% 323_Addendum_Z3Selection.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1 (corrected), pre-registered gated probe P2. A322 gated in one
% corpus-faithful object for the corrected problem: the three families as the
% omega^k eigenspaces of the layer-cycle operator T_cycle (P18 S3.5, P20). This
% addendum tests, at the representation-theory level and not as any overlap, the
% candidate "why": does the corpus layer-cycle Z_3 grade the degree-k Taylor term
% of the density rho with character k mod 3, so that Schur selection forces each
% family to couple only to one degree? The corpus action is the forward fiber
% rotation varphi -> varphi + 2pi/3 (P18 S3.5), which by the Hopf relation sends
% the complex base coordinate z -> omega z and so the degree-k monomial
% x^k -> omega^k x^k; degree-k carries character k mod 3. The Schur sum
% (1/3) sum_j omega^{(a-b)j} = delta_{a==b mod 3} then forces each family to a
% single degree. The selection rule is structurally exact. But the forced pairing
% is a cyclic shift of the corpus assignment: the degrees present {1,2,3} carry
% characters {1,2,0}, so degree 3 lands on character 0 (the trivial family), not
% on level 3, and with the family levels n = character k + 1 (P06/P20: e->0,
% mu->1, tau->2; mass law: e->1, mu->2, tau->3) the character pairing
% (e,mu,tau) -> (16,2,3) is the cyclic shift of the corpus (2,3,16). Verdict
% PARTIAL: Z_3 selection is the mechanism at the character level, but the match to
% the mass law's n<->n labelling needs the level/character offset n=k+1 fixed, which
% the corpus does not yet supply. The overlap-diagonal stays withdrawn; the residual
% goes to P3 (whether c_n is the lens-space Z_3 grading, which would fix the label).
% Verifier: verify_P323.py.

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

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

\title{Addendum 323: The $\mathbb{Z}_3$ Selection Rule\\
\large Whether the layer-cycle character grades the density's degrees, and
what the forced pairing does and does not reach}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} PARTIAL (a structural selection rule; the forced pairing is a
cyclic shift of the assignment) \quad|\quad
\textbf{Addresses:} OI-287-1 (corrected); A322 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P323.py} --- all PASS

\begin{abstract}
A322 gated one corpus-faithful object for the corrected OI-287-1: the three
charged-lepton families as the $\omega^k$ eigenspaces of the layer-cycle
operator $T_{\rm cycle}$, a representation-theoretic grading rather than an
expectation value. This probe tests, at that grading level and forming no
overlap, the candidate reason for the family-to-coefficient assignment: whether
the layer-cycle $\mathbb{Z}_3$ grades the degree-$k$ Taylor term of the density
$\rho$ with character $k \bmod 3$, so that Schur selection forces each family to
couple to a single degree. The corpus action is the forward fiber rotation
$\varphi \mapsto \varphi + 2\pi/3$ of P18~\S3.5, which by the Hopf relation
sends the complex base coordinate $z \mapsto \omega z$ and so the degree-$k$
monomial $x^k \mapsto \omega^k x^k$; the degree-$k$ term of $\rho$ therefore
carries character $k \bmod 3$. The Schur sum
$\tfrac13\sum_j \omega^{(a-b)j} = \delta_{a\equiv b}$ then forces each family to
exactly one degree. The selection rule is structurally exact. The forced pairing,
however, is a cyclic shift of the mass-law assignment: the degrees present
$\{1,2,3\}$ carry characters $\{1,2,0\}$, so degree three lands on character zero,
the trivial family, not on level three, and with the family level
$n = k+1$ (the family characters of P06 and P20 are $0,1,2$ while the mass-law
levels are $1,2,3$) the character pairing sends $(e,\mu,\tau) \mapsto (16,2,3)$,
the cyclic shift of the corpus $(2,3,16)$. The verdict is PARTIAL: the index
matching is a $\mathbb{Z}_3$ selection rule at the character level, the genuine
mechanism the corrected problem sought, but its agreement with the mass law's
$n\leftrightarrow n$ labelling depends on the level-character offset $n=k+1$,
which the corpus does not yet fix. The overlap-diagonal stays withdrawn; the
residual passes to P3.
\end{abstract}

% ----------------------------------------------------------------------
\section{The Character Grading and the Selection Rule}

\begin{proposition}[The layer-cycle grades the density's degrees, and Schur
selection forces one degree per family]
\label{prop:selection}
Let $\omega = e^{2\pi i/3}$ and let $T_{\rm cycle}$ be the layer-cycle generator,
with $T_{\rm cycle}^3 = I$ and eigenvalues the cube roots of unity, the three
families being its $\omega^k$ eigenspaces, $k=0,1,2$ (P18~\S3.5, P20 Family
Origin; the generator is forced by the $L(3,1)$ orientation, A304). The corpus
action of $T_{\rm cycle}$ is the forward fiber rotation
$\varphi \mapsto \varphi + 2\pi/3$ on the Hopf fiber coordinate (P18~\S3.5,
proposition on the canonical selection of $T_{\rm cycle}$). Under the Hopf
fibration $S^3 \subset \mathbb{C}^2$ the fiber phase rotates the complex base
coordinate, $z \mapsto e^{i\varphi} z$, so the forward generator sends
$z \mapsto \omega z$ and the degree-$k$ monomial transforms as
$x^k \mapsto \omega^k x^k$. The degree-$k$ Taylor term of
$\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ therefore carries $\mathbb{Z}_3$
character $k \bmod 3$: the degrees present, $1,2,3$, carry characters $1,2,0$.

