% 304_Addendum_LayerCycleForced.tex
% P18-T2, sub-result (i), one sector made concrete. Per-category operator
% uniqueness is abstract in general; for the layer-cycle (Z_3 family)
% sector it is checkable and verified. The Z_3 action has exactly two
% nontrivial generators, the forward fiber rotation T_cycle and its
% inverse T_cycle^2; their quotients are L(3,1) and L(3,2) = -L(3,1)
% (Reidemeister-Franz). A fixed orientation invariant distinguishes them,
% and the orientation Paper 06 and Paper 28 fix selects T_cycle uniquely.
% T_cycle is therefore the forced categorical operator for its sector.
% This is a verified partial on sub-result (i) for ONE sector; the other
% sectors remain open and A294's re-typing of the whole P18-T2 stands.
% Verifier: verify_P304.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
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\newtheorem{proposition}{Proposition}[section]
\newtheorem{remark}[proposition]{Remark}

\title{Addendum 304: The Layer-Cycle Operator is Forced\\
\large A verified result on P18-T2 sub-result (i) for one sector}
\author{TOE Corpus}
\date{2026-06-13}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case C (partial: one sub-result advanced for one sector)
\quad|\quad
\textbf{Addresses:} P18-T2 sub-result (i); A294 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P304.py} --- all PASS

\begin{abstract}
P18-T2 (assembly uniqueness) requires three independent sub-results; A294
re-typed the question as pure mathematics and offered no new tools, and
A302 gave the third sub-result a first-order local result. This addendum
advances the first sub-result, that each sector's named operator is the
only operator in its category, for one sector where the forcing is
demonstrable rather than asserted: the $\mathbb{Z}_3$ layer-cycle sector
$\gamma T_{\rm cycle}$ of P18 \S3.4. The argument P18 sketches in
Proposition (canonical selection) is made machine-checkable. The
$\mathbb{Z}_3$ action, built as its regular representation, has exactly
two nontrivial generators: the forward fiber rotation $\varphi \mapsto
\varphi + 2\pi/3$, realized by the cyclic shift $C$ carrying eigenvalue
$\omega = e^{2\pi i/3}$ on the primitive character, and its inverse
$C^2 = C^{-1}$, the backward rotation carrying $\omega^2 = \overline{\omega}$.
Both satisfy $X^3 = I$ with spectrum the cube roots of unity, supplying the
three-family assignment $k = 0, 1, 2$. The two generators are inverse,
hence complex-conjugate, representations: $C C^2 = I$ and $C^2 = C^{\mathsf T}$.
A single signed orientation invariant, the imaginary part of the
eigenvalue carried on the fixed $k=1$ Fourier character, takes value
$+\sin(2\pi/3)$ on $C$ and $-\sin(2\pi/3)$ on $C^2$, and the two are
exchanged by orientation reversal. Their quotients are the lens spaces
$L(3,1)$ and $L(3,2) \cong -L(3,1)$ (Reidemeister--Franz). Paper~06
requires $L(3,1)$ and Paper~28 fixes the positive orientation, so the
orientation invariant selects $C = T_{\rm cycle}$ uniquely and excludes
$T_{\rm cycle}^2$. For this sector the categorical operator is forced. The
result is an honest partial: it addresses sub-result (i) for the
layer-cycle sector only; the remaining sectors are untouched and A294's
re-typing of the whole P18-T2 stands.
\end{abstract}

% ----------------------------------------------------------------------
\section{The Forcing}

