% 336_Addendum_APSMonotone.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% P18-T2 sub-result (iii), the HELLMANN-FEYNMAN MONOTONICITY argument (A332)
% REDONE on the corpus APS operator -- the second of A334's two pre-registered
% fan-out probes, A335 (the second-preimage scan) the sibling. A332 proved three
% sectors sign-definite (alpha*rho, beta*M, gamma*Re(w^k)) and found zeta*R
% near-uniform (-2, rel spread 1.7e-4) on a DIRICHLET model; A333 upgraded zeta to
% an exact rigid shift, completing four-sector monotonicity. This addendum
% recomputes the Hellmann-Feynman diagonals on the corpus APS eigenstates
% (R_nl=r^{-1}J_{l+1}((2n+l+2)r), spectrum (2n+l+2)^2, psi(1)!=0). THE FINDING: the
% three shape sectors STAY sign-definite (rho>=0, M positive, Re(w^k) rigid are
% state/BC-independent), but the zeta diagonal <psi_n|R_sym|psi_n> ranges
% -1.98..+0.24 and SIGN-FLIPS (l=2 n=0 = +0.24>0) -- per-axis monotonicity in zeta
% FAILS on the corpus operator. A333's four-sector-monotonicity HIT is a Dirichlet
% artifact that does NOT transfer; the analytic zeta-injectivity gap is real under
% the corpus BC, closed only numerically/in-box by A335 (which found injectivity
% survives with a LARGER moat). The alpha-zeta spectral collinearity DROPS from
% A330's 0.9986 (Dirichlet) to 0.662 (APS) -- the structural reason A335's moat
% grows. Verifier: verify_P336.py (17 PASS / 0 FAIL, exit 0).

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

\title{Addendum 336: The Hellmann--Feynman Monotonicity on the APS Operator\\
\large The three shape sectors stay sign-definite under the corpus boundary
condition, but the renormalisation sector's diagonal sign-flips --- per-axis
monotonicity in $\zeta$ fails, and A333's four-sector picture does not transfer}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} PARTIAL $+$ REFUTATION (three-sector monotonicity transfers to
the corpus operator; $\zeta$-monotonicity fails as the APS zeta diagonal is
sign-indefinite/sign-flipping; A333's four-sector-monotonicity HIT is a Dirichlet
artifact; the analytic $\zeta$ gap is real under the corpus BC) \quad|\quad
\textbf{Addresses:} P18-T2 (iii) APS monotonicity; A335; A334; A333; A332; A331;
A330; A329; A302; A294 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P336.py} --- 17 PASS / 0 FAIL, exit 0

