% 302_Addendum_CoefficientRigidity.tex
% P18-T2, sub-result (iii), first-order local form. Treating the four
% assembly coefficients (alpha, gamma, zeta, beta) as free weights on
% their sectors of P18's radial operator, the spectrum-response Jacobian
% at the canonical point has full rank 4: the coefficients are LOCALLY
% the unique spectrum-matching assignment. Local uniqueness strengthens
% as angular sectors are added (coverage = rigidity). The density (alpha)
% and renormalization (zeta) sectors are ~99.9% spectrally collinear --
% the soft direction where coefficient-uniqueness is fragile. This is a
% first-order local result on one of three sub-results; A294's re-typing
% stands, sub-results (i) and (ii) remain open.
% Verifier: verify_P302.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 302: Coefficient Rigidity\\
\large A first-order local result on P18-T2 sub-result (iii),
and the soft direction}
\author{TOE Corpus}
\date{2026-06-13}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case C (partial advance: first-order local uniqueness
proven for one of three sub-results; soft direction named) \quad|\quad
\textbf{Addresses:} P18-T2 sub-result (iii); A294 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P302.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.
This addendum makes a first-order local advance on the third sub-result,
that the four coefficients $(\alpha, \gamma, \zeta, \beta)$ are the only
spectrum-matching assignment. Treating the four as free assembly weights
on their sectors of P18's radial operator (density $\rho$, layer-cycle
$\omega^k$, renormalization $r\partial_r$, moment $\mu_n/\mu_0$), with
the fixed sectors held canonical, the spectrum-response Jacobian
$J_{ij} = \partial\lambda_i/\partial c_j$ at the canonical point has full
rank $4$: the smallest normalized singular value is bounded away from
zero ($\sigma_{\min}/\sigma_{\max} = 9.3\times10^{-3}$ on the $l=0$
sector), so there is no flat direction and the canonical coefficients are
the locally unique spectrum-matching assignment. The result strengthens
as angular sectors are added, $9.3\times10^{-3} \to 2.3\times10^{-2} \to
3.5\times10^{-2}$ for $l \le 0, 1, 2$, the spectral image of P18's
coverage requirements increasing rigidity. One structural feature
persists and is recorded as the soft direction: the density coupling
$\alpha$ and the renormalization generator $\zeta$ are spectrally
collinear to $0.9986$, both acting as nearly mode-independent shifts.
Local uniqueness is carried by the ground-state mode, where the density
response separates from the renormalization response. This is local and
first order; it does not establish global uniqueness, and sub-results
(i) and (ii) remain exactly as A294 left them.
\end{abstract}

\noindent\textbf{Erratum (verifier sweep, 2026-06-15):} The verifier
\texttt{verify\_P302.py} builds the radial operator on a \emph{Dirichlet wall}
($r \in \mathrm{linspace}(dr, 1{-}dr, N{-}1)$), not the corpus APS boundary
condition (A329). The full-rank-$4$ \emph{local} coefficient-rigidity result
SURVIVES under APS: A330/A335/A336 redo the Jacobian on the corpus operator and
local uniqueness stands. However, the $\alpha$--$\zeta$ collinearity reported
here at $0.9986$ (and the ``soft direction'' fragility it implies) is a
\emph{Dirichlet artifact}: under the corpus APS operator the collinearity is
$0.662$ (A336), and the soft-direction fragility is DISSOLVED by the corpus tie
$\zeta = \alpha^{5/4}$ (A330). The numbers in the abstract, Proposition, and
Remark below are retained as the historical Dirichlet record; for the
corpus-faithful (APS) treatment see A330/A335/A336.

% ----------------------------------------------------------------------
\section{The Rigidity Test}

\begin{proposition}[Local coefficient uniqueness on the spectrum]
Write P18's radial operator with the four named sectors carrying free
weights,
\[
\hat O_{l,k}(c) = -\tfrac{d^2}{dr^2} - \tfrac{3}{r}\tfrac{d}{dr}
+ \tfrac{l(l+2)}{r^2} + V_{\rm self}(r)
+ \alpha\,\rho(r) + \zeta\, r\tfrac{d}{dr}
+ \gamma\,\mathrm{Re}\,\omega^k + \beta\,\tfrac{\mu_n}{\mu_0},
\]
$\omega = e^{2\pi i/3}$, with the kinetic and self-lensing sectors fixed.
Let $\lambda_{n,k}(c)$ be the spectrum and $J = \partial\lambda/\partial
c$ the response Jacobian over modes $(n,k)$ and coefficients
$c = (\alpha, \gamma, \zeta, \beta)$, evaluated at the canonical
$(\alpha, \gamma, \zeta, \beta) = (1/137.036,\ 3/4,\ \alpha^{5/4},\
3\pi/20)$. The column-normalized $J$ has four nonzero singular values
with $\sigma_{\min}/\sigma_{\max} = 9.3\times10^{-3} > 0$ on $l=0$; the
rank is exactly $4$. No coefficient combination leaves the spectrum
invariant to first order: the canonical assignment is locally unique.
Including $l=1$ and $l=2$ raises the ratio monotonically to
$3.5\times10^{-2}$.
\end{proposition}

\begin{remark}[The soft direction, and the limits of this result]
The four sector responses are not equally distinct. The density coupling
and the renormalization generator produce nearly the same spectral
signature: their normalized response columns have inner product
$0.9986$, stable across the added angular sectors. Both act as
almost mode-independent shifts of the spectrum, and what separates them,
hence what carries local uniqueness, is the ground-state mode alone,
where the density response is lower ($\partial\lambda_0/\partial\alpha
\approx 117$ against $\approx 136$ for excited modes). The honest reading:
coefficient uniqueness holds locally but is soft along the
density/renormalization plane, and a proof of the third sub-result will
have to control exactly that pair. Three caveats bound the claim. It is
\emph{local}, a statement about the Jacobian at the canonical point, not
global injectivity. It is \emph{first order}, silent on second-order
coincidences. It addresses only sub-result (iii); the per-category
operator uniqueness (i) and the additive-composition uniqueness (ii)
are untouched, and A294's re-typing of the whole question stands.
\end{remark}

\paragraph{Status.} P18-T2 remains open. Its third sub-result now has a
first-order local result in its favor: the coefficients are locally the
unique spectrum-matching assignment, strengthening with sector coverage,
soft along the $\alpha$--$\zeta$ plane. Sub-results (i) and (ii) are
unchanged. No published number changes.

\paragraph{Verifier.} \texttt{verify\_P302.py}: rebuilds the response
Jacobian on the $l=0,1,2$ sectors, asserts full rank $4$ with
$\sigma_{\min}/\sigma_{\max} > 10^{-3}$, asserts the ratio strengthens
with added sectors, and asserts the $\alpha$--$\zeta$ collinearity at
$0.99{+}$ as the soft direction. All PASS.

\end{document}