\begin{proposition}[The layer-cycle operator is the unique categorical
generator, given the fixed orientation]
Realize the $\mathbb{Z}_3$ layer-cycle action of P18 \S3.4 as its regular
representation, with the cyclic shift
\[
C = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix},
\qquad C^2 = C^{-1} = C^{\mathsf T},
\]
cycling $\text{bulk} \to \text{boundary} \to \text{fiber} \to \text{bulk}$.
Then:
\begin{enumerate}
\item[\textnormal{(i)}] $\mathbb{Z}_3$ has exactly two nontrivial generators,
$C$ and $C^2$, both of order three: $C^3 = (C^2)^3 = I$.
\item[\textnormal{(ii)}] The spectrum of either generator is the set of cube
roots of unity $\{1, \omega, \omega^2\}$, $\omega = e^{2\pi i/3}$, carrying
the three-family eigenspace assignment $k = 0, 1, 2$ of P18.
\item[\textnormal{(iii)}] $C$ and $C^2$ are inverse, hence
complex-conjugate, representations: $C C^2 = I$ and $C^2 = C^{\mathsf T}$;
on the fixed primitive character $v_1 = (1, \omega, \omega^2)/\sqrt 3$ the
generator $C$ carries $\omega$ and $C^2$ carries $\overline{\omega}$.
\item[\textnormal{(iv)}] The signed orientation invariant
$\iota(X) = \operatorname{Im}\langle v_1, X v_1\rangle$ satisfies
$\iota(C) = +\sin(2\pi/3)$ and $\iota(C^2) = -\sin(2\pi/3)$; orientation
reversal (complex conjugation) exchanges the two, matching
$L(3,2) \cong -L(3,1)$ under Reidemeister--Franz.
\end{enumerate}
Paper~06 \S3 requires the positively oriented quotient $L(3,1)$ and
Paper~28 \S3.2 fixes the positive orientation on $S^3 \subset \mathbb{C}^2$.
The condition $\iota(X) > 0$ therefore selects exactly one generator,
$C = T_{\rm cycle}$, and excludes $C^2 = T_{\rm cycle}^2$. The named
operator of the layer-cycle sector is forced.
\end{proposition}

\begin{remark}[What is forced, what remains open]
The forcing is exact and finite: it rests on the order-three structure of
$\mathbb{Z}_3$, the cube-root spectrum, the conjugacy of the two
generators, and a single orientation invariant whose sign is fixed by two
prior papers. No coefficient is tuned and no spectrum is fitted. The scope
is narrow and stated as such. The result establishes per-category operator
uniqueness for the layer-cycle sector \emph{alone}, conditional on the
orientation Papers~06 and~28 fix; without that orientation the two
generators are interchangeable. The remaining sectors named in P18 \S3 are
not closed here. The kinetic operator $D^2_{B^4}$, the Laplacians
$\Delta_{S^3}$ and $\Delta_{S^1}$, the self-lensing potential, the density
coupling $\rho$, and the moment hierarchy each require a separate
categorical-uniqueness argument that this addendum does not supply. The
renormalization generator $\mathcal{R} = r\,\partial_r + \Delta_\psi$ was
examined and is reported as partial: the dilation generator $r\,\partial_r$
is the unique scale-transformation generator on the radial half-line up to
scale, with eigenfunctions $r^p$ and eigenvalue the scaling dimension, but
the additive $\Delta_\psi$ is a $c$-number representation choice rather
than an independent operator, so strict categorical uniqueness for that
sector is not established. Sub-results (ii) additive composition and (iii)
coefficient uniqueness are unchanged; (iii) carries A302's first-order
local result and (ii) is untouched. A294's re-typing of the whole P18-T2
stands.
\end{remark}

\paragraph{Status.} P18-T2 remains open. Its first sub-result now has a
verified result for one sector: the layer-cycle operator $T_{\rm cycle}$ is
the unique categorical generator of the $\mathbb{Z}_3$ family sector, given
the orientation Papers~06 and~28 fix, with $T_{\rm cycle}^2$ excluded as
the opposite-orientation quotient. The remaining sectors of sub-result (i)
are open. Sub-results (ii) and (iii) are unchanged. No published number
changes.

\paragraph{Verifier.} \texttt{verify\_P304.py}: builds the $\mathbb{Z}_3$
regular representation from scratch, asserts exactly two order-three
generators with $C^3 = (C^2)^3 = I$, asserts the cube-root spectrum and the
three-family assignment, asserts the conjugacy $C C^2 = I$ and
$C^2 = C^{\mathsf T}$, asserts the orientation invariant flips sign between
the two generators, and asserts the fixed positive orientation selects
exactly one. Eight checks, all PASS.

\end{document}
