% 335_Addendum_APSScan.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% P18-T2 sub-result (iii), the BOUNDED-BOX SECOND-PREIMAGE SCAN redone on the
% CORPUS APS operator (the first of A334's two pre-registered fan-out probes).
% A331 found in-box global injectivity (no second preimage, moat 1.67e-3) on a
% DIRICHLET model; A334 built the corpus APS operator and showed the zeta sector
% is genuinely soft and state-dependent there. This addendum redoes A331's scan
% on the corpus APS operator. THE FINDING (option a): in-box global injectivity
% SURVIVES -- the zeta-valley floor stays strictly positive (alpha* pinned at
% canonical), the multi-start finds no second preimage, and the MOAT is LARGER
% under APS (6.2e-3, ~3.7x A331's 1.67e-3), because the live state-dependent zeta
% diagonal BREAKS the alpha-zeta degeneracy that flattened the Dirichlet valley.
% A331's conclusion transfers, with a larger moat; the analytic global proof
% stays open (A336). Verifier: verify_P335.py (16 PASS / 0 FAIL, exit 0).

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

\title{Addendum 335: The Second-Preimage Scan on the Corpus APS Operator\\
\large A331's bounded-box global injectivity redone on the corpus boundary
condition: it survives the soft zeta sector, and the moat is larger}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} GROUNDED $+$ TRANSFER (A331's in-box second-preimage scan redone
on the corpus APS operator; no second preimage; the moat is \emph{larger} under
APS; in-box global injectivity survives the genuinely soft zeta sector) \quad|\quad
\textbf{Addresses:} P18-T2 (iii) APS injectivity; A334; A331; A332; A330; A329;
A302; A294 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P335.py} --- 16 PASS / 0 FAIL, exit 0

\begin{abstract}
P18-T2 sub-result (iii) asks whether $(\alpha,\gamma,\zeta,\beta)$ are the unique
spectrum-matching assignment on P18's master operator. A331 answered the global
half numerically: a bounded-box second-preimage scan found no coefficient
assignment distinct from canonical with a machine-zero residual --- in-box global
injectivity, with a moat (the minimum off-canonical residual) of $1.67\times
10^{-3}$. But A331's radial operator was the interior-mesh \textbf{Dirichlet
model} the whole (iii) thread used, whose $l=0$ spectrum is the squared Bessel
zeros $(14.68,49.22,\dots)$, not the corpus APS ladder $(4,16,36,\dots)$ that A329
established (P18 line 171). A334 built the corpus APS operator
$R_{nl}(r)=r^{-1}J_{l+1}(kr)$, $k=2n+l+2$, and showed that on it the $\zeta$ sector
is \emph{genuinely soft and state-dependent}: the zeta diagonal
$\langle R_{\mathrm{sym}}\rangle=-2+\tfrac12|\psi(1)|^2/\langle\psi|\psi\rangle$
ranges $-1.98$ to $+0.24$ across modes (std $\approx0.6$), not the near-uniform
$-2$ of the Dirichlet model. \textbf{So A331's scan must be redone on the corpus
operator}, where the live boundary term could in principle open a second preimage
the Dirichlet model hid. That is this probe. We build the spectrum-matching
residual on the corpus APS eigenstates --- $\lambda_{nl}=(2n+l+2)^2+\langle
V_{\mathrm{self}}\rangle+\alpha\langle\rho\rangle+\zeta\langle R_{\mathrm{sym}}
\rangle$, with the additive sectors $\gamma$ (layer-cycle $\mathrm{Re}\,\omega^k$)
and $\beta$ (moment) re-optimised at each point, exactly as A331 --- the sole
corpus-faithful change being that $\langle R_{\mathrm{sym}}\rangle$ is now the
live, state-dependent APS quantity. \textbf{The finding} is option (a): in-box
global injectivity \emph{survives}. The zeta-valley floor (minimised over
$\alpha$) stays strictly positive everywhere $\zeta\neq\zeta_{\mathrm{canon}}$
(shallowest $2.6\times10^{-4}$ at $M=6$); the residual-minimising $\alpha$ stays
\emph{pinned} at canonical along the whole valley; a dense $\alpha$--$\zeta$ grid
plus a 150-seed multi-start over the admissible box ($\alpha\in[0.6,1.6]A_0$,
$\zeta\in[0.2,4]Z_0$) finds \emph{no} second preimage, and the canonical
assignment is the unique in-box machine-zero ($4.4\times10^{-16}$). \textbf{The
moat is larger under APS}, not collapsed: $6.2\times10^{-3}$ at $M=6$,
$\approx\!3.7\times$ A331's Dirichlet moat. The mechanism is the opposite of a
threat: the live state-dependent zeta diagonal A334 measured \emph{breaks} the
near-perfect $\alpha$--$\zeta$ degeneracy that made the Dirichlet valley flat, so
off-canonical residuals rise \emph{faster} under the corpus BC, deepening the
moat. Robust across $M\in\{4,6,8\}$ (moat $5.5\times10^{-3}\dots7.2\times10^{-3}$)
and box size. So A331's in-box global-injectivity conclusion is corpus-faithful:
it \emph{transfers} to the corpus APS operator, with a larger moat. This is a
genuine numerical in-box result, not an analytic global proof --- the $\zeta$
sector is genuinely soft (A334), and the analytic statement is A336's monotonicity
probe; the corpus tie $\zeta=\alpha^{5/4}$ remains the clean closure for the
corpus-posed family. Verifier \texttt{verify\_P335.py}: 16 PASS / 0 FAIL. No
published number changes.
\end{abstract}

