% 342_Addendum_DetSignTotalPositivity.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% P18-T2 sub-result (iii): the ANALYTIC attempt to make A341's CONSTANT-SIGN
% det J_F(c) (=> local injectivity everywhere) rigorous, via total positivity /
% generalized-Vandermonde theory (Karlin, Gantmacher-Krein) on the GENUINE
% eigh-diagonalized corpus APS operator-with-potential. A341 established that the
% 4x4 difference-functional Jacobian J_F(c) has a constant-sign, bounded-away-from-
% zero determinant over the box -- a NUMERICAL fact -- and named the pre-registered
% conditional follow-up: make it analytic. A342 runs that attack. THE FINDING: the
% briefed primary attack (total positivity) FAILS -- the kernel is not totally
% positive (the character row is not a function of the radial node k; the genuine
% APS moment columns are non-monotone in k, so the 2x2 minors are sign-indefinite).
% A DIFFERENT analytic handle -- a pure-mode / mixing split -- controls the sign:
% det J_F^pure is basis-converged and negative, the character column is exactly
% c-independent, the mixing perturbation is column-localized, and a first-order
% multilinear perturbation bound (<= 1.728 over the box) stays below |det_pure| =
% 3.812, so the sign cannot flip => constant-sign det => LOCAL injectivity
% everywhere. CEILING (explicit): local, NOT global (det!=0 is necessary not
% sufficient); global in-box still rests on A335's scan. The pure-mode sign is
% semi-analytic; the mixing bound is basis-converged + structurally explained but
% NUMERICAL, not closed-form. Verdict: PARTIAL -- the TP route is closed off as a
% dead end, the A341 floor is sharpened to a single named perturbation inequality.
% Verifier: verify_P342.py (18 PASS / 0 FAIL, exit 0).

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\newtheorem{theorem}{Theorem}[section]
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\title{Addendum 342: The Determinant-Sign Question for the
$\zeta$-Sector\\
\large Total positivity fails on the genuine \texttt{eigh}-diagonalized corpus APS
operator; a pure-mode / mixing split controls A341's constant-sign determinant down
to one numerical perturbation inequality, giving local injectivity everywhere ---
a PARTIAL}
\author{TOE Corpus}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
\noindent\textbf{Outcome:} PARTIAL (the briefed total-positivity route fails; a
pure-mode / mixing split controls the sign, modulo one numerical bound).\\
\textbf{Addresses:} P18-T2 sub-result (iii), the analytic attempt to prove
$\det J_F(c)$ has constant sign over the admissible box --- A341's pre-registered
conditional follow-up, via total positivity / generalized-Vandermonde theory
(Karlin; Gantmacher--Krein), on the \emph{genuine} \texttt{eigh}-diagonalized
corpus APS operator-with-potential.\\
\textbf{Verifier:} \texttt{verify\_P342.py} (18 PASS / 0 FAIL, exit 0).

\medskip
A341 established, on a genuine \texttt{eigh}-diagonalization of
$H(c)=H_0+\alpha\hat\rho+\zeta R_{\mathrm{sym}}+\beta\hat M+V_{\mathrm{self}}$, that
the $4\times4$ difference-functional Jacobian $J_F(c)$ is \emph{not} a P-matrix but
has a \emph{constant-sign}, bounded-away-from-zero determinant
($|\det|\in[4.38,5.44]$) over the box --- a \emph{numerical} fact giving local
injectivity everywhere. A341's pre-registered conditional follow-up: make that
constant-sign claim \emph{analytic}. The natural tool is total positivity: a
determinant $\det[f_i(k_j)]$ of a totally-positive kernel evaluated at \emph{ordered}
nodes has a definite sign (Gantmacher--Krein, Karlin). \textbf{The finding is a
PARTIAL.} The briefed primary attack \emph{fails}: the kernel is \emph{not} totally
positive --- (a) the character row $\operatorname{Re}(\omega^{l})$ is not a function
of the radial node $k$ (it depends on $l\bmod3$), so the generalized-Vandermonde
hypothesis does not even apply to it; (b) the genuine APS moment columns
$\langle\hat\rho\rangle$, $\langle r\rangle$ are \emph{non-monotone} in $k$ (A336's
shift-model monotonicity does not survive the true mixing), so the kernel's
$2\times2$ minors are sign-indefinite (20 positive, 16 negative). What \emph{does}
work is a pure-mode / mixing split: $\det J_F^{\mathrm{pure}}$ (diagonals on the
unmixed APS modes, no \texttt{eigh}) is basis-converged ($-3.81210$ at
$\mathrm{NM}=8/10/14$) and negative, matching A341's true sign; the character
column is \emph{exactly} $c$-independent; the mixing perturbation is
column-localized; and the first-order multilinear perturbation bound ($\le1.728$
over the box) stays below $|\det J_F^{\mathrm{pure}}|=3.812$, so the sign cannot
flip --- a constant-sign nonvanishing determinant, hence \emph{local} injectivity at
every box point. \textbf{Ceiling (explicit):} this is local injectivity everywhere
--- an upgrade over A341's purely numerical determinant --- \emph{not} global in-box
injectivity ($\det\neq0$ is necessary, not sufficient; cf.\ $(e^x\cos y,e^x\sin y)$);
global in-box still rests on A335's no-second-preimage scan. The pure-mode sign is
semi-analytic; the mixing bound is basis-converged and structurally explained but
\emph{numerical}, not closed-form. Verdict: PARTIAL --- the total-positivity route is
closed off as a dead end, and the A341 floor is \emph{sharpened} from ``a
constant-sign determinant with no analytic handle'' to ``a constant-sign determinant
reduced to a single explicit perturbation inequality, modulo one numerical bound.''
Honesty over closure: a PARTIAL is reported because the TP route fails and the
surviving bound is numerical --- no fourth overclaim on this thread.
\end{abstract}

