% 324_Addendum_Z3Decomposition.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1 (corrected), probe P3. Within the one object A322 gated in,
% O-Z3-grading, this addendum decomposes the density rho itself into its three
% Z_3 character components, at the structural level, with no wavefunction. The
% layer-cycle action of P18 S3.5 rotates the Hopf fiber phase by 2 pi/3 with
% eigenspaces carrying omega^k; its graded reading on a base monomial sends x^d
% to character d mod 3. Under the standard character projectors the three
% nonzero terms of rho split one-to-one into the three Z_3 characters, and their
% coefficients are exactly the set (2,3,16). The family-to-character map then
% reproduces the A287 assignment family n draws coefficient c_n under a mod-3
% reindexing, char(n) = n mod 3, in which character zero carries the top degree
% three rather than degree zero. Verdict PARTIAL: the coefficient set is the
% Z_3 character components of rho (structurally clean), and the assignment is
% recovered under the stated reindexing. The overlap-diagonal of A299 through
% A320 stays withdrawn (A321); no wavefunction overlap is formed. The open piece
% for P4 is whether this grading pairing reproduces the mass-law beta_n itself.
% Verifier: verify_P324.py.

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

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

\title{Addendum 324: The $\mathbb{Z}_3$ Decomposition of the Density\\
\large The density's own character components are the coefficient set}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} PARTIAL (a structural derivation step on the gated object;
the coefficient set is the $\mathbb{Z}_3$ character components of $\rho$, and
the family assignment is recovered under a stated mod-3 reindexing) \quad|\quad
\textbf{Addresses:} OI-287-1 (corrected); A322 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P324.py} --- all PASS

\begin{abstract}
A322 gated one corpus-faithful object for the corrected selection rule, the
$\mathbb{Z}_3$ grading of the three families as the $\omega^k$ eigenspaces of
the layer-cycle operator of P18 and P20. This addendum works inside that object
at the structural level: it decomposes the density
$\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ into its three $\mathbb{Z}_3$
character components, with no wavefunction and no overlap. The layer-cycle
action rotates the Hopf fiber phase by $2\pi/3$ with eigenspaces carrying
$\omega^k$; its graded reading on a base monomial sends $x^d$ to character
$d \bmod 3$. Under the standard character projectors
$P_k = \tfrac13\sum_{j} \omega^{-kj} g^j$ the three nonzero terms of $\rho$
split one to one into the three characters: character zero carries the degree
three term $16\pi^3 x^3$, character one the degree one term $2\pi x$, character
two the degree two term $3\pi^2 x^2$. The coefficients of these components are
exactly the set $(2,3,16)$, the integers $c_n = \rho^{(n)}(0)/(\pi^n n!)$ read
directly. The family to character map reproduces the A287 assignment, family $n$
draws coefficient $c_n$, under the reindexing $\mathrm{char}(n) = n \bmod 3$, in
which character zero carries the top degree three rather than degree zero. The
verdict is PARTIAL: the coefficient set is the $\mathbb{Z}_3$ character
components of $\rho$, structurally clean, and the corpus assignment is recovered
under the stated reindexing. The overlap-diagonal of A299 through A320 stays
withdrawn; no wavefunction overlap is formed. Whether this grading pairing
reproduces the mass-law $\beta_n$ is the standing piece for P4.
\end{abstract}

% ----------------------------------------------------------------------
\section{The Decomposition}

\begin{proposition}[The three terms of $\rho$ are its three $\mathbb{Z}_3$
character components, with coefficients the set $(2,3,16)$]
\label{prop:decomp}
Let $\omega = e^{2\pi i/3}$ and let $g$ be the $\mathbb{Z}_3$ generator whose
graded action on a base monomial is $g\,x^d = \omega^d x^d$, the reading on
homogeneous components of the layer-cycle phase rotation
$\varphi \mapsto \varphi + 2\pi/3$ on the Hopf fiber of P18 \S3.5. The character
projectors are $P_k = \tfrac13\sum_{j=0,1,2} \omega^{-kj} g^j$, $k=0,1,2$, and
satisfy
\[
P_k\,x^d \;=\; \tfrac13\sum_{j=0,1,2}\omega^{\,j(d-k)}\,x^d
          \;=\; [\,d \equiv k \!\!\pmod 3\,]\;x^d ,
\]
verified for $k=0,1,2$ and $d=0,\dots,5$. Applied to
$\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$, whose nonzero degrees are
$1,2,3$ covering the residues $1,2,0$, the projectors split $\rho$ one to one:
\[
P_0\rho = 16\pi^3 x^3,\qquad
P_1\rho = 2\pi x,\qquad
P_2\rho = 3\pi^2 x^2 .
\]
In units $\pi^d$ the coefficients of these components are $16$, $2$, $3$ in
character order $k=0,1,2$; as an unordered set they are exactly $(2,3,16)$, the
integers $c_n = \rho^{(n)}(0)/(\pi^n n!)$ computed from $\rho'(0)=2\pi$,
$\rho''(0)=6\pi^2$, $\rho'''(0)=96\pi^3$. The $\mathbb{Z}_3$ character
decomposition of $\rho$ is therefore the coefficient set itself.
\end{proposition}