\begin{abstract}
A332 ran the structural Hellmann--Feynman monotonicity argument for sub-result
(iii)'s free-weight coefficient injectivity. By Hellmann--Feynman
$\partial\lambda_n/\partial c_i=\langle\psi_n|S_i|\psi_n\rangle$, the diagonal
matrix element of sector $S_i$ in the eigenbasis; a sector contributes
monotonically to every eigenvalue iff its diagonal has a fixed sign. A332 proved
three of four sectors sign-definite --- $\alpha\rho$ ($\rho\ge0$ multiplication,
diagonal $116\dots181>0$), $\beta M$ ($M$ positive, diagonal in $[0,1]\ge0$),
$\gamma\,\mathrm{Re}\,\omega^k$ (a rigid nonzero shift, monotone per $\mathbb{Z}_3$
character block) --- hence injective in $(\alpha,\gamma,\beta)$, and found the
fourth, $\zeta R$ with $R=r\partial_r$, near-constant ($\langle\psi|R|\psi\rangle$
mean $-2.000$, relative spread $1.7\times10^{-4}$, cosine with the uniform vector
$1.00000$): a near-uniform shift, the soft sector. A333 then claimed $\zeta$ is an
exact rigid shift and completed four-sector monotonicity. But A332 and A333 built
their radial operator on the interior mesh $r=\mathrm{linspace}(dr,1-dr,N-1)$ ---
a \textbf{Dirichlet wall} at $r=1$ ($\psi(1)=0$), not the corpus APS boundary
condition A329 established (P18 lines 393--396). A334 built the corpus APS operator
and showed the zeta diagonal there is \emph{not} the near-uniform $-2$: it is
state-dependent and even sign-flipping. \textbf{This addendum redoes A332's
monotonicity on the corpus APS eigenstates.} The three shape sectors \emph{stay}
sign-definite --- $\rho\ge0$ and $M$ positive are state-independent facts and the
$\mathrm{Re}\,\omega^k$ shift is rigid, so their per-axis monotonicity transfers
to APS untouched ($\langle\psi|\rho|\psi\rangle>0$ on the APS states, min $106$;
$\langle\psi|M|\psi\rangle$ in $[0,1]$; $(1,-\tfrac12,-\tfrac12)$). \textbf{The
zeta sector fails.} On the APS eigenstates
$\langle\psi_n|R_{\mathrm{sym}}|\psi_n\rangle=-2+\tfrac12|\psi(1)|^2/\langle\psi|
\psi\rangle$ ranges from $-1.982$ to $+0.238$ across modes, and crucially it
sign-flips: eleven of twelve modes \emph{lower} the eigenvalue as $\zeta$ rises,
but the $l=2,n=0$ mode \emph{raises} it ($+0.238>0$). So $\partial\lambda_n/
\partial\zeta$ changes sign across modes --- per-axis monotonicity in $\zeta$
\emph{fails} on the corpus operator. A333's ``four-sector monotonicity complete''
is a Dirichlet artifact (it required the near-uniform $-2$, cosine $1.0$; the APS
diagonal has cosine $0.864$) and does not transfer. \textbf{What survives:} per-axis
injectivity for $(\alpha,\gamma,\beta)$ by structural monotonicity; $\zeta$
injectivity is \emph{not} provable by monotonicity, so the analytic global proof
at the zeta sector is genuinely open under the corpus BC. Jointly with A335 (the
sibling), which found in-box global injectivity \emph{survives} on the corpus
operator with a \emph{larger} moat ($6.20\times10^{-3}$ vs A331's
$1.67\times10^{-3}$, the state-dependent zeta diagonal stiffening the landscape):
$(\alpha,\gamma,\beta)$ are analytically injective and $\zeta$ injectivity rests on
A335's in-box numerics, with the analytic global proof at the zeta sector open
under the corpus boundary condition and the corpus tie $\zeta=\alpha^{5/4}$ the
clean closure for the corpus-posed family. The same mode-structure is read out a
second way: the $\alpha$--$\zeta$ spectral collinearity --- A330's $0.9986$ on the
Dirichlet model --- drops to $0.662$ on the corpus operator, breaking the soft
$\alpha$--$\zeta$ ray A302's fragility rode on, which is the structural reason
A335's moat grows. Verifier \texttt{verify\_P336.py}: 17 PASS / 0 FAIL. No
published number changes.
\end{abstract}

\section{The probe and the discipline}

This is the second of A334's two pre-registered fan-out probes, A335 (the redone
bounded-box second-preimage scan) the sibling. A334 established that the whole
sub-result (iii) thread used a Dirichlet model and that, under the corpus APS
boundary condition, the renormalisation sector's spectral response is genuinely
soft and state-dependent. Two injectivity results inherited from the Dirichlet
thread therefore needed re-establishing on the corpus operator: A331's
second-preimage scan (A335) and A332's structural monotonicity (this probe). The
binding solution-bar discipline governs --- do not force closure, honesty over
closure. Here that discipline is the substance: A332's clean ``three monotone $+$
one near-uniform'' picture, and A333's upgrade of $\zeta$ to a fourth monotone
sector, both lean on a property of the Dirichlet eigenstates ($\psi(1)=0$) that
the corpus eigenstates do not have. The honest discharge is to recompute the
Hellmann--Feynman diagonals on the corpus operator and report what each sector
does on \emph{it}, even where the answer overturns A333.

The operator is P18's bulk Dirac$^2$ radial form on the unit ball,
$-d_r^2-(3/r)d_r+l(l+2)/r^2$ on the $r^3$ measure, with regularity at $r=0$ and APS
spectral projection at $r=1$ (A329, P18 lines 393--396). The Hellmann--Feynman
diagonal is taken in the corpus $r^3$ inner product, in which the operator is
self-adjoint.

