% 332_Addendum_CoefficientMonotone.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% P18-T2 sub-result (iii), the STRUCTURAL MONOTONICITY argument for free-weight
% coefficient injectivity: the SECOND of A330's two pre-registered fan-out parts,
% the analytic complement to A331's bounded-box second-preimage scan. By
% Hellmann-Feynman the per-coefficient spectral response is the sector's diagonal
% matrix element. Three of the four sectors are sign-definite or rigid-monotone
% (alpha*rho with rho>=0; beta*M with M positive; gamma*Re(omega^k) a rigid
% nonzero shift, monotone per Z_3 character), so the coefficient-to-spectrum map
% is injective in the (alpha,gamma,beta) directions. The fourth sector, zeta*R
% with R=r d/dr, is a pure first-order generator whose Hellmann-Feynman diagonal
% is near-CONSTANT (a near-uniform spectral shift) -- the single soft/degenerate
% direction, the only obstruction to a clean monotonicity proof, and EXACTLY
% A330's soft direction, A331's zeta valley, and the freedom the corpus tie
% zeta=alpha^{5/4} removes. The near-uniform value has an ANALYTIC origin: the
% symmetric part of r d/dr in the r^3 measure is the state-independent constant
% -(1/2)(4r^3)/r^3 = -2, so zeta's shift is structurally (not just numerically)
% uniform. HONEST scope: PARTIAL -- monotonicity proves injectivity for three
% sectors and isolates the obstruction to zeta; it does not by itself give a full
% analytic global proof. Verifier: verify_P332.py (16 PASS / 0 FAIL, exit 0).

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\newtheorem{proposition}{Proposition}[section]
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\title{Addendum 332: Coefficient Injectivity by Structural Monotonicity\\
\large Three sectors are sign-definite-monotone; the obstruction to a clean
global injectivity proof is isolated to the single renormalization direction}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} PARTIAL (the second of A330's two pre-registered fan-out parts;
the structural-monotonicity complement to A331's scan; injective in
$(\alpha,\gamma,\beta)$, the obstruction isolated to $\zeta$)
\quad|\quad
\textbf{Addresses:} P18-T2 sub-result (iii); A330; A331; A302; A312 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P332.py} --- 16 PASS / 0 FAIL, exit 0

\begin{abstract}
P18-T2 sub-result (iii) asks whether the four coefficients
$(\alpha,\gamma,\zeta,\beta)$ are the only spectrum-matching assignment on P18's
now-forced radial operator (A328/A329). A330 reproduced A302's first-order local
rank-$4$ result, upgraded it to a strict second-order local minimum, and located
one genuine residual in the free-weight version: a near-flat $\zeta$ valley along
the $\alpha$--$\zeta$ near-degenerate ray, the soft direction. It pre-registered
two parallel fan-out parts on the free-weight global injectivity question: a
bounded-box second-preimage scan (A331, no second preimage in the box) and a
structural monotonicity argument. This addendum is the second part. By
Hellmann--Feynman, the per-coefficient spectral response $\partial\lambda_n/
\partial c_i$ is the diagonal matrix element $\langle\psi_n|\,\text{Sector}_i\,|
\psi_n\rangle$, and a sector gives a strictly monotone single-coefficient map
(hence injective) iff that diagonal has a fixed sign. \textbf{Three of the four
sectors qualify.} The density sector $\alpha\rho$ is multiplication by a
nonnegative function ($\rho\ge 0$ on $[0,1]$), so $\langle\psi|\rho|\psi\rangle>0$
($116$ to $181$); the moment sector $\beta\mathcal M$ is a positive operator
($\mu_n\ge 0$, $P_n$ projections), so $\langle\psi|\mathcal M|\psi\rangle\in[0,1]
\ge 0$; the layer-cycle sector $\gamma\,\mathrm{Re}\,\omega^k$ contributes a rigid
state-independent nonzero shift, monotone per $\mathbb Z_3$ character block. So
the coefficient-to-spectrum map is injective in the $(\alpha,\gamma,\beta)$
directions. \textbf{The fourth is the obstruction.} The renormalization sector
$\zeta\mathcal R$ with $\mathcal R=r\partial_r$ is a pure first-order generator
(its operator diagonal is exactly zero, no multiplication part), and its
Hellmann--Feynman diagonal $\langle\psi|\mathcal R|\psi\rangle$ is near-constant
across all modes and $l$-sectors (mean $-2.000$, relative spread $1.7\times
10^{-4}$, cosine with the uniform vector $1.00000$): $\zeta$ contributes a
near-\emph{uniform} spectral shift, the soft/degenerate direction. This single
sector is the only obstruction to a clean global monotonicity proof, and it is
exactly A330's soft direction, A331's $\zeta$ valley, and the freedom the corpus
tie $\zeta=\alpha^{5/4}$ removes --- one obstruction, three views.
\textbf{Honest scope: PARTIAL.} Monotonicity proves injectivity for three sectors
and isolates the obstruction to $\zeta$; it does not by itself give a full
analytic global proof, because $\zeta$ breaks the fixed-sign multiplication
structure the other three enjoy. The honest joint statement, combining with
A331, is: injective in $(\alpha,\gamma,\beta)$ by structural monotonicity, no
second preimage along the residual $\zeta$ direction in the admissible box, and
the corpus tie as the clean closure for the corpus-posed family. Verifier
\texttt{verify\_P332.py}: 16 PASS / 0 FAIL. No published number changes.
\end{abstract}

