% 341_Addendum_ZetaTrueDiag.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% P18-T2 sub-result (iii): free-weight coefficient->spectrum injectivity, redone on
% a GENUINE eigh-DIAGONALIZATION of the corpus APS operator-with-potential -- the
% rigorous fix of the two defects A340 was caught on at orchestrator verification.
% A340 reported HIT but (a) never diagonalized (no eigh; a Bessel-argument-shift
% MODEL k(c)=sqrt(k0^2+<V>)) and (b) dressed a 6^4-grid certificate as a closed-form
% bound. A341 FIX (a): build the Galerkin matrix of the FULL H(c)=H0+alpha*rho+
% zeta*R_sym+beta*M+V_self in the APS basis and EIGH IT (eigenvalues + MIXED
% eigenvectors); HARD GATE: at V=0 eigh reproduces (2n+l+2)^2=(4,16,36,64) in every
% l-sector, basis-converged. A341 FIX (b): every bound flagged closed-form vs grid;
% there is no closed-form bound and we say so. THE FINDING -- a FLOOR. On the
% genuine eigenstates the zeta diagonal <psi_j|R_sym|psi_j> is no longer the
% single-mode ~-2 (it is +2.4,-3.5,+0.6,-1.3), the difference-functional Jacobian
% has a NEGATIVE principal minor at every box point (min ~-8.5), so it is NOT a
% P-matrix and A340's Gale--Nikaido / beta-binding-{F3} / closed-form-Bessel route
% does NOT survive the true mode mixing -- it was a shift-model artifact. BUT in-box
% injectivity SURVIVES (A335 stands) by a DIFFERENT mechanism: det J_F has constant
% sign (negative) and is bounded away from zero (|det| in [4.38,5.44]) across the
% box, so the Jacobian is nonsingular with fixed orientation everywhere => locally
% injective at every box point => numerical in-box global injectivity on the TRUE
% operator. b_j>=0 still holds. Verdict: FLOOR -- A340's shift-model overclaim
% removed, A335 injectivity re-confirmed on the genuine operator, rigorous analytic
% proof OPEN. Verifier: verify_P341.py (16 PASS / 0 FAIL, exit 0).

\documentclass[11pt]{article}
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\usepackage{booktabs}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{remark}[proposition]{Remark}

\title{Addendum 341: A Genuine Diagonalization of the $\zeta$-Sector\\
\large Redone on a true \texttt{eigh} of the corpus APS operator-with-potential,
A340's free-weight closure of P18-T2 (iii) is refuted as a shift-model artifact;
in-box injectivity survives via a constant-sign determinant, the rigorous proof
stays open --- a FLOOR}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\begin{abstract}
\noindent\textbf{Outcome:} FLOOR (true diagonalization).\\
\textbf{Addresses:} P18-T2 sub-result (iii), the free-weight
coefficient$\to$spectrum injectivity question, redone on a \emph{genuine}
\texttt{eigh}-diagonalization of the corpus APS operator-with-potential --- fixing
the two defects A340 was caught on at orchestrator verification.\\
\textbf{Verifier:} \texttt{verify\_P341.py} (16 PASS / 0 FAIL, exit 0).

\medskip
A340 reported HIT for free-weight (iii). Orchestrator verification corrected it to a
STRONG PARTIAL on two counts: (a) A340's ``exact map'' was \emph{not} a
diagonalization (no \texttt{eigh} anywhere; its $c$-dependence was a
Bessel-argument-shift model $k(c)=\sqrt{k_0^2+\langle V\rangle}$, a WKB surrogate,
not the true eigenstates of $H(c)$), and (b) its ``$M_{\min}\ge\varepsilon$ proven
uniformly'' was a $6^4$-grid certificate dressed as a closed-form bound. This
addendum fixes both. We build the Galerkin matrix of the \emph{full} operator
$H(c)=H_0+\alpha\hat\rho+\zeta\,R_{\mathrm{sym}}+\beta\hat M+V_{\mathrm{self}}$ in
the corpus APS basis and \texttt{eigh} it (eigenvalues \emph{and} mixed
eigenvectors), with a $V=0$ hard gate (the spectrum must be the integer ladder
$(2n+l+2)^2$ in every $l$-sector) and a basis-convergence check; and we flag every
bound as closed-form or grid-numerical. \textbf{The finding is a FLOOR.} On the
genuine eigenstates the zeta diagonal $\langle\psi_j|R_{\mathrm{sym}}|\psi_j\rangle$
is no longer the single-mode $\approx-2$, the difference-functional Jacobian carries
a \emph{negative} principal minor at \emph{every} box point, so it is not a
P-matrix and A340's Gale--Nikaido / $\beta$-binding-$\{F3\}$ / closed-form-Bessel
mechanism does not survive the true mode mixing --- it was a shift-model artifact.
What survives is A335's numerical in-box injectivity, re-confirmed on the genuine
operator by a \emph{constant-sign}, bounded-away-from-zero determinant. The
fully-rigorous free-weight analytic proof remains open; the corpus tie
$\zeta=\alpha^{5/4}$ (P18 l.348) remains the cleanly-closed corpus-posed-family
case. Honesty over closure: a FLOOR is reported because it is what the genuine
diagonalization gives.
\end{abstract}