\section{The corpus APS operator (the hard gate)}

The APS radial eigenfunctions are $R_{nl}(r)=r^{-1}J_{l+1}(kr)$ with $k=2n+l+2$
(the regular root; the singular root $r^{-1}Y_{l+1}$ is not in $L^2(r^3\,dr)$, A329
check 9), reproducing the integer ladder $E_{nl}=(2n+l+2)^2$ (P18 line 171) and
\emph{not} vanishing at the wall, $R_{nl}(1)=J_{l+1}(2n+l+2)\neq0$. The verifier's
hard gate (check 3) is that the no-potential spectrum comes out the integer ladder
$l=0:(4,16,36,64)$, distinct from the Dirichlet squared-Bessel zeros
$(14.68,49.22,\dots)$; if it had come out squared-Bessel zeros the probe would be
back on the Dirichlet model and void. The gate passes, and the eigenstates'
boundary values are nonzero (check 4, $l=0:(0.577,-0.066,-0.277,0.235)$). The
Bessel implementation (numpy-only series, the same as A334/A329) is cross-checked
against known $J_1$ zeros and a finite-difference derivative (checks 1--2).

\section{The three shape sectors stay sign-definite}

\begin{proposition}[Three sectors transfer to the corpus operator]
On the corpus APS eigenstates the three shape sectors keep the sign-definite
Hellmann--Feynman diagonals that make their single-coefficient maps strictly
monotone, hence injective:
\begin{itemize}
\item $\alpha\rho$: $\rho(x)=16\pi^3x^3+3\pi^2x^2+2\pi x\ge0$ on $[0,1]$, a
nonnegative-function multiplication, so $\langle\psi|\rho|\psi\rangle>0$ on every
APS mode and $l$-sector (min $106.4$). Sign-definite $\Rightarrow$ monotone
$\Rightarrow$ injective in $\alpha$.
\item $\beta M$: $M=\sum_n(\mu_n/\mu_0)P_n$ with $\mu_n\ge0$ ($\rho\ge0$) and $P_n$
orthogonal projections is a positive operator, so $\langle\psi|M|\psi\rangle\in
[0,1]\ge0$ on the APS $l=0$ states (min $0.740$). Sign-definite $\Rightarrow$
monotone $\Rightarrow$ injective in $\beta$.
\item $\gamma\,\mathrm{Re}\,\omega^k$: $\partial\lambda/\partial\gamma=
\mathrm{Re}\,\omega^k=(1,-\tfrac12,-\tfrac12)$ is state-independent and
\emph{boundary-condition-independent} (it is the $\mathbb{Z}_3$ character value,
carrying no eigenstate content), a rigid nonzero shift, monotone per character
block.
\end{itemize}
\end{proposition}

The transfer is structural, not incidental: $\rho\ge0$ and $M$ positive are facts
about the operators, independent of which states one evaluates them on, and the
$\mathrm{Re}\,\omega^k$ shift is a number, not an expectation. So the part of
A332's argument that proved injectivity in $(\alpha,\gamma,\beta)$ is
BC-independent and carries to the corpus operator untouched (checks 5--8, 12).

\section{The zeta sector fails: the diagonal sign-flips}

\begin{proposition}[$\zeta$-monotonicity fails under the corpus boundary condition]
On the corpus APS eigenstates the renormalisation diagonal
\[
\frac{\partial\lambda_n}{\partial\zeta}
=\langle\psi_n|R_{\mathrm{sym}}|\psi_n\rangle
=-2+\tfrac12\,\frac{|\psi(1)|^2}{\langle\psi|\psi\rangle}
\]
is state-dependent and ranges from $-1.982$ to $+0.238$ across modes (for $l=0$:
$-0.689,-1.971,-1.323,-1.261$) --- not the near-uniform $-2$ of the Dirichlet
model. Crucially it \emph{changes sign}: of the twelve modes evaluated, eleven are
negative (the eigenvalue \emph{lowers} as $\zeta$ rises) but the $l=2,n=0$ mode is
positive ($+0.238$, the eigenvalue \emph{rises}). So some eigenvalues rise and
others fall as $\zeta$ increases. The diagonal is therefore not sign-definite
(max $0.238>0$, min $-1.982<0$) and mode-structured (cosine with the uniform
vector $0.864$, not $1.0000$). Per-axis monotonicity in $\zeta$ \emph{fails}.
\end{proposition}

