$ equals the
ground-relative $\rho$-degree of the coefficient it draws,
\[
\boxed{\;k \;=\; (\mathrm{degree}-1)\bmod 3\;,}
\]
with the edge layer (degree $1$, the $\rho(0)=0$ Dirichlet ground) taken as the index
origin. This identifies P20's Family-Origin labelling (the family is the $\omega^k$
eigenspace of $T_{\rm cycle}$, line~201) with A287's level assignment (the family draws
the coefficient of its layer, lines~60--64): $k=0$ draws degree $1$ (edge, $c=2$, the
anchor), $k=1$ draws degree $2$ (boundary, $c=3$), $k=2$ draws degree $3$ (bulk,
$c=16$).
\end{proposition}

This addendum asks how much of Proposition~\ref{prop:bridge} the corpus forces, and what
single statement remains to be posited. Table~\ref{tab:bridge} sets the joined
labelling out.

\begin{table}[h]
\centering
\begin{tabular}{lccc}
\toprule
\textbf{Family} & $e$ & $\mu$ & $\tau$ \\
\midrule
P20 eigenvalue index $k$ (line~201) & $0$ & $1$ & $2$ \\
Layer drawn (A287, lines~60--64) & edge & boundary & bulk \\
$\rho$-degree of that layer & $1$ & $2$ & $3$ \\
Coefficient $c_n$ & $2$ & $3$ & $16$ \\
\midrule
Bridge: $(\mathrm{degree}-1)\bmod 3$ & $0$ & $1$ & $2$ \\
\bottomrule
\end{tabular}
\caption{The Family--Degree Bridge joins the two corpus labellings. The eigenvalue
index $k$ of P20 equals the ground-relative degree $(\mathrm{degree}-1)\bmod 3$ of the
A287 coefficient, edge as ground. The bottom row reproduces the P20 index exactly.}
\label{tab:bridge}
\end{table}

% ----------------------------------------------------------------------
\section{What the Corpus Forces}

A325 left three items in play. Two of them, and the sign of the twist, close here.

\paragraph{The fiber is the edge (A325 flag (i), resolved).} A325 flagged that P18
\S3.5 names the third layer of the cycle ``fiber'' ($\text{bulk}\to\text{boundary}\to
\text{fiber}$), while A287 and the kernel name the degree-1 layer ``edge'', and that the
bridge depends on whether these are the same layer. They are, and the corpus states it.
P05's figure caption reads ``the edge $S^1$ fibers support observational structure
($2.3\%$)''; its section ``The Observational Edge'' identifies the linear term $2\pi x$
with ``the fibers of observation''; P18 line~127 lists ``Fiber $S^1$: observational
mode (dimension~$1$)'' and line~575 reads $\Delta_{S^1}=$ Observation. All coincide:
the fiber is the $S^1$ Hopf layer, is the degree-1 $2\pi x$ term, is observation, is the
share $2.3\% = \pi/\alpha^{-1}$, is the edge. P18's $\text{bulk}\to\text{boundary}\to
\text{fiber}$ is therefore the degree cycle $3\to2\to1$ with fiber $=$ edge $=$ degree
$1$. This is a corpus identity, not a posit.

\paragraph{The anchor is forced (A325 flag (ii), resolved).} The electron is the trivial
representation: P20 gives it $k=0$, eigenvalue $1$. Because $\rho(0)=0$, the density has
no degree-zero term, so the present degrees are $\{1,2,3\}$ and the ground (lowest
present layer) is the edge at degree $1$. The trivial character is the one whose lowest
\emph{present} representative is not the minimal degree --- its lowest present term is
degree $3$, while characters $1,2$ first appear at degrees $1,2$. So the trivial family
is the anchor, and the anchor sits at the ground, the edge. The identification
$k=0\leftrightarrow\text{edge}\leftrightarrow\text{degree }1$ is forced by P20, $\rho(0)=0$,
and the fiber-equals-edge identity together; it is not a convention.

\paragraph{The handedness sign is forced.} The base grading on monomials is
$x^d\mapsto\omega^d x^d$, under which the character \emph{increases} with degree. The
layer cycle of P18 runs $3\to2\to1$, \emph{decreasing} degree. Both are the one
$\mathbb{Z}_3$ acting through the same forward rotation $\varphi\mapsto\varphi+2\pi/3$,
which A304 forces (the backward generator $C^2$, root $\omega^2$, is excluded by the
$L(3,1)$ positive orientation of Papers~06 and~28). One forward layer-cycle step
therefore changes the graded character by a uniform $-1\pmod 3$: the residual is a fixed
$-1$, not a choosable $\pm1$. This is the ``one unit of $\omega$'' of A325's
Resolution~C, now with its sign pinned to the forced rotation.

\begin{proposition}[Given the projection premise, the bridge is forced with no residual]
\label{prop:noresidual}
Let $G1$ be the premise that the mass-law coefficient a family draws is the
character-resonant layer of its eigenspace --- the layer whose ground-relative grading
character equals the family's eigenvalue index. Given $G1$, the forced anchor
$k=0\leftrightarrow$ degree~$1$, and the forced forward unit step, the assignment of the
two non-anchor families is determined uniquely: of the two bijections
$\{1,2,3\}\to\{0,1,2\}$ that fix the anchor, exactly one advances by the forced unit
step, namely $k=(\mathrm{degree}-1)\bmod 3$. This coincides with the unique reconciler of
A325's convention scan under the A304-forced generator, and the literal grading
($\mathrm{char}=d$) and the naive Dirichlet shift ($\mathrm{char}=d+1$) both fail. There
is no residual freedom once $G1$ is granted.
\end{proposition}