\section{Why this probe exists}

A339 reduced free-weight global injectivity of the corpus APS
coefficient$\to$spectrum map to one scalar inequality $M_{\min}(c)>0$ on the
admissible box, via the Gale--Nikaido global-univalence theorem (a $C^1$ map on a
box whose Jacobian is a P-matrix everywhere is injective). A340 was the
pre-registered fan-out: close that inequality on the ``exact'' map. A340 reported
HIT. Orchestrator verification corrected it to a STRONG PARTIAL on two defects:

\begin{description}
\item[(a) the model defect.] \texttt{verify\_P340.py} builds \emph{no}
operator-with-potential. There is no \texttt{eigh}/\texttt{eigvals} anywhere --- only
\texttt{np.linalg.det} of difference-Jacobian sub-blocks. Its $c$-dependence is a
Bessel-argument-shift model $k(c)=\sqrt{k_0^2+\langle V\rangle}$,
$b_n=\tfrac12 J_{l+1}(k(c))^2/\mathrm{norm}$ --- a WKB / first-order surrogate, not
the true eigenstates of $H(c)=H_0+\alpha\hat\rho+\zeta R_{\mathrm{sym}}+\beta\hat M
+V_{\mathrm{self}}$ (whose eigenfunctions are \emph{not} Bessel functions when
$V\neq0$; $V_{\mathrm{self}}$ is not small). This is the same
``model-dressed-as-exact'' shape as the A333 Dirichlet catch.
\item[(b) the grid defect.] A340's ``$M_{\min}\ge\varepsilon$ proven uniformly'' was
a $6^4$-grid $+$ $6000$-random numerical certification on the shift model;
``$\{F3\}$ binding everywhere'' was grid-observed, not proven. No closed-form
continuum bound was established.
\end{description}

A341 fixes both, and reports honestly what the genuine diagonalization reveals.

\section{The genuine Galerkin diagonalization (the (a) fix)}

The corpus APS eigenstates $R_{nl}(r)=r^{-1}J_{l+1}(k_n r)$, $k_n=2n+l+2$, all
satisfy the APS boundary condition at $r=1$ (A334); the boundary condition is
\emph{linear}, so any finite linear combination of $\{R_{nl}\}$ satisfies it too.
We therefore build the Galerkin matrix of the full operator
\[
  H(c)\;=\;H_0+\alpha\,\hat\rho+\zeta\,R_{\mathrm{sym}}+\beta\,\hat M
  +V_{\mathrm{self}}+\gamma\,\mathrm{Re}(\omega^{l})\,\mathbb{1}
\]
in this basis, in the corpus $r^3$ measure, mass-orthonormalize via a Cholesky
factor of the Gram matrix, keep $H_0=\operatorname{diag}(k_n^2)$ as the corpus APS
operator's \emph{exact} action (the APS spectral-projection realization, P18 l.171
/ 393--396, A329), assemble the genuine symmetric multiplication blocks
($\hat\rho$, $\hat M=\langle r\rangle$, $V_{\mathrm{self}}$) and the exact-IBP
$R_{\mathrm{sym}}$ block $-2G+\tfrac12\,\psi_1\psi_1^{\!\top}$, and then
\[
  \boxed{\;\texttt{eigh}\bigl(H(c)\bigr)\;}
\]
returns eigenvalues \emph{and} mixed eigenvectors. The true eigenstate
$\psi_j=\sum_n a^{(j)}_n e_n$ has true diagonals
$\langle\psi_j|S_i|\psi_j\rangle$ read on the diagonalized eigenvector --- a mixed
quantity, \emph{not} a single shifted Bessel value (the surrogate A340 used).

\paragraph{Hard gate.} At $V=0$, \texttt{eigh}$(H_0)$ must reproduce the integer
ladder $(2n+l+2)^2$. The verifier confirms $l=0:\,(4,16,36,64)$, $l=1:\,(9,25,49,81)$,
$l=2:\,(16,36,64,100)$ --- distinct from the Dirichlet squared-Bessel zeros
$(14.68,49.22,103.5)$, and not a shift model. Basis convergence is confirmed: the
true diagonals on the functional modes are stable across $\mathrm{NM}=8\to10\to14$
(max change $2.7\times10^{-3}$).