\section{The fan-out part and the discipline}

This is the second of the two parallel fan-out parts A330 pre-registered on the
free-weight global injectivity question. A331 ran the first, a bounded-box
second-preimage scan, and found no second preimage in the admissible box (the
$\zeta$ valley floor strictly positive, the residual-minimizing $\alpha$ pinned at
canonical, the moat bounded below by $\varepsilon>10^{-4}$): a numerical, in-box
result. This part runs the analytic complement. The two parts test the same
residual --- the free-weight $\zeta$ valley --- from opposite sides: A331 by direct
search, this part by asking whether the coefficient-to-spectrum map is
\emph{structurally} injective, a statement stronger than any scan where it holds.

The binding solution-bar discipline carried from Heart 1 governs: do not force
closure, do not manufacture a faithful object, do not overclaim a structural
isolation as a full analytic proof. The likely verdict is PARTIAL, and it is the
honest one: monotonicity closes three of the four directions cleanly and isolates
the entire obstruction to the one sector A330 and A331 both already named. That
isolation, made structural rather than numerical, is the value of the probe.

\section{The Hellmann--Feynman criterion}

The four sectors enter P18's radial operator additively with coefficients
$(\alpha,\gamma,\zeta,\beta)$:
\[
\hat O_{l,k}=-\frac{d^2}{dr^2}-\frac{3}{r}\frac{d}{dr}+\frac{l(l+2)}{r^2}
+V_{\text{self}}+\alpha\rho+\gamma\,\mathrm{Re}\,\omega^k+\zeta\,r\partial_r
+\beta\sum_n\frac{\mu_n}{\mu_0}P_n .
\]
For a coefficient $c_i$ multiplying a sector operator $S_i$, the Hellmann--Feynman
theorem gives the first-order spectral response as the diagonal matrix element in
the eigenbasis,
\[
\frac{\partial\lambda_n}{\partial c_i}=\big\langle\psi_n\big|\,S_i\,\big|\psi_n
\big\rangle ,
\]
computed in the corpus $r^3$ measure. The verifier confirms this identity against
finite differences for $\alpha$ and $\zeta$ to $4\times10^{-2}$ and $1\times
10^{-3}$ respectively (checks 1--2). A sector contributes \emph{monotonically} to
every eigenvalue if and only if its diagonal $\langle\psi_n|S_i|\psi_n\rangle$ has
a fixed sign on the relevant subspace: then by Weyl monotonicity increasing $c_i$
moves all eigenvalues one way, the single-coefficient map $c_i\mapsto\lambda(c_i)$
is strictly monotone, and a strictly monotone map is injective. We assess each
sector against this criterion.

\section{The three sign-definite sectors}

\begin{proposition}[Density: $\alpha\rho$ is sign-definite-monotone]
The density $\rho(x)=16\pi^3x^3+3\pi^2x^2+2\pi x$ is nonnegative on $[0,1]$
(minimum $0.011$ on the interior mesh), so $\alpha\rho$ is multiplication by a
nonnegative function and $\langle\psi|\rho|\psi\rangle>0$ on every mode and
$l$-sector ($116$ to $181$). The single-coefficient map $\alpha\mapsto\lambda$ is
strictly increasing, hence injective.
\end{proposition}