\section{The probe and the discipline}

A334 settled the orientation of sub-result (iii): the whole (iii) thread
(A302/A330/A331/A332/A333) built its radial operator on the interior mesh
$r=\mathrm{linspace}(dr,1-dr,N-1)$, a Dirichlet wall at $r=1$, and under the
\emph{corpus} APS boundary condition the $\zeta$ sector is genuinely soft and
state-dependent rather than the rigid $-2$ shift A333 claimed. A334 therefore
pre-registered two faithful fan-out probes to re-establish the (iii) injectivity
results on the corpus operator, not transfer them from the Dirichlet model. This
addendum is the first: A331's bounded-box second-preimage scan, redone on the
corpus APS operator.

The binding solution-bar discipline governs: do not force closure, do not
manufacture a fan-out, honesty over closure. The honest outcomes here are sharply
opposed --- either the live boundary term opens a second preimage the Dirichlet
model hid (the moat collapses, the free weights are non-injective on the corpus
operator, the tie load-bearing), or in-box injectivity survives on the corpus
operator (A331's conclusion transfers). We recompute every number from scratch
(numpy only; scipy unavailable) and report whichever the corpus operator gives.
It gives option (a) --- but not weakly: the moat is \emph{larger} under APS, for a
structural reason A334's own measurement predicts.

\section{The corpus APS residual, built from scratch}

The corpus operator is the one A334 built and validated: P18's bulk Dirac$^2$
radial form $-d_r^2-(3/r)d_r+l(l+2)/r^2$ on the $r^3$ measure, regularity at
$r=0$, APS spectral projection at $r=1$ (A329, P18 lines 393--396), with radial
eigenfunctions $R_{nl}(r)=r^{-1}J_{l+1}((2n+l+2)r)$ and spectrum $(2n+l+2)^2$
(P18 line 171). The verifier reproduces the hard gate (check 3): the no-potential
APS spectrum is the integer ladder ($l=0$: $4,16,36,64$; $l=1$: $9,25,49,81$),
distinct from the Dirichlet squared-Bessel zeros A331 actually scanned (check 4).
If the spectrum had come out squared-Bessel zeros we would be back on Dirichlet
--- the failure mode the whole reconciliation exists to escape.

\begin{proposition}[The spectrum-matching residual on the APS operator]
The coefficient sectors act on the APS eigenstates through their first-order
(Hellmann--Feynman) diagonal matrix elements --- exactly the machinery A334 used
to measure the $\zeta$ sector. The perturbed eigenvalue is
\[
\lambda_{nl}(\alpha,\zeta)
= (2n+l+2)^2 + \langle V_{\mathrm{self}}\rangle_{nl}
+ \alpha\,\langle\rho\rangle_{nl}
+ \zeta\,\langle R_{\mathrm{sym}}\rangle_{nl},
\qquad
\langle R_{\mathrm{sym}}\rangle_{nl}
= -2 + \tfrac12\frac{|\psi_{nl}(1)|^2}{\langle\psi_{nl}|\psi_{nl}\rangle},
\]
with $\langle\cdot\rangle_{nl}$ the expectation on the $r^3$ measure. The additive
structural sectors $\gamma$ (layer-cycle $\mathrm{Re}\,\omega^k$) and $\beta$
(moment) are re-optimised linearly at each $(\alpha,\zeta)$, and the
spectrum-matching residual is
$r(\alpha,\zeta)=\sqrt{\mathrm{mean}\,(\lambda_{\mathrm{low}\,M}-\lambda_{\mathrm
{target}})^2}$ over the lowest $M$ eigenvalues of $l=0,1,2$, the target the
canonical assignment value. The \emph{sole} corpus-faithful change from A331 is
that $\langle R_{\mathrm{sym}}\rangle_{nl}$ is the live, state-dependent APS
quantity ($-1.98\dots+0.24$, std $0.61$; check 5), not a near-uniform $-2$. The
APS eigenfunctions are genuine $L^2(r^3\,dr)$ states (finite positive norms,
check 6), so the residual landscape is well-defined.
\end{proposition}

