% 339_Addendum_ZetaGaleNikaido.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% P18-T2 sub-result (iii): the GALE--NIKAIDO global-univalence route to a full
% ANALYTIC free-weight global-injectivity proof at the zeta sector, on the corpus
% APS operator. This is the lone live NON-FLOOR gap across both open hearts: after
% A334 reconciled (iii) to the corpus operator, A335 (numerical) found in-box
% injectivity survives with a larger moat and A336 (analytic, per-axis) found three
% shape sectors monotone but the zeta sector NON-monotone (its Hellmann-Feynman
% diagonal sign-flips). This addendum converts the global question into a pointwise
% P-matrix condition via Gale--Nikaido, shows the offset+difference functionals make
% the 4x4 Jacobian a P-matrix (min principal minor +0.0305) by exploiting the exact
% IBP split (zeta column = -2*ones + b, b_n>=0), and -- honestly -- finds the result
% is a sharp PARTIAL: in the affine first-order model the box proof goes through but
% is rank-trivial, and the load-bearing EXACT claim reduces to ONE explicit
% principal-minor inequality M_min(c) > 0 on the box, true at canonical and
% numerically in-box (A335) but not closed because the APS boundary values are
% Bessel-valued. Verifier: verify_P339.py (17 PASS / 0 FAIL, exit 0).

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[margin=1in]{geometry}
\usepackage{booktabs}
\usepackage{hyperref}

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

\title{Addendum 339: The Gale--Nikaid\^o Route to $\zeta$-Sector Global
Injectivity\\
\large On the corpus APS operator the offset--difference functionals make the
coefficient Jacobian a P-matrix; the lone non-floor gap sharpens to a single
principal-minor inequality on the box}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} PARTIAL (Gale--Nikaid\^o reduces free-weight $\zeta$-sector
global injectivity to a pointwise P-matrix condition; the offset--difference
functionals achieve the P-matrix at canonical, min principal minor $+0.0305$; in
the affine first-order model this is a P-matrix on the whole box, giving
Gale--Nikaid\^o injectivity, but that model is rank-injective trivially, so the
load-bearing exact claim reduces to one named principal-minor inequality on the
box --- true at canonical, numerical in-box, open in closed form) \quad|\quad
\textbf{Addresses:} P18-T2 (iii) $\zeta$-sector global injectivity; A336; A335;
A334; A333; A332; A331; A330; A302; A294 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P339.py} --- 17 PASS / 0 FAIL, exit 0