\begin{proposition}[Moment: $\beta\mathcal M$ is sign-definite-monotone]
The moment operator $\mathcal M=\sum_n(\mu_n/\mu_0)P_n$ has nonnegative
coefficients ($\mu_n=\int_0^1 x^n\rho\,dx\ge 0$ because $\rho\ge 0$) and
orthogonal projections $P_n$, so $\mathcal M$ is a positive operator and
$\langle\psi|\mathcal M|\psi\rangle\in[0,1]\ge 0$ on every mode. The
single-coefficient map $\beta\mapsto\lambda$ is monotone, hence injective.
\end{proposition}

\begin{proposition}[Layer-cycle: $\gamma\,\mathrm{Re}\,\omega^k$ is rigid-monotone
per character]
The layer cycle $T_{\text{cycle}}$ has eigenvalues $1,\omega,\omega^2$; the real
spectrum-matching reduction reads $\mathrm{Re}\,\omega^k=1,-\tfrac12,-\tfrac12$.
The response $\partial\lambda/\partial\gamma=\mathrm{Re}\,\omega^k$ is
\emph{state-independent} (a rigid additive shift) and nonzero in every block, so
$\gamma\mapsto\lambda$ is strictly monotone --- injective --- within each character
block. The sign splits by character (the trivial character raises, the two
complex characters lower); it is not a single global sign, but it is a fixed
nonzero shift per block, which is exactly what injectivity in $\gamma$ requires
once the character assignment is fixed.
\end{proposition}

These are checks 3--8. Three of the four sectors give strictly monotone
single-coefficient maps, so the coefficient-to-spectrum map is injective in the
$(\alpha,\gamma,\beta)$ directions: no two distinct assignments differing only in
$\alpha$, only in $\gamma$, or only in $\beta$ produce the same spectrum.

\section{The renormalization sector is the obstruction}

\begin{proposition}[Renormalization: $\zeta\mathcal R$ is the soft, near-uniform
sector]
The renormalization generator $\mathcal R=r\partial_r$ is a pure first-order
operator: its matrix diagonal is exactly zero (no zeroth-order multiplication
part), in contrast to $\rho$ and $\mathcal M$, whose diagonals are nonnegative.
So $\langle\psi|\mathcal R|\psi\rangle$ is not a fixed-sign multiplication
readout. The decisive numerical fact is that $\langle\psi|\mathcal R|\psi\rangle$
is near-\emph{constant} across all modes and $l$-sectors: mean $-2.000$, relative
spread $1.7\times10^{-4}$, and cosine with the uniform vector $1.00000$. Thus
$\zeta$ contributes a near-uniform spectral shift. By contrast the density
response is mode-structured (relative spread $0.087$, cosine with the uniform
vector $0.996$). The near-uniform value is not a numerical accident: the
symmetric part of $r\partial_r$ in the $r^3$ measure is the state-independent
constant
\[
\tfrac12\big\langle\psi\big|\,r\partial_r+(r\partial_r)^\dagger\,\big|\psi\big\rangle
=-\tfrac12\,\frac{1}{r^3}\frac{d}{dr}\!\big(r^3\cdot r\big)
=-\tfrac12\,\frac{4r^3}{r^3}=-2 ,
\]
so to leading order $\langle\psi|\mathcal R|\psi\rangle$ is the exact constant
$-2$ regardless of the eigenstate --- a structural, not just numerical, reason the
$\zeta$ shift is uniform (check 15).
\end{proposition}

A near-uniform shift is precisely the degenerate direction. A rigid shift of all
eigenvalues by a common amount is what the other additive sectors can nearly
absorb: it is collinear, to high precision, with the $\gamma$ block shifts and
with the flat part of the mode-structured $\alpha$ and $\beta$ responses. This is
the structural origin of the soft direction. The monotonicity argument that
cleanly closes $\alpha$, $\beta$, and $\gamma$ does not give a fixed-sign
multiplication readout for $\zeta$ that separates it from the rest; the near-
uniformity is exactly the failure mode (checks 9--11).

\section{The unification}

The three views of the same obstruction now coincide:
\begin{itemize}
\item A330 found, in normalized coordinates, an $\alpha$--$\zeta$ soft plane and,
in raw coordinates, that the soft direction is almost exactly ``move $\zeta$
alone'' with $\zeta$ spectrally cheap.
\item A331 found, by direct scan, a near-flat $\zeta$ valley whose floor stays
strictly positive, with the residual-minimizing $\alpha$ pinned at canonical ---
no second preimage in the box.
\item This addendum finds, by Hellmann--Feynman, that $\zeta$'s response is a
near-uniform spectral shift: the analytic reason $\zeta$ is cheap, the valley is
flat, and the soft plane exists.
\end{itemize}
A near-uniform Hellmann--Feynman diagonal is spectrally cheap (small structured
response, hence A330's $71\times$ magnitude gap), it makes the residual flat along
$\zeta$ (hence A331's shallow valley), and it is exactly the freedom the corpus
tie $\zeta=\alpha^{5/4}$ removes by slaving $\zeta$ to the stiff, mode-structured
$\alpha$ (A330's $820\times$ tied response). One obstruction, three views (check
14).

