% 334_Addendum_APSReconcile.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% P18-T2 sub-result (iii), the APS BOUNDARY-CONDITION RECONCILIATION: the whole
% (iii) thread (A302/A330/A331/A332/A333) built its radial operator on
% r=linspace(dr,1-dr,N-1) -- a DIRICHLET wall at r=1 (psi(1)=0 by construction),
% NOT the corpus APS BC (A329, P18 l.393-396). A333 measured |psi(1)|^2 ~ 1e-9
% and concluded the boundary term vanishes, hence an "exact rigid shift" R_sym=-2I;
% that vanishing is a DIRICHLET ARTIFACT. This addendum builds the CORPUS APS
% operator (R_nl=r^{-1}J_{l+1}(k r), k=2n+l+2, spectrum (2n+l+2)^2, eigenstates
% that do NOT vanish at r=1) and measures the zeta-residual on it. THE DECISIVE
% MEASUREMENT: under the corpus APS BC the boundary term is LARGE and state-
% dependent (range ~0.02..2.2), <psi_n|R_sym|psi_n> ranges -1.98..+0.24 across
% modes (NOT the rigid -2, not sign-definite), state-dependence std ~0.70 = ~420x
% A331's moat 1.67e-3. So A333's HIT does NOT transfer: zeta is GENUINELY soft and
% state-dependent under the corpus boundary condition, the original A332
% obstruction is real, and (iii)'s free-weight injectivity has a real analytic gap
% at the zeta sector under the corpus operator. Verifier: verify_P334.py (16 PASS /
% 0 FAIL, exit 0).

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\newtheorem{proposition}{Proposition}[section]
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\title{Addendum 334: The APS Boundary-Condition Reconciliation\\
\large The whole sub-result (iii) thread used a Dirichlet model; under the corpus
APS boundary condition the renormalisation sector is genuinely soft, and A333's
exact rigid shift is a Dirichlet artifact}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} GROUNDED $+$ REFUTATION (the corpus APS operator built and
validated; the zeta sector is genuinely soft and state-dependent under the corpus
BC; A333's exact-rigid-shift HIT is a Dirichlet artifact that does \emph{not}
transfer) \quad|\quad
\textbf{Addresses:} P18-T2 (iii) APS reconciliation; A333; A332; A331; A330; A329;
A302; A294 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P334.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. A329 established that the
corpus boundary condition is APS spectral projection at $r=1$ (P18 lines
393--396), whose no-potential spectrum is the integer ladder $E_{nl}=(2n+l+2)^2$
(P18 line 171). But every numerical addendum in the (iii) thread --- A302, A330,
A331, A332, and crucially A333 --- built its radial operator on the interior mesh
$r=\mathrm{linspace}(dr,1-dr,N-1)$, which forces $\psi=0$ at \emph{both} endpoints:
a \textbf{Dirichlet wall} at $r=1$. The Dirichlet $l=0$ spectrum is the squared
Bessel zeros $j_{1,n}^2=(14.68,49.22,\dots)$, which \emph{disagree} with the
corpus APS ladder $(4,16,36,\dots)$ at every mode (A329). So the (iii) thread is a
Dirichlet \emph{model}, not the corpus operator. A333 rested its central claim on
this: it proved the exact integration-by-parts identity
$\langle\psi|R_{\mathrm{sym}}|\psi\rangle=-2+\tfrac12|\psi(1)|^2/\langle\psi|\psi
\rangle$ (correct), measured $|\psi(1)|^2\sim10^{-9}$ on its eigenstates, and
concluded the boundary term vanishes, hence $R_{\mathrm{sym}}=-2I$ exactly --- an
``exact rigid shift'', $\zeta$ the cleanest sector. But $|\psi(1)|^2\sim10^{-9}$
is an \emph{artifact of the Dirichlet construction}, not a property of the corpus
APS eigenstates. \textbf{We build the corpus APS operator and measure the
zeta-residual on it.} The APS radial eigenfunctions are
$R_{nl}(r)=r^{-1}J_{l+1}(kr)$ with $k=2n+l+2$ (the APS quantisation), reproducing
the integer ladder $(2n+l+2)^2$ and distinct from the Dirichlet squared-Bessel
spectrum; they do \emph{not} vanish at the wall, $R_{nl}(1)=J_{l+1}(2n+l+2)\neq0$.
The exact identity is verified directly on these eigenstates (the numeric
$\langle R|r\partial_r|R\rangle/\!\langle R|R\rangle$ matches $-2+\tfrac12
R(1)^2/\!\langle R|R\rangle$ to $10^{-5}$). \textbf{The decisive measurement.}
Under the corpus APS BC the boundary term is \emph{large and state-dependent}
(range $0.018\dots2.238$, mean $0.79$, not $\sim10^{-9}$), and
$\langle\psi_n|R_{\mathrm{sym}}|\psi_n\rangle$ ranges from $-1.98$ to $+0.24$
across modes --- emphatically not the rigid $-2$, and not even sign-definite as a
shift (the $l=2,n=0$ mode gives $+0.24$). The state-dependence (std $0.70$) is
$\sim\!420\times$ A331's moat $1.67\times10^{-3}$. So A333's exact rigid shift is a
\textbf{Dirichlet artifact}: the boundary term A333 measured as $\sim10^{-9}$ and
killed is the \emph{live} term under the corpus boundary condition. The decoupling
$\Phi=(\text{rigid }-2\zeta)\oplus(\text{shape})$ \emph{fails} (the zeta diagonal
is mode-structured, cosine with the uniform vector $0.864$, not $1.0000$); $\zeta$
is genuinely the soft, state-dependent sector --- the original A332 obstruction,
not cleaned. The consequence for (iii) is honest: A333's HIT does not transfer;
free-weight injectivity rests on A331's in-box numerical scan with a \emph{real}
analytic gap at the zeta sector under the corpus operator, and the corpus tie
$\zeta=\alpha^{5/4}$ remains the clean closure for the corpus-posed family.
Verifier \texttt{verify\_P334.py}: 16 PASS / 0 FAIL. No published number changes.
\end{abstract}