\begin{abstract}
After A334 reconciled the whole sub-result (iii) thread to the corpus APS operator,
two fan-outs settled both halves \emph{on} that operator: A335 (numerical) found
in-box global injectivity survives with a larger moat ($6.20\times10^{-3}$ vs
A331's Dirichlet $1.67\times10^{-3}$), and A336 (analytic, per-axis) found the three
shape sectors $\alpha\rho,\beta M,\gamma\,\mathrm{Re}\,\omega^k$ stay sign-definite
hence per-axis monotone/injective, but the renormalisation sector $\zeta R$,
$R=r\partial_r$, is \emph{non}-monotone --- its Hellmann--Feynman diagonal
$\langle\psi_n|R_{\mathrm{sym}}|\psi_n\rangle$ ranges $-1.98$ to $+0.24$ and
\emph{sign-flips}. That sign-flip is the single obstruction to a clean analytic
free-weight global-injectivity proof, and it is the lone live \emph{non-floor} gap
across both open hearts. This addendum attacks it via the \textbf{Gale--Nikaid\^o
global univalence theorem}: a $C^1$ map on a closed box whose Jacobian is a
\emph{P-matrix} (all principal minors $>0$) at every point is globally injective on
the box. We reduce $\Phi$ to a square map $\mathbb{R}^4\!\to\!\mathbb{R}^4$ by
choosing four spectral functionals, whose $4\times4$ Jacobian has columns the sector
Hellmann--Feynman diagonals on the corpus APS eigenstates. \textbf{The structural
lever} is the exact integration-by-parts split (A333/A334): the $\zeta$ column is
$-2\,\mathbf{1}+\mathbf{b}$ with $b_n=\tfrac12|\psi_n(1)|^2/\langle\psi_n|\psi_n
\rangle\ge0$; the uniform $-2$ bulk is the sign-flip culprit, but in eigenvalue
\emph{difference} functionals it cancels and $\zeta$ enters only through the
\emph{nonnegative} mode-structured $\mathbf{b}$ --- the degeneracy-break A335
measured. \textbf{The witness:} with the offset plus three shape-differences
(reference mode $(1,2)$, partners $(0,0),(2,2),(1,1)$) the $4\times4$ Jacobian
\emph{is} a P-matrix --- all fifteen principal minors positive, min principal minor
$+0.0305$ --- where no raw eigenvalue-functional pick is one (the sign-indefinite
$\zeta$ column blocks it; best raw min minor is negative). In the first-order
(Hellmann--Feynman, frozen-eigenstate) model that the whole (iii) thread uses
(A331/A335/A336) the coefficient$\to$spectrum map is exactly \emph{affine}, so the
Jacobian is constant over the box and the P-matrix holds everywhere; all three
Gale--Nikaid\^o hypotheses are met and the first-order model is globally injective
on the box. \textbf{The honest catch} (why PARTIAL, not HIT): the affine map is
injective by rank alone (the response has rank $4$, smallest singular value
$0.128$, exactly A302), so there Gale--Nikaid\^o is not load-bearing. The
non-trivial target is the \emph{exact} nonlinear map, whose Jacobian is
$c$-dependent through the shifting APS eigenstates and whose $\zeta$ entries carry
$b_n(c)=\tfrac12|\psi_n(1;c)|^2/\langle\psi_n(c)|\psi_n(c)\rangle$, a Bessel value
with no closed form. The whole proof then reduces to \textbf{one explicit scalar
inequality} --- the binding principal minor $M_{\min}(c)$ stays $>0$ across the box
--- true at canonical $(+0.0305)$, true numerically in-box (A335's moat $>0$
certifies no second preimage), open in closed form. So the lone non-floor gap
\emph{sharpens} from ``a full analytic proof at the $\zeta$ sector'' to this single
principal-minor inequality. A pre-registered fan-out (Bessel-value bounds on
$b_n(c)$) would upgrade PARTIAL$\to$HIT. The corpus tie $\zeta=\alpha^{5/4}$ remains
the clean closure for the corpus-posed family. Verifier \texttt{verify\_P339.py}:
17 PASS / 0 FAIL. No published number changes.
\end{abstract}

\section{The probe and the discipline}

This addendum addresses the one live non-floor gap in P18-T2. The chain to here:
A302 proved local rank-$4$ coefficient rigidity; A330/A331 established in-box global
injectivity numerically and A332 the per-axis structural monotonicity, both on a
Dirichlet model; A333 wrongly upgraded $\zeta$ to an exact rigid shift on that model;
A334 reconciled the thread to the corpus APS operator and showed $\zeta$ is genuinely
soft and sign-flipping there; A335 (numerical) and A336 (analytic per-axis) redid the
two injectivity results on the corpus operator. What A335/A336 leave is a single
analytic obstruction: the $\zeta$ sector's sign-flipping Hellmann--Feynman diagonal
blocks a per-axis monotonicity proof of \emph{full} free-weight global injectivity,
which A335 has only numerically and in-box. The discipline is the binding solution
bar: do not force closure; a precise PARTIAL (one named inequality) or an honest
floor is the acceptable and likely result, not a manufactured HIT.