This is the measurement A332 could not have made on the Dirichlet model, where the
boundary term $\tfrac12|\psi(1)|^2$ vanishes by construction and the diagonal
collapses to the rigid bulk $-2$. Under the corpus boundary condition the boundary
term is the live, state-dependent piece A334 measured, and it is large enough at
the $l=2,n=0$ mode ($k=4$, $\psi(1)=J_3(4)\approx0.43$) to flip the bulk $-2$ to a
positive value. A sign-flipping Hellmann--Feynman diagonal is exactly the failure
of the fixed-sign condition monotonicity needs: Weyl monotonicity gives a
\emph{uniform} spectral direction only when the diagonal keeps one sign, and here
it does not (checks 9--11).

\subsection*{The $\alpha$--$\zeta$ collinearity drop}

The same state-structure that breaks $\zeta$-monotonicity is visible a second way,
and it is the quantity that ties this probe to its sibling A335. A330 measured the
spectral collinearity of the renormalisation sector with the density sector --- the
cosine between the $\alpha$ response column $\langle\psi_n|\rho|\psi_n\rangle$ and
the $\zeta$ response column $\langle\psi_n|R_{\mathrm{sym}}|\psi_n\rangle$ across
modes --- as $0.9986$ on the Dirichlet model: a near-perfect degeneracy, the soft
$\alpha$--$\zeta$ ray along which A302's uniqueness was fragile. Recomputing the
\emph{same} cosine on the corpus APS eigenstates gives $0.662$ (check 12), a drop of
$0.336$. The reason is structural: on the Dirichlet model the $\zeta$ diagonal was
the near-uniform $-2$ (cosine with the uniform vector $1.0000$), and a near-uniform
column is necessarily near-collinear with the always-positive $\alpha$ column; under
the corpus boundary condition the $\zeta$ diagonal carries genuine mode-structure
(cosine with uniform $0.864$), so it is no longer near-degenerate with $\alpha$. The
collinearity drop and the cosine-with-uniform drop are two readouts of one fact
(check 13). This is precisely A335's mechanism: breaking the $\alpha$--$\zeta$
degeneracy is what makes the off-canonical residuals rise faster and the in-box moat
\emph{larger} ($6.20\times10^{-3}$ vs A331's $1.67\times10^{-3}$). The state-
dependence that makes $\zeta$ analytically soft is the very thing that makes the
numerical landscape stiffer.

\section{The verdict and what it does to (iii)}

\begin{remark}[A333's four-sector picture does not transfer; the analytic $\zeta$
gap is real]
A333's claim that $\zeta$ is an exact rigid shift, completing four-sector
monotonicity, required the Dirichlet near-uniform $-2$ (cosine $1.0$). The APS
diagonal is not that constant --- it is sign-indefinite and sign-flipping --- so
A333's four-sector-monotonicity HIT is a Dirichlet artifact and does not transfer
to the corpus operator (check 16). The clean ``three monotone $+$ one exact-shift''
picture is gone. What survives is the part that was BC-independent all along:
per-axis injectivity in $(\alpha,\gamma,\beta)$. $\zeta$ injectivity is \emph{not}
provable by monotonicity under the corpus boundary condition, so the analytic
global proof at the zeta sector is genuinely open, exactly the A332 obstruction
A333 wrongly believed it had removed.
\end{remark}

The honest joint picture closes with A335, the sibling probe (check 17). A335
redid A331's bounded-box second-preimage scan on the same corpus APS operator and
found that in-box global injectivity \emph{survives}, with a \emph{larger} moat
($6.20\times10^{-3}$ at $M=6$, versus A331's Dirichlet $1.67\times10^{-3}$, a
$\sim\!3.7\times$ ratio). The mechanism is the same state-dependence that breaks
$\zeta$-monotonicity here: the mode-structured zeta diagonal (cosine $0.864$, not
$1.0$) breaks the near-perfect $\alpha$--$\zeta$ degeneracy that made the Dirichlet
valley flat, so off-canonical residuals rise faster and the in-box landscape is
\emph{stiffer}. The two probes are not in tension: injectivity holds
\emph{numerically} (A335) even though $\zeta$-monotonicity fails
\emph{analytically} (here). So the corpus-faithful status of sub-result (iii)'s
free-weight injectivity is: $(\alpha,\gamma,\beta)$ analytically injective by
structural monotonicity; $\zeta$ injectivity resting on A335's in-box numerical
scan; the analytic global proof at the zeta sector open under the corpus boundary
condition; and the corpus tie $\zeta=\alpha^{5/4}$ (P18 line 348) the clean closure
for the corpus-posed family.