% ----------------------------------------------------------------------
\section{The Family Map and Its Scope}

\begin{remark}[The assignment under a mod-3 reindexing, and what remains]
\label{rem:map}
The mass correction law of A287 assigns family $n$ the degree-$n$ coefficient
$c_n$, that is family $1,2,3$ to coefficients $2,3,16$ at degrees $1,2,3$. The
$\mathbb{Z}_3$ grading assigns character $k$ the degree-($k\bmod 3$) term, so
character $0,1,2$ carries coefficients $16,2,3$ at degrees $3,1,2$. The two
orderings differ. P20 labels the families $e,\mu,\tau$ by layer-cycle
eigenvalue $1,\omega,\omega^2$, that is characters $0,1,2$ in order; the
identity labelling, family $n$ to character $n-1$, assigns $(16,2,3)$ and does
not reproduce the law. The corpus assignment is recovered by reading the family
index as the degree it draws: family $n$ draws degree $n$, and degree $n$ sits
in character $n\bmod 3$, so the family to character map is
$\mathrm{char}(n) = n\bmod 3$. Under this map the character components are
$(2,3,16)$ in family order, reproducing the law. The reindexing is the mod-3
subtlety: character zero carries the top degree three, not degree zero, because
the residue of degree three is zero. The step is structural and honest in its
scope. It shows that the coefficient set is not an external input but the
$\mathbb{Z}_3$ character content of $\rho$ on the lens space, read with no
wavefunction; it does not yet show that the family that carries $\omega^k$ in
the mass law is the same family that the grading attaches to character
$k\bmod 3$ under the degree reading, which is the content P4 must supply against
the A287 $\beta_n$ form. A316 found that the real reduction $\mathrm{Re}(\omega^k)$
folds characters one and two; the present decomposition uses the full complex
character, where the three are distinct, consistent with that finding. The
overlap-diagonal of A299 through A320 is not revived; no expectation value of any
$\psi_n$ enters.
\end{remark}

\paragraph{Status.} OI-287-1 remains open in its corrected form. The gated
object yields a structural step: the three $\mathbb{Z}_3$ character components of
$\rho$ are its three terms, with coefficients the set $(2,3,16)$, and the corpus
family-$n$ to coefficient-$c_n$ assignment is recovered under the mod-3
reindexing $\mathrm{char}(n) = n\bmod 3$. The derivation of why the family
carrying $\omega^k$ is the one drawing the degree matched to it, against the
mass-law $\beta_n$, is the standing piece for P4. No published number changes;
the A287 lepton law and its $10^{-4}$ accuracy are untouched. The corpus
$\mathbb{Z}_3$ action is the Hopf fiber phase rotation of P18 \S3.5; the action
$g\,x^d = \omega^d x^d$ used here is its graded reading on homogeneous
components, not a separately stated radial action.

\paragraph{Verifier.} \texttt{verify\_P324.py}: recomputes $\omega = e^{2\pi i/3}$
with $\omega^3 = 1$; the projector identity $P_k x^d = [d\equiv k \bmod 3]\,x^d$
for $k=0,1,2$ and $d=0,\dots,5$; the coefficients $c_n = (2,3,16)$; the three
character components $P_0\rho = 16\pi^3 x^3$, $P_1\rho = 2\pi x$,
$P_2\rho = 3\pi^2 x^2$ with the set $(2,3,16)$ recovered as components; the
identity family map assigning $(16,2,3)$ and failing the law; the mod-3
reindexed map $\mathrm{char}(n)=n\bmod 3$ assigning $(2,3,16)$ and reproducing
it; and the PARTIAL verdict. Nine checks, all PASS.

\end{document}