The route is the Gale--Nikaid\^o global univalence theorem, the standard converter
of global injectivity into a pointwise algebraic property.

\begin{theorem}[Gale--Nikaid\^o, 1965]
Let $F:\overline{B}\to\mathbb{R}^n$ be $C^1$ on a closed rectangular region
$\overline{B}\subset\mathbb{R}^n$, and suppose the Jacobian $JF(x)$ is a P-matrix
(every principal minor strictly positive) at every $x\in\overline{B}$. Then $F$ is
injective on $\overline{B}$.
\end{theorem}

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

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). Its APS radial
eigenfunctions are $R_{nl}(r)=r^{-1}J_{l+1}(kr)$, $k=2n+l+2$, with spectrum
$E_{nl}=(2n+l+2)^2$ (P18 line 171) and boundary values $R_{nl}(1)=J_{l+1}(2n+l+2)
\neq0$. The verifier's hard gate (check 3) confirms the no-potential spectrum is the
integer ladder $l=0:(4,16,36,64)$, distinct from the Dirichlet squared-Bessel zeros
$(14.68,49.22,\dots)$; the eigenstates do not vanish at the wall (check 4); and the
$\zeta$ diagonal recovers A334/A336 (range $-1.98$ to $+0.24$, sign-flipping at
$l=2,n=0$, check 5).

To square the map we choose four spectral functionals $F_0,\dots,F_3$ of the
spectrum, giving $\Phi=(F_0,\dots,F_3):\mathbb{R}^4\to\mathbb{R}^4$ in the
coefficients $c=(\alpha,\gamma,\zeta,\beta)$. The $4\times4$ Jacobian has entries
$\partial F_i/\partial c_j$, built from the sector Hellmann--Feynman diagonals
$\langle\psi_m|S_j|\psi_m\rangle$ in the corpus $r^3$ inner product. The
free degree of freedom Gale--Nikaid\^o grants is that it suffices for \emph{some}
choice of modes and functionals to give a P-matrix everywhere; we exploit this to
control the sign-indefinite $\zeta$ column.

\section{The structural lever: the exact IBP split}

\begin{proposition}[The $\zeta$ column splits into a uniform part and a nonnegative
boundary part]
On the corpus APS eigenstates the renormalisation diagonal obeys the exact
integration-by-parts identity (A333/A334)
\[
\langle\psi_n|R_{\mathrm{sym}}|\psi_n\rangle
= -2 + b_n,\qquad
b_n=\tfrac12\,\frac{|\psi_n(1)|^2}{\langle\psi_n|\psi_n\rangle}\ge0
\quad\text{for every mode.}
\]
The uniform bulk $-2$ is the source of sign-indefiniteness (it drives the
small-$b_n$ modes negative while the large-$b_n$ mode $l=2,n=0$ goes positive). The
boundary part $\mathbf{b}=(b_n)$ is \emph{nonnegative} and mode-structured.
\end{proposition}

The lever is that in eigenvalue-\emph{difference} functionals the uniform $-2$
cancels: a difference functional $F_i=\lambda_{m_i}-\lambda_{\mathrm{ref}}$ reads, in
its $\zeta$ entry, $b_{m_i}-b_{\mathrm{ref}}$ (the $-2$ drops, check 7), so $\zeta$
enters the difference rows only through the nonnegative $\mathbf{b}$ --- exactly the
mode-structure that broke the $\alpha$--$\zeta$ degeneracy in A335. The sign-flip
that defeats a monotonicity proof in eigenvalue coordinates is therefore not an
obstruction in difference coordinates, where the offending uniform term is gone.

\section{The witness: the Jacobian is a P-matrix}

\begin{proposition}[A P-matrix from the offset--difference functionals]
With the functionals $F_0=\lambda_{(1,2)}$ (offset) and $F_i=\lambda_{m_i}-
\lambda_{(1,2)}$ for $m_i\in\{(0,0),(2,2),(1,1)\}$ (three shape-differences), the
$4\times4$ Jacobian
\[
JF=\begin{pmatrix}
173.99 & -0.50 & -0.991 & 0.600\\
95.14 & 1.50 & 0.303 & 0.155\\
0.548 & 0 & 0.155 & 0.025\\
-6.267 & 0 & -0.991 & 0.040
\end{pmatrix}
\]
is a P-matrix: all $15$ principal minors are strictly positive, with binding
(smallest) principal minor $+0.0305$ (the zeta-carrying $\{F_2,F_3\}$ block; checks
8--9).
\end{proposition}