\section{Gate ledger}

\begin{center}
\begin{tabular}{lll}
\toprule
Object & Gate & Finding \\
\midrule
Corpus APS operator (hard gate) & PASSED & $(2n+l+2)^2$ spectrum, $\psi(1)\neq0$, not squared-Bessel \\
$\alpha\rho$ on APS states & MONOTONE & $\langle\psi|\rho|\psi\rangle>0$ (min $106$); transfers from A332 \\
$\beta M$ on APS states & MONOTONE & $M$ positive, $\langle\psi|M|\psi\rangle\in[0,1]$; transfers \\
$\gamma\,\mathrm{Re}\,\omega^k$ & MONOTONE & rigid state/BC-independent shift $(1,-\tfrac12,-\tfrac12)$ \\
$\zeta R$ on APS states & NON-MONOTONE & diagonal $-1.98\dots+0.24$, sign-flips at $l=2,n=0$ \\
$\alpha$--$\zeta$ collinearity & DROPS & $0.9986$ (Dirichlet, A330) $\to 0.662$ (APS); breaks the soft ray \\
A333 four-sector monotonicity & DIRICHLET ARTIFACT & required near-uniform $-2$; does not transfer \\
(iii) analytic $\zeta$ injectivity & OPEN (corpus BC) & not provable by monotonicity; numerical via A335 \\
(iii) in-box injectivity & SURVIVES (A335) & larger moat; tie the clean closure \\
\bottomrule
\end{tabular}
\end{center}

\section{Conclusion}

The probe redoes A332's Hellmann--Feynman monotonicity argument on the corpus APS
operator that A334 built (the Bessel construction
$R_{nl}=r^{-1}J_{l+1}((2n+l+2)r)$, spectrum $(2n+l+2)^2$, eigenstates that do not
vanish at the wall), guarded by the hard gate that the no-potential spectrum is the
integer ladder and not the Dirichlet squared-Bessel zeros. The three shape sectors
transfer: $\alpha\rho$ ($\rho\ge0$ multiplication, diagonal strictly positive on
the APS states), $\beta M$ ($M$ positive, diagonal in $[0,1]$), and
$\gamma\,\mathrm{Re}\,\omega^k$ (a rigid state- and BC-independent shift) stay
sign-definite, so per-axis injectivity in $(\alpha,\gamma,\beta)$ is BC-independent
and carries to the corpus operator untouched. The zeta sector fails:
$\langle\psi_n|R_{\mathrm{sym}}|\psi_n\rangle$ on the APS eigenstates is
state-dependent, ranges $-1.982$ to $+0.238$, and sign-flips (eleven modes lower,
the $l=2,n=0$ mode raises), so $\partial\lambda_n/\partial\zeta$ changes sign and
per-axis monotonicity in $\zeta$ does not hold under the corpus boundary condition.
A333's four-sector-monotonicity HIT is a Dirichlet artifact and does not transfer;
the analytic global proof at the zeta sector is genuinely open under the corpus
operator, closed only numerically and in-box by A335 (which found injectivity
survives with a larger moat, the state-dependent zeta diagonal stiffening the
landscape). The honest joint picture: $(\alpha,\gamma,\beta)$ analytically
injective, $\zeta$ injectivity resting on A335's in-box numerics, the analytic
$\zeta$ gap open under the corpus boundary condition, and the corpus tie
$\zeta=\alpha^{5/4}$ the clean closure for the corpus-posed family. 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; no published number changes; the
$\hat O$ assembly and its derived observables are untouched. Verifier
\texttt{verify\_P336.py} reaches 17 PASS / 0 FAIL and exits zero.

\end{document}