\section{Why this probe exists}

A341 redid A340's free-weight (iii) probe on a \emph{genuine}
\texttt{eigh}-diagonalization of the corpus APS operator-with-potential and reported
an honest FLOOR: A340's P-matrix / Gale--Nikaido / $\beta$-binding mechanism was a
shift-model artifact (the Jacobian is not a P-matrix on the true mixed eigenstates),
but in-box injectivity \emph{survived} by a different mechanism --- the determinant
of the $4\times4$ difference-functional Jacobian $J_F(c)$ has \emph{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, hence locally
injective at every box point. That determinant check was \emph{numerical}. A341's
single pre-registered conditional follow-up named the present probe: try to make
``$\det J_F(c)\neq0$ (constant sign) for all $c$ in the box'' \emph{analytic}.

The natural analytic tool is total positivity. Classical theory (Karlin,
\emph{Total Positivity}; Gantmacher--Krein) says that a determinant
$\det[f_i(k_j)]$ of a totally-positive (sign-regular) kernel evaluated at
\emph{ordered} nodes $k_1<\dots<k_m$ has a \emph{definite} sign --- a generalized
Vandermonde. If the columns of $J_F$ were moment-type functions of an ordered radial
mode $k$, and that kernel were totally positive, the constant sign would be
\emph{forced} closed-form. This addendum runs that attack on the genuine operator.

\paragraph{The honest ceiling, stated up front.} Proving $\det J_F(c)\neq0$
everywhere gives \emph{local} injectivity everywhere --- an upgrade over A341's
numerical determinant --- \emph{not} global in-box injectivity. A nonvanishing
Jacobian is necessary but not sufficient for global univalence: the standard
counterexample $(x,y)\mapsto(e^x\cos y,\,e^x\sin y)$ has nowhere-vanishing Jacobian
yet is not injective. So the \emph{maximum} this probe can deliver is ``local
injectivity everywhere, rigorously''; the global in-box no-second-preimage statement
still rests on A335's scan. We do \emph{not} claim (iii) closes globally from a
determinant-sign result.

\section{The genuine operator (re-grounded from A341)}

We carry A341's construction verbatim. The corpus APS eigenstates
$R_{nl}(r)=r^{-1}J_{l+1}(k_n r)$, $k_n=2n+l+2$, all satisfy the linear APS boundary
condition (A334), so any finite combination does; we build the Galerkin matrix of
the full $H(c)$ in this basis (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 exact action (A329 / P18 l.171), 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 \texttt{eigh} it. The verifier confirms the
$V=0$ hard gate (spectrum $(2n+l+2)^2$ in every $l$-sector, distinct from the
Dirichlet squared-Bessel zeros), a live off-diagonal coupling (max off-diag $1.13$),
mixed eigenvectors, and A341's constant-sign determinant over the box. The
difference functionals are A339's: offset $F_0=\lambda_{(1,2)}$ and three shape
differences against partners $(0,0)$, $(2,2)$, $(1,1)$.

\section{The briefed primary attack: total positivity --- it fails}

For the generalized-Vandermonde sign theorem the kernel $K=[g_i(k_j)]$ must be a
function of the ordered node $k$ and totally positive. Two obstructions, both fatal.

\paragraph{(a) The character row is not a function of $k$.} The character functional
returns $\operatorname{Re}(\omega^{l})\in\{1,-\tfrac12,-\tfrac12\}$ by $l$-sector
($l\bmod3$), \emph{not} by the radial mode $k$. So $K$ is not even of the form
$[f_i(k_j)]$ in its character row, and the Karlin / Gantmacher--Krein hypothesis
does not apply to it at all.

\paragraph{(b) The moment columns are non-monotone in $k$.} On the genuine APS
moments, over the $k$-ordered modes $k\in\{2,5,7,8\}$,
\[
  \langle\hat\rho\rangle=(269.16,\,235.98,\,218.74,\,229.81),\qquad
  \langle r\rangle=(0.754,\,0.689,\,0.650,\,0.678),
\]
both \emph{non-monotone} (each rises at the last node). A336's shift-model
monotonicity of $\langle r\rangle$ in $k$ does \emph{not} survive the genuine mode
structure. A totally-positive kernel needs monotone (more precisely, sign-regular)
generators; these are not. Consequently the kernel's $2\times2$ minors are
sign-indefinite (20 positive, 16 negative): $K$ is \emph{not} totally positive, and
the classical generalized-Vandermonde sign theorem gives \emph{no} closed-form
pure-mode sign. The briefed primary attack fails --- recorded honestly.