\section{Gate ledger}

\begin{center}
\begin{tabular}{lll}
\toprule
Sector & Hellmann--Feynman diagonal & Monotonicity verdict \\
\midrule
$\alpha\rho$ (density) & $\langle\rho\rangle\in[116,181]>0$ & MONOTONE (sign-definite) \\
$\beta\mathcal M$ (moment) & $\langle\mathcal M\rangle\in[0,1]\ge 0$ & MONOTONE (sign-definite) \\
$\gamma\,\mathrm{Re}\,\omega^k$ (cycle) & rigid shift $1,-\tfrac12,-\tfrac12$ & MONOTONE per character block \\
$\zeta\,r\partial_r$ (renorm) & near-uniform ($-2.000$, spread $10^{-4}$) & NOT clean: the obstruction \\
\bottomrule
\end{tabular}
\end{center}

\begin{remark}[The assembled claim and its honest scope]
Three of the four sectors give strictly monotone single-coefficient maps, so the
coefficient-to-spectrum map is injective in the $(\alpha,\gamma,\beta)$
directions, and the entire non-injectivity risk is confined to the single $\zeta$
direction. This is a structural result, stronger where it holds than any scan.
But it is PARTIAL: it does not by itself give a full analytic global proof of the
free-weight version, because $\zeta$ breaks the fixed-sign multiplication
structure the other three enjoy --- the near-uniformity is shallow, not a clean
sign, and a near-uniform direction is exactly where a far-away $\zeta$ with
re-optimized companions can nearly reproduce the spectrum. The $\zeta$ gap is
closed only numerically and in-box by A331 (no second preimage), and analytically
by the corpus tie (which removes the $\zeta$ freedom). The honest joint statement:
injective in $(\alpha,\gamma,\beta)$ by structural monotonicity, plus no second
preimage along the residual $\zeta$ direction in the admissible box (A331), equals
free-weight global injectivity established to the honesty floor; a full analytic
proof of the free-weight version stays open at the $\zeta$ sector, and the corpus
tie $\zeta=\alpha^{5/4}$ is the clean closure for the corpus-posed family (checks
12--13, 16).
\end{remark}

\section{Conclusion}

The structural monotonicity argument closes three of the four coefficient
directions cleanly. By Hellmann--Feynman the per-coefficient spectral response is
the sector's diagonal matrix element; the density $\alpha\rho$ and moment
$\beta\mathcal M$ sectors are sign-definite (nonnegative diagonals), and the
layer-cycle $\gamma\,\mathrm{Re}\,\omega^k$ sector is a rigid nonzero shift
monotone per $\mathbb Z_3$ character block, so the coefficient-to-spectrum map is
injective in the $(\alpha,\gamma,\beta)$ directions. The renormalization sector
$\zeta\,r\partial_r$ is a pure first-order generator whose Hellmann--Feynman
diagonal is near-uniform across modes, a near-rigid spectral shift; it is the one
sector that breaks the fixed-sign multiplication structure, and it is exactly
A330's soft direction, A331's $\zeta$ valley, and the freedom the corpus tie
removes. The entire non-injectivity risk is therefore isolated to the single
$\zeta$ direction, structurally rather than numerically. We do not claim a full
analytic global proof: the verdict is PARTIAL, and the $\zeta$ gap is closed in
the admissible box by A331's scan and analytically by the corpus tie. With A330
(local and second-order rigidity, the free-vs-tied resolution), A331 (no second
preimage in the box), and this addendum (three sectors structurally injective, the
obstruction isolated), free-weight global injectivity is established to the
honesty floor and its residual is named precisely. A294's re-typing of the whole
P18-T2 stands; sub-results (i) (A328/A329) and (ii) (A310) are unchanged. No canon
source files are edited; this is a standalone addendum. No published number
changes; the $\hat O$ assembly and its derived observables are untouched. Verifier
\texttt{verify\_P332.py} reaches 16 PASS / 0 FAIL and exits zero.

\end{document}