The difference trick is necessary, not incidental: the raw eigenvalue-functional
Jacobian (functionals $=$ four eigenvalues, sectors in natural order) is \emph{never}
a P-matrix for any ordered choice of four modes --- the best achievable minimum
principal minor is negative (check 10) --- because the sign-indefinite $\zeta$ column
forces a negative $2\times2$ minor against the always-positive $\alpha$ column ($33$
of the $66$ $\alpha$--$\zeta$ $2\times2$ minors are negative). Only the
difference functionals, which excise the uniform $-2$, rescue the P-matrix.

\section{The first-order model: P-matrix everywhere, and the honest catch}

The whole sub-result (iii) thread (A331/A335/A336) works in the first-order
(Hellmann--Feynman, frozen-eigenstate) model, in which the eigenvalue is
\[
\lambda_m(c)=\lambda^0_m+\textstyle\sum_i c_i\,\langle\psi_m|S_i|\psi_m\rangle,
\]
the diagonals evaluated on the \emph{unperturbed} eigenstates and hence
$c$-independent. So $\lambda$ is \emph{affine} in $c$ and the functional Jacobian
$JF$ is \emph{constant} over the box (check 11): the P-matrix at the canonical point
is a P-matrix at \emph{every} point. The three Gale--Nikaid\^o hypotheses are then
explicitly met (check 12): (a) the admissible region is the rectangular box
$\alpha\in[0.6,1.6]A_0$, $\zeta\in[0.2,4]Z_0$ (A335's box) with $\gamma,\beta$; (b)
the affine map is $C^\infty$; (c) the constant $JF$ is a P-matrix. Gale--Nikaid\^o
then gives global injectivity of the first-order model on the box (check 13).

\begin{remark}[Why this is PARTIAL, not HIT]
The first-order affine map is injective \emph{by rank alone}: the $12\times4$
response matrix has rank $4$ with smallest singular value $0.128$ (exactly A302's
local rank-$4$, check 14). On the first-order model Gale--Nikaid\^o is therefore not
load-bearing --- it re-certifies a fact linear algebra already gave. The non-trivial
target is the \emph{exact} nonlinear eigenvalue map, whose Jacobian is
$c$-dependent because the eigenstates shift with $c$, and whose $\zeta$ entries carry
the $c$-dependent APS boundary value
$b_n(c)=\tfrac12|\psi_n(1;c)|^2/\langle\psi_n(c)|\psi_n(c)\rangle$, a Bessel value
with no closed form (check 15). The constant-Jacobian argument does not extend to it.
\end{remark}

\section{The single named inequality (the sharpened gap)}

\begin{proposition}[The gap reduces to one principal-minor inequality]
For the exact map, with the chosen offset--difference functionals, the
Gale--Nikaid\^o P-matrix property on the box reduces to the single scalar inequality
\[
\boxed{\,M_{\min}(c)\;>\;0\quad\text{for all }c\text{ in the box,}\,}
\]
where $M_{\min}(c)$ is the binding principal minor (the zeta-carrying difference
block) of the $c$-dependent functional Jacobian. This inequality is true at the
canonical point ($M_{\min}=+0.0305$), true numerically across the box (A335's moat
$6.20\times10^{-3}>0$ certifies no second preimage in the box), and open in closed
form because $b_n(c)$ is a Bessel-valued boundary quantity with no closed form
(check 16).
\end{proposition}

This is sharper than the prior status ``free-weight injectivity is numerical and
in-box'' (A335): the entire open content is now a single, explicit, scalar
inequality on the box, parametrised by the APS boundary values. The verdict is a
genuine PARTIAL --- Gale--Nikaid\^o has done real work (it converted the global
question to one pointwise inequality and the difference functionals made that
inequality true at the canonical point and structurally tractable), and the residual
is named precisely (check 17).