\section{What does work: a pure-mode / mixing split}

Even though $K$ is not totally positive, the determinant sign is controlled, by a
different route.

\paragraph{The pure-mode determinant is basis-converged and negative.} Reading the
sector diagonals on the \emph{unmixed} APS basis modes $e_n$ of $H_0$ (no
\texttt{eigh}) gives a pure-mode Jacobian $J_F^{\mathrm{pure}}$ with
$\det J_F^{\mathrm{pure}}=-3.81210$, stable to five digits across
$\mathrm{NM}=8/10/14$ --- a genuine continuum value (a semi-analytic quantity: a
combination of Bessel-weighted moment integrals, no diagonalization needed). Its
sign matches A341's true determinant sign (negative): the unmixed structure carries
the sign, and the mixing is a perturbation, not a sign-reverser.

\paragraph{The character column is exactly $c$-independent.} The mixing perturbation
$\Delta(c)=J_F(c)-J_F^{\mathrm{pure}}$ leaves the character column \emph{exactly}
unchanged (max $|\Delta_{\mathrm{char}}|=0$ over the box). Only the moment columns
move; among those, $\|\Delta_{\mathrm{rsym}}\|\le0.52$,
$\|\Delta_{\mathrm{mom}}\|\le0.07$, and $\|\Delta_{\rho}\|$ is large in norm but acts
on a column of magnitude $O(220$--$270)$, a small \emph{relative} move (eigenvector
off-axis component $\le0.157$).

\paragraph{The first-order multilinear bound stays below $|\det J_F^{\mathrm{pure}}|$.}
By multilinearity of the determinant in columns,
$\det(J_F^{\mathrm{pure}}+\Delta)=\det J_F^{\mathrm{pure}}+(\text{column-replacement
terms})$. The first-order term, summed in absolute value
$\sum_j\bigl|\det(J_F^{\mathrm{pure}}\text{ with column }j\to\Delta_j(c))\bigr|$,
is $\le1.728$ over the box, \emph{below} $|\det J_F^{\mathrm{pure}}|=3.812$.
Equivalently $\max_c|\det J_F(c)-\det J_F^{\mathrm{pure}}|=1.630<3.812$. So
$\det J_F(c)$ keeps the pure-mode negative sign and is nonzero everywhere: a
constant-sign, nonvanishing determinant, hence local injectivity at every box point.

\section{Verdict}

\begin{proposition}[PARTIAL]
On the genuine \texttt{eigh}-diagonalized corpus APS operator-with-potential (V=0
gate passed; A341's constant-sign determinant re-confirmed), the total-positivity /
generalized-Vandermonde route \emph{fails}: the difference-functional kernel is not
totally positive --- its character row is not a function of the radial node $k$, and
the genuine APS moment columns are non-monotone in $k$ (so the $2\times2$ minors are
sign-indefinite, 20 positive / 16 negative). A pure-mode / mixing split instead
controls the sign: $\det J_F^{\mathrm{pure}}$ is basis-converged and negative, the
character column is exactly $c$-independent, and the first-order multilinear
perturbation bound ($\le1.728$ over the box) stays below
$|\det J_F^{\mathrm{pure}}|=3.812$, so the determinant is constant-sign and
nonvanishing --- giving \emph{local} injectivity at every box point. The pure-mode
sign is semi-analytic; the perturbation bound is basis-converged and structurally
explained but \emph{numerical}, not closed-form.
\end{proposition}

So sub-result (iii) advances as follows: A341's constant-sign determinant (local
injectivity everywhere) is now \emph{analytically structured} --- the total-positivity
route is closed off as a dead end (the kernel is provably not totally positive), and
the surviving constant-sign claim is reduced to one explicit perturbation inequality
(first-order multilinear bound $<|\det J_F^{\mathrm{pure}}|$ over the box). That
inequality is currently certified \emph{numerically} (basis-converged grid), not in
closed form. The ceiling stands: this is \emph{local} injectivity everywhere, not
global; the global in-box no-second-preimage still rests on A335's scan. 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).} Bound the first-order
multilinear perturbation term in \emph{closed form} --- e.g.\ via a Cauchy--Schwarz
estimate of the eigenvector off-axis component times the column moment spreads, using
the bounded mixing ($\le0.157$) and the frozen character column. A closed-form bound
below $|\det J_F^{\mathrm{pure}}|$ would upgrade PARTIAL to the rigorous
local-injectivity proof; if the perturbation term resists a closed form, settle the
mixing bound 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; A341's constant-sign determinant unchanged; no canon \texttt{.tex} edited;
no published number changes.

\end{document}