\section{The probe and the discipline}

This is the corpus-faithfulness fix for the whole sub-result (iii) thread. The
cross-thread inconsistency it resolves is real and was flagged on verification of
A333: Heart 2 sub-result (i) settled the boundary condition on APS (A329, from P18
lines 393--396), while sub-result (iii) silently used Dirichlet. The binding
solution-bar discipline governs: do not force closure, do not manufacture a
fan-out, honesty over closure. Here that discipline is the substance --- the
honest discharge of the probe is to build the operator the corpus actually
specifies and report what the zeta sector does on \emph{it}, even though the answer
refutes A333's HIT. The brief-authoring discipline of re-grounding against the
canonical source (here A329 and P18 lines 393--396, not the convenient interior
mesh) is exactly what was missed across the (iii) thread and is corrected here.

The operator is P18's bulk Dirac$^2$ radial form on the unit ball, the same one
A329 fixed: $-d_r^2-(3/r)d_r+l(l+2)/r^2$ on the $r^3$ measure, with regularity at
$r=0$ and APS spectral projection at $r=1$ (P18 lines 393--396). Sub-result (iii)
is a question about \emph{this} operator; the Dirichlet model the (iii) thread used
is a different operator with a different spectrum.

\section{The corpus APS operator, built and validated}

The radial equation $[-d_r^2-(3/r)d_r+l(l+2)/r^2]R=ER$ is solved by the regular
root $R(r)=r^{-1}J_{l+1}(\sqrt{E}\,r)$ (the singular root $r^{-1}Y_{l+1}$ is not in
$L^2(r^3\,dr)$, A329 check 9). The quantisation is what distinguishes the two
boundary conditions, and it is the whole point.

\begin{proposition}[The corpus APS spectrum and its eigenstates]
Under the APS spectral projection at $r=1$ the admissible radial wavenumbers are
the integer ladder $k=2n+l+2$ (boundary Dirac base $l+2$, the integerised
$|D_{S^3}|=l+3/2$, plus radial overtone $2n$ in steps of two; A329 check 12), so
the spectrum is $E_{nl}=(2n+l+2)^2$: $l=0$ gives $4,16,36,64$; $l=1$ gives
$9,25,49,81$. This is exactly P18 line 171. The eigenfunctions
$R_{nl}(r)=r^{-1}J_{l+1}((2n+l+2)r)$ do \emph{not} vanish at the wall:
$R_{nl}(1)=J_{l+1}(2n+l+2)$, e.g.\ for $l=0$ the values are
$(0.577,-0.066,-0.277,0.235)$ --- $O(10^{-2}\dots10^{-1})$, not the $\sim10^{-9}$
A333 measured on the Dirichlet model.
\end{proposition}

The verifier reproduces this from scratch (checks 3--6): the integer ladder, the
APS quantisation structure, the nonzero boundary values, and the finite positive
$L^2(r^3\,dr)$ norms that make the residual a well-defined expectation. The Bessel
implementation (numpy-only series, the same as A329) is cross-checked against
known $J_1$ zeros and against a finite-difference derivative (checks 1--2). The
guardrail of the whole probe is that the APS spectrum \emph{must} come out
$(2n+l+2)^2$ and \emph{must} differ from the Dirichlet squared-Bessel zeros; both
are verified (checks 7--8). If the spectrum had come out as squared Bessel zeros we
would be back on Dirichlet --- the failure mode this probe exists to escape.