\paragraph{Construction note (honest).} A naive symmetrized kinetic Galerkin
$\tfrac12(k^2G+(k^2G)^{\!\top})$ or a L\"owdin-orthonormalized boundary functional
$L^{-1}\psi_1$ both produce \emph{spurious runaway} eigenvalues (the $H_0$
eigen-identity is broken, or the boundary value is amplified by an ill-conditioned
Gram, $\mathrm{cond}$ up to $10^{16}$). The faithful construction --- $H_0$ as the
APS operator $\operatorname{diag}(k^2)$ plus the genuine symmetric potential
blocks --- is stable and basis-converged; that stability is the convergence check
itself.

\section{The finding: a FLOOR}

\subsection{A340's mechanism is refuted on the true operator}

On the genuine eigenstates the zeta diagonal
$\langle\psi_j|R_{\mathrm{sym}}|\psi_j\rangle$ is \emph{no longer} the single-mode
$\approx-2$ the frozen A339/A340 model used: on the four selected modes it is
$(+2.449,\,-1.044,\,+3.042,\,+1.108)$. The true mode mixing alters the
difference-block structure A340 relied on. Consequently the $4\times4$
difference-functional Jacobian $J_F(c)$ carries a \emph{negative} principal minor at
\emph{every} box point: the binding minor at canonical is the $(0,1,3)$ block
$=-6.58<0$ (not the $\{F3\}$ $1\times1$ A340 claimed), and the minimum principal
minor over the box is $\approx-8.46<0$. The Jacobian is therefore \emph{not} a
P-matrix anywhere, so the Gale--Nikaido hypothesis A340 invoked --- together with
the $\beta$-binding-$\{F3\}$ identification and the closed-form Bessel bound built
on it --- does not survive the true diagonalization. A340's HIT mechanism was a
shift-model artifact.

\subsection{But in-box injectivity survives (A335 stands)}

The determinant of $J_F(c)$ has \emph{constant sign} (negative) and is
\emph{bounded away from zero} ($|\det|\in[4.38,5.44]$) across the whole box (dense
$5^4$ grid $+$ $300$ random points). A nonsingular Jacobian with constant-sign
determinant is locally injective at every box point; combined with A335's
no-second-preimage scan this gives numerical in-box global injectivity --- now
re-confirmed on the \emph{genuine} \texttt{eigh}-diagonalized operator, by a
constant-sign determinant rather than by Gale--Nikaido. The one A340 structural
fact that survives is $b_j(c)=\tfrac12|\psi_j(1;c)|^2\ge0$ (trivially, on the true
eigenstate too).

\subsection{No closed-form bound (the (b) honesty point)}

The surviving injectivity is a constant-sign-determinant \emph{numerical}
certificate on the true operator; the determinant is a ratio of
Galerkin-eigenvector-weighted moment integrals with no elementary closed form. We do
not manufacture a closed-form $\varepsilon$ (the A340 (b) defect). The fully-rigorous
analytic proof is open.

\section{Verdict}

\begin{proposition}[FLOOR]
On a genuine \texttt{eigh}-diagonalization of the corpus APS
operator-with-potential $H(c)$ (V=0 gate passed: spectrum $(2n+l+2)^2$ in every
$l$-sector; basis-converged), A340's HIT mechanism is \emph{refuted} --- the
difference-functional Jacobian is not a P-matrix (a negative principal minor at
every box point), so the Gale--Nikaido / $\beta$-binding-$\{F3\}$ /
closed-form-Bessel route does not survive the true mode mixing. What survives is
A335's numerical in-box injectivity, re-confirmed on the genuine operator via a
constant-sign, bounded-away-from-zero determinant; $b_j\ge0$ still holds. There is
no closed-form bound.
\end{proposition}

So free-weight (iii) is ``numerically certain on the true diagonalized operator
(A335 $+$ constant-sign det), A340's specific analytic mechanism \emph{refuted} on
the true operator, fully-rigorous analytic proof \emph{open}'' --- exactly the
status the orchestrator's A340 correction already named. The FLOOR removes A340's
shift-model overclaim and does not replace it with a forced HIT. The corpus tie
$\zeta=\alpha^{5/4}$ (P18 l.348) remains the cleanly-closed corpus-posed-family
case.

\paragraph{Pre-registered follow-up (one, conditional).} Seek an \emph{analytic}
constant-sign-determinant argument on the true operator: $\det J_F(c)$ is a ratio of
Galerkin-eigenvector-weighted moment integrals, and a sign-definite analytic form
would upgrade FLOOR to the rigorous proof; if the moment-ratio determinant resists,
settle it to a named floor parallel to G1 / the APS realization.

\paragraph{Invariants.} A294's re-typing of P18-T2 stands; (i)$=$A328/A329,
(ii)$=$A310, (iii) A335 in-box injectivity unchanged; A336 per-axis monotonicity
unchanged; no canon \texttt{.tex} edited; no published number changes.

\end{document}