\section{The zeta-valley floor under APS}

A331's central probe was the $\zeta$-valley: sweep $\zeta$ over a wide admissible
range and minimise the residual over $\alpha$ (and the additive $\gamma,\beta$);
if a second preimage existed on the soft $\alpha$--$\zeta$ ray, the valley floor
would touch zero somewhere off canonical. On the corpus operator it does not.

\begin{proposition}[The APS valley floor is strictly positive, $\alpha^\ast$ pinned]
Minimising the residual over $\alpha$ at each $\zeta$, the valley floor stays
strictly positive everywhere $\zeta\neq\zeta_{\mathrm{canon}}$: the shallowest
off-canonical floor is $2.60\times10^{-4}$ at $\zeta=1.16\,Z_0$ ($M=6$). The
residual-minimising $\alpha$ stays \emph{pinned} at canonical ($1.000\,A_0$) along
the whole valley (drifting only to $0.975\,A_0$ at the extreme $\zeta=4\,Z_0$).
So even trading $\alpha$ against the soft $\zeta$ along the near-degenerate ray,
the residual cannot be driven to zero off canonical: there is no second preimage
on the soft ray, despite the live state-dependent zeta diagonal (checks 9--11).
\end{proposition}

\section{The bounded-box scan: the moat is larger under APS}

\begin{proposition}[No second preimage; the moat deepens under APS]
A dense $\alpha$--$\zeta$ grid ($\alpha\in[0.6,1.6]A_0$, $\zeta\in[0.2,4]Z_0$)
plus a 150-seed multi-start over the admissible box finds no second preimage. The
\emph{moat} --- the minimum residual at any point outside a $0.10$ relative ball
around canonical --- is $6.20\times10^{-3}$ at $M=6$, bounded below by
$\varepsilon>10^{-4}$; the canonical assignment is the unique in-box machine-zero
($4.36\times10^{-16}$); the multi-start finds zero second-preimage hits (best
off-canonical residual $8.40\times10^{-4}$) (checks 12--14). \emph{The APS moat is
$\approx\!3.7\times$ A331's Dirichlet moat $1.67\times10^{-3}$ --- larger, not
collapsed.}
\end{proposition}

The reason is structural and is exactly A334's measurement read forward. A331's
valley was so flat because, on the Dirichlet model, the $\zeta$ diagonal was a
near-uniform $-2$ (cosine with the uniform vector $1.00000$, A332): a uniform
shift is nearly degenerate with the $\alpha$ direction along the $\alpha$--$\zeta$
ray, which is why A330 measured a $99.86\%$ collinearity and A331 a shallow valley.
A334 showed that under the corpus BC the $\zeta$ diagonal is \emph{mode-structured}
(cosine with the uniform vector $0.864$, range $-1.98\dots+0.24$): it carries
genuine shape, so it is no longer near-degenerate with $\alpha$. The same
state-dependence that makes the $\zeta$ sector ``soft'' as an analytic object
(A332's lone non-sign-definite sector, A334's live boundary term) makes the
\emph{numerical} second-preimage landscape \emph{stiffer}, because off-canonical
the structured zeta shift no longer cancels against an $\alpha$ re-optimisation
the way a uniform shift does. The soft sector, far from opening the second
preimage the Dirichlet model might have hidden, deepens the moat.

\section{Robustness and the verdict}

\begin{remark}[A331's conclusion transfers, with a larger moat]
Across $M\in\{4,6,8\}$ the moat stays bounded below and larger than A331's at
every $M$ ($7.18,\ 6.20,\ 5.51\times10^{-3}$) and the valley floor stays strictly
positive ($2.92,\ 2.60,\ 2.50\times10^{-4}$, check 15); extending $\zeta$ to
$6\,Z_0$ still leaves the residual finite and far from zero ($8.12\times10^{-3}$,
check 16). So A331's in-box global-injectivity conclusion is corpus-faithful: it
transfers to the corpus APS operator, with a larger moat. This is a genuine
numerical in-box injectivity result for the free-weight version on the right
operator; it is \emph{not} an analytic global proof. The $\zeta$ sector is
genuinely soft (A334), and the analytic statement is A336's monotonicity probe;
the corpus tie $\zeta=\alpha^{5/4}$ (P18 line 348) remains the clean closure for
the corpus-posed family.
\end{remark}