\section{The exact identity, verified on the APS eigenstates}

A333's integration-by-parts decomposition is correct and is reproduced here.
Under the measure $w=r^3$, for the first-order generator $R=r\partial_r$,
\[
\langle\psi|R_{\mathrm{sym}}|\psi\rangle_w
= -2 + \tfrac12\,\frac{|\psi(1)|^2}{\langle\psi|\psi\rangle_w},
\]
a state-independent bulk shift $-2$ plus the APS boundary term. What A333 got
wrong is not the identity but the value of $\psi(1)$. We verify the identity
\emph{directly} on the corpus APS eigenstates: the numeric
$\langle R|r\partial_r|R\rangle_w/\langle R|R\rangle_w$, computed from the
eigenfunction and its analytic derivative with no $-2$ assumed, equals
$-2+\tfrac12 R(1)^2/\langle R|R\rangle_w$ to $1.1\times10^{-5}$ across all modes
(check 9), and the residual is mesh-converged (Richardson-stable to
$10^{-5}$ under refinement, check 10) --- these are continuum values, not
discretisation.

\section{The decisive measurement: zeta is soft under APS}

\begin{proposition}[The zeta-residual under the corpus boundary condition]
On the corpus APS eigenstates the boundary term
$\tfrac12|\psi(1)|^2/\langle\psi|\psi\rangle$ is large and state-dependent: it
ranges from $0.018$ (the $l=0,n=1$ mode, $k=4$) to $2.238$ (the $l=1,n=0$ mode,
$k=3$), with mean $0.79$ --- not the $\sim10^{-9}$ A333 found on the Dirichlet
model. Consequently $\langle\psi_n|R_{\mathrm{sym}}|\psi_n\rangle$ ranges from
$-1.98$ to $+0.24$ across modes (for $l=0$: $-0.69,-1.97,-1.32,-1.26$),
emphatically \emph{not} the rigid $-2$, and not even sign-definite as a shift
(the $l=2,n=0$ mode gives $+0.24$). The state-dependence (standard deviation
across modes) is $0.70$, roughly $420\times$ A331's moat $1.67\times10^{-3}$, and
the zeta diagonal is mode-structured (cosine with the uniform vector $0.864$, not
$1.0000$).
\end{proposition}

This is the measurement A333 never ran on the right operator (checks 11--15). It
inverts A333's finding term by term. A333: boundary term $\sim10^{-9}$ (empirically
zero); APS: boundary term $0.02\dots2.2$ (large, state-dependent). A333:
$R_{\mathrm{sym}}=-2$ exactly for all $n$ (rigid shift); APS:
$R_{\mathrm{sym}}\in[-1.98,+0.24]$ across $n$ (state-dependent, not even a
fixed-sign shift). A333: residue $\to0$ in the continuum (discretisation); APS:
residual is $O(1)$, mesh-converged, far above the moat. The boundary term A333
killed --- $\psi(1)=J_{l+1}(2n+l+2)\neq0$ --- is precisely the live term the brief
hypothesised, and it is \emph{not} refuted under the corpus boundary condition.

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

\begin{remark}[The Dirichlet rigid-shift picture does not transfer]
Under the corpus APS boundary condition the zeta sector is genuinely soft and
state-dependent: $\langle\psi_n|R_{\mathrm{sym}}|\psi_n\rangle$ varies by $O(1)$
across modes, $\sim\!420\times$ A331's moat, and the rigid-offset decoupling
$\Phi=(\text{rigid }-2\zeta)\oplus(\text{shape})$ fails. A333's exact rigid shift
holds only for the Dirichlet model. So A332's lone non-sign-definite sector ---
the zeta obstruction --- is \emph{real} under the corpus operator, not cleaned;
A333's HIT does not transfer. Sub-result (iii) is less closed than A333 implied:
free-weight injectivity rests on A331's in-box numerical scan (itself a Dirichlet
model, also requiring an APS redo) with a genuine analytic gap at the zeta sector
under the corpus operator. The corpus tie $\zeta=\alpha^{5/4}$ (P18 line 348)
remains the clean closure for the corpus-posed family; the \emph{free}-weight
analytic global proof stays open, and is harder than A333 implied.
\end{remark}