For a family of character $a$ and a density term of character $b$, the
$\mathbb{Z}_3$-invariant pairing is
\[
P(a,b) \;=\; \frac{1}{3}\sum_{j=0}^{2} \omega^{(a-b)j} \;=\; \delta_{a \equiv b \,(\mathrm{mod}\,3)},
\]
by Schur orthogonality. A coupling between a family and a density term is
therefore nonzero only when their characters agree, and each family is forced to
couple to exactly one of the three degrees, the one whose character matches. The
index matching is a $\mathbb{Z}_3$ selection rule, not a coincidence: family of
character $k$ couples only to the degree of character $k$.
\end{proposition}

% ----------------------------------------------------------------------
\section{The Forced Pairing and the Offset It Leaves}

\begin{remark}[A cyclic shift, the level-character offset, and where the residual
goes]
\label{rem:offset}
The selection rule fixes which degree each family couples to by character, but the
resulting pairing is a cyclic shift of the assignment the mass law uses. The
family characters are $e \mapsto 0$, $\mu \mapsto 1$, $\tau \mapsto 2$ (P06~\S3,
P20 Family Origin), while the mass-law levels are $e \mapsto 1$, $\mu \mapsto 2$,
$\tau \mapsto 3$, so the level and the character of a family differ by one,
$n = k+1$. Matching by character then sends the character-zero family to the
degree-three term, since degree three carries character zero, and gives
$(e,\mu,\tau) \mapsto$ coefficients $(16,2,3)$. The corpus mass law assigns
$(e,\mu,\tau) \mapsto (2,3,16)$. The two are related by a cyclic shift, not the
identity: the selection rule is exact at the character level, but it does not
reproduce the literal $n\leftrightarrow n$ pairing, because degree three lands on
character zero rather than on level three.

This is the honest content of the PARTIAL verdict. The candidate reason is real:
the family-to-coefficient assignment is a $\mathbb{Z}_3$ selection rule, the
degree-$k$ term of $\rho$ carrying character $k \bmod 3$ and Schur orthogonality
forcing one degree per family. What it does not yet settle is the labelling,
the offset $n = k+1$ between the family level the mass law indexes and the
character the grading assigns, so that the trivial family pairs with the cubic
term rather than with level three. The corpus states the family characters and
states the mass-law levels but does not fix their alignment, which is what would
turn the cyclic shift into the identity. The overlap-diagonal of A299 through
A320 remains withdrawn and is not revived here; no overlap or expectation value
is formed, only the character of each degree and the Schur selection sum. The
residual passes to P3: whether the identity $c_n = \rho^{(n)}(0)/(\pi^n n!)$ is
itself a reading of the $\mathbb{Z}_3$-graded structure of $\rho$ on the lens
space $L(3,1)$, which would fix the level-character alignment from the geometry
rather than leaving it to be assumed. P4 stays conditional on a faithful pairing
with the labelling settled.
\end{remark}

\paragraph{Status.} OI-287-1 remains open in its corrected A287 and A296 form.
The probe establishes a $\mathbb{Z}_3$ selection rule as the mechanism of the
family-to-coefficient assignment at the character level, a derivation step on the
gated object, and identifies the standing residual as the level-character offset
$n = k+1$ that the corpus does not yet fix. No published number changes; the A287
lepton law and its $10^{-4}$ accuracy are untouched.

\paragraph{Verifier.} \texttt{verify\_P323.py}: recomputes the $\mathbb{Z}_3$
structure ($T_{\rm cycle}^3 = I$, eigenvalues $\{1,\omega,\omega^2\}$, cite
A304); the Schur identity $\tfrac13\sum_j \omega^{(a-b)j} = \delta_{a\equiv b}$
and that it forces one degree per family; the character of each density degree,
$1,2,3 \mapsto 1,2,0$, under the corpus action $x^k \mapsto \omega^k x^k$; the
forced pairing $(e,\mu,\tau) \mapsto (16,2,3)$ against the corpus $(2,3,16)$ and
that it is a cyclic shift, not the identity; and the PARTIAL verdict's defining
condition, that selection is structurally exact while the forced pairing is the
cyclic reindex of the assignment with $n = k+1$ and degree three on character
zero. Nine checks, all PASS.

\end{document}