% ----------------------------------------------------------------------
\section{What Stays Unforced, and Why the Verdict Is PARTIAL}

\begin{remark}[The eigenspaces are layer-spread; $G1$ is a projection choice]
\label{rem:g1}
The forcing reaches exactly as far as Proposition~\ref{prop:noresidual} and no further,
and it is honest to stop there. The family eigenspaces are not individual layers. In the
regular representation that A304 fixes, $\psi^{(k)}$ is the $\mathbb{Z}_3$ Fourier
combination $(\text{bulk}+\omega^{-k}\,\text{boundary}+\omega^{-2k}\,\text{fiber})/\sqrt3$,
which carries weight $\tfrac13$ on \emph{each} of the three layers. ``Family $k$ draws a
single degree'' is therefore a statement about a \emph{projection} of a layer-spread
state, not an identity between a family and a layer. Which functional of the spread
eigenspace selects the drawn coefficient is the content of $G1$. The character-resonant
diagonal is the unique $\mathbb{Z}_3$-equivariant choice that, once adopted, leaves no
freedom (Proposition~\ref{prop:noresidual}); but $\mathbb{Z}_3$-equivariance alone does
not pin it --- the conjugate equivariant diagonal $k\mapsto-k$ gives $(0,2,1)$ and does
\emph{not} reconcile. Selecting the resonant diagonal over the alternatives is the
posited premise. It is precisely the route A287 flagged (``layer-projected perturbation
of $\hat O$, each family's state drawing a correction proportional to its layer's
coefficient''), and the withdrawn overlap-diagonal of A299 through A320 was the attempt
to \emph{compute} this projection from the operator; it overshot and was withdrawn,
which is independent evidence that the projection is not fixed by representation theory
without further input. The corpus does not derive $G1$. The verdict is therefore PARTIAL.
\end{remark}

\paragraph{The pin the corpus must add.} One statement upgrades PARTIAL to HIT:
\emph{the mass-law coefficient a charged-lepton family draws is the character-resonant
layer of its $T_{\rm cycle}$ eigenspace} (premise $G1$). Given $G1$, the Family--Degree
Bridge $k=(\mathrm{degree}-1)\bmod 3$ follows as a theorem
(Proposition~\ref{prop:noresidual}), and OI-287-1's why-question closes. The pin is
sharper than A325's: A325 posited the bare equality $k=(\mathrm{degree}-1)\bmod 3$; A326
shows that equality decomposes into the forced anchor, the forced handedness sign, and
the forced no-residual propagation, leaving exactly one premise to posit --- the
resonant-diagonal projection rule.

\paragraph{End to end.} Table~\ref{tab:e2e} carries the bridge through the A287 law. The
bridge fixes \emph{which} coefficient each family draws; the A287 law then reproduces the
mass ratios at the published $10^{-4}$ accuracy.

\begin{table}[h]
\centering
\begin{tabular}{lcccc}
\toprule
\textbf{Family} & index $k$ & degree $k{+}1$ & coeff $c_n$ & $m_n/m_e$ (pred / meas) \\
\midrule
$e$    & $0$ & $1$ & $2$  & $1$ (anchor) \\
$\mu$  & $1$ & $2$ & $3$  & $206.80\ /\ 206.768$ \\
$\tau$ & $2$ & $3$ & $16$ & $3477.4\ /\ 3477.4$ \\
\bottomrule
\end{tabular}
\caption{End-to-end reproduction. The bridge $k\mapsto\text{degree }k{+}1\mapsto c_n$
selects the coefficient; the A287 law $m_n/m_e=(\lambda_n/\lambda_1)^{\beta_n}$,
$\beta_n = 6\mu_1/\mu_0 - c_n\,\alpha^{1/4}/\lambda_1$, reproduces the ratios to
$\sim 10^{-4}$.}
\label{tab:e2e}
\end{table}

\paragraph{Status.} OI-287-1 remains open in its corrected form, but its residual is now
reduced to a single irreducible premise. The mechanism is a $\mathbb{Z}_3$ character
selection rule (A323); the coefficient set is the density's own $\mathbb{Z}_3$ character
content (A324); the family-to-coefficient labelling reconciles under a unique convention
(A325); and the bridge joining the eigenvalue labelling to the drawn degree decomposes,
here, into three corpus-forced pieces and one posited premise --- the resonant-diagonal
projection of the layer-spread eigenspace. The fiber-equals-edge identification and the
electron anchor, which A325 left open, are now corpus-forced. The overlap-diagonal of
A299 through A320 stays withdrawn; no wavefunction or overlap magnitude is formed. No
published number changes; the A287 lepton law and its $10^{-4}$ accuracy are untouched.

\paragraph{Verifier.} \texttt{verify\_P326.py}: recomputes $\alpha^{-1}=4\pi^3+\pi^2+\pi$
and the layer fractions; the layer-degree map and $c_n=(2,3,16)$ with $\rho(0)=0$; the
fiber-equals-edge-equals-$S^1$-equals-degree-1 identity at share $2.3\%$; the two
labellings; the A304 forward generator and its orientation invariant; the opposite
handedness of the layer cycle and the grading as a fixed $-1$; the forced anchor; the
convention scan and unique reconciler; the no-residual propagation given $G1$; the
layer-spread eigenspaces and the honest unforcedness of $G1$ (the conjugate diagonal
fails); the end-to-end reproduction $(e,\mu,\tau)\leftrightarrow(2,3,16)$ through the
A287 law to $m_\mu/m_e$ and $m_\tau/m_e$; the PARTIAL verdict; and the exact pin.
Twenty-five checks, all PASS.

\end{document}