\section{Gate ledger}

\begin{center}
\begin{tabular}{lll}
\toprule
Object & Gate & Finding \\
\midrule
Corpus APS operator built & DONE & $(2n+l+2)^2$ spectrum, $\psi(1)\neq0$, distinct from Dirichlet \\
Exact IBP identity on APS states & DERIVED & $\langle R|r\partial_r|R\rangle/\!\langle R|R\rangle=-2+\tfrac12 R(1)^2/\!\langle R|R\rangle$ \\
APS boundary term & LIVE & large \& state-dependent ($0.02\dots2.2$), not $\sim10^{-9}$ \\
$\langle\psi_n|R_{\mathrm{sym}}|\psi_n\rangle$ & STATE-DEP & ranges $-1.98\dots+0.24$, not the rigid $-2$ \\
A333 exact rigid shift & DIRICHLET ARTIFACT & does not transfer to the corpus operator \\
Zeta-residual vs A331 moat & ABOVE MOAT & std $0.70\approx420\times$ moat; threatens injectivity \\
(iii) free-weight injectivity & OPEN (corpus BC) & real analytic gap at zeta under APS; tie clean \\
\bottomrule
\end{tabular}
\end{center}

\section{Pre-registered fan-out}

The probe decisively settles the orientation question: under the corpus BC the
zeta sector is soft, so the (iii) injectivity results must be re-established on the
APS operator, not transferred from the Dirichlet model. Two faithful fan-out
probes follow; we do not manufacture beyond them.

\begin{enumerate}
\item \textbf{A335 --- the bounded-box second-preimage scan on the APS operator.}
Redo A331's free-weight scan (the $\alpha$--$\zeta$ ray with $\gamma,\beta$
re-optimised, the moat, the zeta-valley floor) on the corpus APS operator built
here, where $\psi(1)\neq0$ and the boundary residual is live. The question is
whether the in-box global result (no second preimage, canonical the unique
machine-zero) survives the boundary term, or whether the genuinely soft zeta
sector opens a second preimage the Dirichlet model hid.
\item \textbf{A336 --- the Hellmann--Feynman monotonicity on the APS operator.}
Redo A332's structural-monotonicity argument on the 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 this 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 under the
corpus BC or whether the soft sector breaks it outright.
\end{enumerate}

If A335 finds no second preimage and A336 re-establishes the three shape sectors,
(iii) stands as a corpus-faithful in-box numerical injectivity result with the
zeta sector an honest soft direction (the tie the clean closure), genuinely
parallel to where A331/A332 left the Dirichlet model --- but now on the right
operator. If A335 finds a second preimage, the free-weight version is
non-injective under the corpus BC and the tie is load-bearing, not merely clean.
We report whichever is true.

\section{Conclusion}

The probe builds the corpus APS radial operator the whole sub-result (iii) thread
should have used --- $R_{nl}(r)=r^{-1}J_{l+1}((2n+l+2)r)$, spectrum $(2n+l+2)^2$
(P18 line 171), eigenstates that do not vanish at the wall --- validates it against
the integer ladder and against its distinctness from the Dirichlet squared-Bessel
spectrum, and measures the zeta-sector residual on it. The exact integration-by-
parts identity $\langle\psi|R_{\mathrm{sym}}|\psi\rangle=-2+\tfrac12|\psi(1)|^2/
\langle\psi|\psi\rangle$ (A333's, correct) is verified directly on the APS
eigenstates. Under the corpus boundary condition the boundary term is large and
state-dependent ($0.02\dots2.2$, not $\sim10^{-9}$), and
$\langle\psi_n|R_{\mathrm{sym}}|\psi_n\rangle$ ranges from $-1.98$ to $+0.24$
across modes --- not the rigid $-2$, not sign-definite, with $O(1)$
state-dependence some $420\times$ A331's moat. So A333's exact rigid shift is a
Dirichlet artifact that does not transfer: the boundary term A333 killed is the
live term, $\zeta$ is genuinely the soft, state-dependent sector under the corpus
operator, and the rigid-offset decoupling fails. Sub-result (iii) is therefore
less closed than A333 implied: its free-weight injectivity has a real analytic gap
at the zeta sector under the corpus boundary condition, A331's in-box scan and
A332's monotonicity must be redone on the APS operator (A335/A336 pre-registered),
and the corpus tie $\zeta=\alpha^{5/4}$ remains the clean closure for the
corpus-posed family. The cross-thread inconsistency --- Heart 2 (i) on APS (A329)
versus (iii) on Dirichlet --- is resolved in favour of APS: the corpus operator
carries the APS BC, and (iii) must be argued on it. 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\_P334.py} reaches
16 PASS / 0 FAIL and exits zero.

\end{document}