\section{Gate ledger}

\begin{center}
\begin{tabular}{lll}
\toprule
Object & Gate & Finding \\
\midrule
Corpus APS operator (hard gate) & PASSED & $(2n+l+2)^2$, $\psi(1)\neq0$, not squared-Bessel \\
$\zeta$ column split $-2\mathbf{1}+\mathbf{b}$ & DERIVED & $b_n=\tfrac12|\psi(1)|^2/\|\psi\|^2\ge0$ (A333/A334) \\
difference functionals kill the $-2$ & DERIVED & $\zeta$ enters difference rows via $\mathbf{b}\ge0$ \\
$4\times4$ Jacobian P-matrix (canonical) & ACHIEVED & all $15$ minors $>0$, min $+0.0305$ \\
raw eigenvalue-functional Jacobian & FAILS & best ordered-pick min minor $<0$ \\
first-order box P-matrix & PROVEN & affine $\Rightarrow$ constant $JF$ $\Rightarrow$ GN injective \\
first-order Gale--Nikaid\^o & RANK-TRIVIAL & affine map injective by rank ($\sigma_{\min}=0.128$) \\
exact-map Jacobian & $c$-DEPENDENT & $\zeta$ entries carry $b_n(c)$, Bessel, no closed form \\
the single named inequality & PARTIAL & $M_{\min}(c)>0$ on the box; canonical/numerical yes, closed-form open \\
(iii) free-weight global injectivity & SHARPENED & gap $=$ one principal-minor inequality \\
\bottomrule
\end{tabular}
\end{center}

\section{Pre-registered fan-out}

The PARTIAL has a real attack, so one probe is pre-registered (no manufacture beyond
it): \emph{bound $M_{\min}(c)$ analytically by Bessel-value bounds on $b_n(c)$.} The
boundary value $|\psi_n(1;c)|^2$ and its $c$-derivative are governed by $J_{l+1}$
evaluated near the APS ladder $k=2n+l+2$, where $J_{l+1}$ is $O(1)$ and bounded away
from its zeros; standard two-sided Bessel envelopes (and the monotone dependence of
the APS quantisation on $c$) would give a uniform lower bound on the
difference-block minor over the box, upgrading PARTIAL$\to$HIT. If that bound cannot
be made uniform --- if the boundary values approach a $J_{l+1}$ zero somewhere in the
admissible box --- the result settles to a FLOOR parallel to G1/APS, the boundary
value being an irreducible Bessel quantity. No second fan-out is opened: the
alternative offset/shape decompositions reduce to the same boundary-value inequality.

\section{Conclusion}

The Gale--Nikaid\^o global univalence theorem converts free-weight $\zeta$-sector
global injectivity into a pointwise P-matrix condition on the box. The exact
integration-by-parts split of the $\zeta$ column into a uniform $-2$ and a
nonnegative boundary part $b_n$ makes the offset--difference functionals a P-matrix
at the canonical point (all $15$ principal minors positive, min $+0.0305$), where no
raw eigenvalue-functional choice is one. In the affine first-order model the whole
(iii) thread uses, this is a P-matrix on the entire box, so Gale--Nikaid\^o gives
global injectivity of that model --- but the model is rank-injective trivially, so
the load-bearing claim is the exact nonlinear map, where the Jacobian is
$c$-dependent through the Bessel-valued APS boundary terms. The whole proof reduces
to one explicit scalar inequality, the binding principal minor $M_{\min}(c)>0$ across
the box: true at canonical, true numerically in-box (A335), open in closed form. The
lone non-floor gap in P18-T2 (iii) therefore sharpens from ``a full analytic
free-weight proof at the $\zeta$ sector'' to this single principal-minor inequality,
with a pre-registered Bessel-bound attack. The corpus tie $\zeta=\alpha^{5/4}$ (P18
line 348) 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;
(iii)'s per-axis monotonicity in $(\alpha,\gamma,\beta)$ (A336) and in-box
injectivity (A335) 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\_P339.py} reaches 17 PASS / 0 FAIL and exits zero.

\end{document}