\section{Gate ledger}

\begin{center}
\begin{tabular}{lll}
\toprule
Object & Gate & Finding \\
\midrule
Corpus APS operator (hard gate) & PASSED & $(2n+l+2)^2$ spectrum, distinct from Dirichlet, $\psi(1)\neq0$ \\
Zeta diagonal $\langle R_{\mathrm{sym}}\rangle$ & LIVE & state-dependent $-1.98\dots+0.24$, std $0.61$ (A334) \\
Canonical assignment & MACHINE-ZERO & residual $4.4\times10^{-16}$ \\
Zeta-valley floor (APS) & STRICTLY POSITIVE & shallowest $2.6\times10^{-4}$; $\alpha^\ast$ pinned at canonical \\
Second preimage in the box & NONE & multi-start zero hits; canonical the unique zero \\
APS moat vs A331 Dirichlet moat & LARGER ($3.7\times$) & $6.2\times10^{-3}$ vs $1.67\times10^{-3}$; soft zeta deepens it \\
In-box global injectivity & SURVIVES (corpus BC) & A331's conclusion transfers, with a larger moat \\
Analytic global proof & OPEN & zeta soft; A336 monotonicity; tie clean \\
\bottomrule
\end{tabular}
\end{center}

\section{What it does to (iii), and the sibling probe}

This probe re-establishes, on the corpus operator, exactly the in-box numerical
result A331 established on the Dirichlet model --- and strengthens it: the moat is
larger, for the structural reason A334's measurement predicts. The cross-thread
inconsistency A334 flagged (Heart 2 (i) on APS, (iii) on Dirichlet) is now closed
for the second-preimage half of (iii): the bounded-box global injectivity is a
corpus-faithful statement, not a Dirichlet artifact. What stays open is the
\emph{analytic} global proof, and the obstruction is precisely the soft $\zeta$
sector --- the same obstruction A332 isolated and A334 showed is real under the
corpus BC. The sibling probe A336 (running in parallel) redoes A332's
Hellmann--Feynman monotonicity on the corpus APS operator: the three shape sectors
($\alpha\rho$, $\beta M$, $\gamma\,\mathrm{Re}\,\omega^k$) are
multiplication/projection/character operators whose sign-definiteness should be
BC-independent but must be re-verified on the APS eigenstates, and the zeta
diagonal, now state-dependent rather than near-uniform, must be re-characterised
to determine whether per-axis monotonicity in $\zeta$ survives at all. If A336
re-establishes the three shape sectors, (iii) stands as a corpus-faithful in-box
numerical injectivity result (this probe) plus three-sector structural
monotonicity, with the $\zeta$ sector an honest soft direction --- genuinely
parallel to where A331/A332 left the Dirichlet model, but now on the right
operator.

\section{Conclusion}

A335 redoes A331's bounded-box second-preimage scan on the corpus APS operator
A334 built --- $R_{nl}(r)=r^{-1}J_{l+1}((2n+l+2)r)$, spectrum $(2n+l+2)^2$ (P18
line 171), eigenstates that do not vanish at the wall --- where the zeta diagonal
$\langle R_{\mathrm{sym}}\rangle=-2+\tfrac12|\psi(1)|^2/\langle\psi|\psi\rangle$ is
the live, state-dependent APS quantity, not the near-uniform $-2$ of the Dirichlet
model the (iii) thread used. The spectrum-matching residual is built on the APS
eigenstates with the additive $\gamma,\beta$ sectors re-optimised, exactly as
A331. The zeta-valley floor (minimised over $\alpha$) stays strictly positive
everywhere $\zeta\neq\zeta_{\mathrm{canon}}$ (shallowest $2.6\times10^{-4}$ at
$M=6$), the residual-minimising $\alpha$ stays pinned at canonical, and a dense
$\alpha$--$\zeta$ grid plus a 150-seed multi-start over the admissible box finds
no second preimage: canonical is the unique in-box machine-zero. The moat is
larger under APS --- $6.2\times10^{-3}$ at $M=6$, $\approx\!3.7\times$ A331's
Dirichlet moat --- not collapsed, because the live state-dependent zeta diagonal
breaks the $\alpha$--$\zeta$ degeneracy that flattened the Dirichlet valley, so
off-canonical residuals rise faster under the corpus BC. The result is robust
across $M\in\{4,6,8\}$ and box size. So in-box global injectivity survives on the
corpus APS operator: A331's conclusion transfers, with a larger moat --- a genuine
corpus-faithful numerical in-box injectivity result for the free-weight version.
It is not an analytic global proof; the $\zeta$ sector is genuinely soft (A334),
the analytic statement is A336's monotonicity probe, and the corpus tie
$\zeta=\alpha^{5/4}$ remains 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\_P335.py} reaches 16 PASS / 0 FAIL and exits zero.

\end{document}
