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% Addendum 345b (extends Paper 18, The Master Operator -- P18-T2, sub-result (iii)).
% b-thread: assembly-uniqueness / free-weight injectivity.  Sibling of 345a (cosmology); the
% "b" suffix is the zeta-injectivity lane (Joel's a/b numbering, 2026-06).
% Author: Leon Fernando Vlegels.  Text CC BY 4.0.
% Verifier: verify_P345b.py (16/16 pass).
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\newcommand{\Rsym}{R_{\mathrm{sym}}}
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\title{Addendum 345b: The Global-Univalence Line Test\\
\large Upgrading constant-sign-determinant local injectivity to global\\
injectivity on the admissible box --- P18-T2 sub-result (iii), Angle~1\\[0.3em]
\normalsize (Addendum to Paper 18)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
The lone non-floor gap remaining across both hearts of P18-T2 (the assembly-uniqueness
theorem of the master operator) is a fully rigorous proof that the free-weight
coefficient-to-spectrum map $\Phi:c=(\alpha,\gamma,\zeta,\beta)\mapsto$ the four chosen
spectral functionals is \emph{globally} injective on the admissible box, evaluated on the
corpus APS operator. Addendum~A341 established, on a genuine eigendecomposition of the
operator-with-potential $\Hc=H_0+\alpha\rho+\zeta\Rsym+\beta M+V_{\mathrm{self}}$, that the
$4\times4$ difference-functional Jacobian $\Jf(c)$ has a \emph{constant-sign},
bounded-away-from-zero determinant ($|\det|\in[4.38,5.44]$) across the box --- giving local
injectivity \emph{everywhere}. But a constant-sign nonsingular determinant gives local
injectivity \emph{only}: the map $(x,y)\mapsto(e^x\cos y,e^x\sin y)$ is a local diffeomorphism
on all of $\mathbb{R}^2$ with $\det=e^{2x}>0$ everywhere and is not globally injective. This
addendum (Angle~1 of a three-angle attack, with 346b and 347b) tests three global-univalence
theorems on the true operator. (1a)~The Hadamard small-variation route is \textbf{refuted}:
the Jacobian's relative variation along box chords is $\sim\!10^3$, not $<1$. (1b)~The
boundary-injectivity / degree-theory route is \textbf{valid} but does not lower the floor:
the boundary map inherits the same $\zeta$-softness ($\zeta$ moves the image $\sim\!0.037$
while $\alpha$ moves it $\sim\!1.63$, a $\sim\!45\times$ anisotropy). (1c)~The convex-domain
\textbf{line test} is the strong route: $F\in C^1$ on a convex domain, injective on every
segment, is globally injective; on the true operator the within-segment velocity
\emph{direction} spread is tiny (median $\sim\!1.5^\circ$, all chords $<90^\circ$) even
though the velocity \emph{magnitude} swings $\sim\!10^3\times$, so $0$ lies outside the
velocity hull with margin $\sim\!0.998$ on every sampled chord. This supplies exactly the
local-to-global bridge the constant-sign determinant lacks. The honest limit: the clean
closed-form lever (directional-convexity sign-constancy of $v^\top J v$) holds on
$296/300$ chords, not all, so the line-test hypothesis is met \emph{numerically}, with the
closed-form proof open at the same $\zeta$-soft moment-ratio floor named in A341/A342.
\textbf{Verdict: PARTIAL} --- the correct theorem and the missing bridge, met numerically;
no closed-form HIT. All claims pinned by \texttt{verify\_P345b.py} (16/16).
\end{abstract}

\section{The gap, the operator, and the local-only status}
\label{sec:setup}

P18-T2 asserts that the assembly of the master operator $\hat{O}$ is unique across admissible
forms. Sub-result~(iii) reduces one face of that uniqueness to the injectivity of the
free-weight coefficient-to-spectrum map: distinct weight tuples
$c=(\alpha,\gamma,\zeta,\beta)$ must produce distinct spectra, or the assembly would be
ambiguous. The map is taken on the corpus APS operator,
\begin{equation}
  \Hc \;=\; H_0 + \alpha\,\rho + \zeta\,\Rsym + \beta\,M + V_{\mathrm{self}},
  \label{eq:Hc}
\end{equation}
diagonalised genuinely (Galerkin in the APS Bessel basis $R_{nl}=r^{-1}J_{l+1}(k_n r)$,
$k_n=2n+l+2$, all satisfying the APS boundary condition; blocks built in the corpus $r^3$
measure and mass-orthonormalised; $\Rsym$ from its exact integration-by-parts block
$-2G+\tfrac12\psi_1\psi_1^\top$). The construction passes the hard gate at $V=0$: the
spectrum is $(2n+l+2)^2=(4,16,36,64)$ in every $l$-sector --- the corpus APS eigen-identity
(A329, P18~\S171), \emph{not} the Dirichlet operator of A333 and \emph{not} the
argument-shift model of A340.

The chosen functionals are A339's: with reference mode $(1,2)$, the four spectral readouts
are $\lambda_{(1,2)},\lambda_{(0,0)},\lambda_{(2,2)},\lambda_{(1,1)}$, assembled by the fixed
difference matrix $L$ into $F=L\,\lambda$ (an offset plus three shape partners). Its Jacobian
is $\Jf(c)=L\,\partial\lambda/\partial c$, the entries computed by Hellmann--Feynman on the
true (mixed) eigenstates.

\begin{remark}[Why A341 is not enough]
\label{rmk:localonly}
A341 proved $\det\Jf(c)$ has constant sign and is bounded away from zero on the box. By the
inverse function theorem this gives a local diffeomorphism at every point --- local
injectivity everywhere. It does \emph{not} give global injectivity: the planar exponential
$(x,y)\mapsto e^{x}(\cos y,\sin y)$ has $\det=e^{2x}>0$ everywhere yet identifies points
differing by $2\pi$ in $y$. The task of Angle~1 is a genuine global-univalence theorem whose
hypotheses are met on the box.
\end{remark}

\section{Angle 1a (small variation) and 1b (degree theory)}
\label{sec:1a1b}

\paragraph{1a --- Hadamard small variation: refuted as a route.}
The cleanest sufficient condition for a local diffeomorphism to be global is that the
Jacobian vary little relative to its conditioning: $\sup_t \|J(t)-J(0)\|_2/\sigmin(J(0))<1$
along every chord. On the true operator this ratio is $\sim\!10^3$ (maximum $\sim\!3300$)
along box chords. The criterion fails by three orders of magnitude.

\begin{remark}[An honest negative]
\label{rmk:1a}
The constant-sign determinant is therefore \emph{not} a consequence of $J$ being nearly
constant. The Jacobian swings enormously; the determinant is steady for a subtler reason
(developed in~\S\ref{sec:1c}). Recording the refutation prevents a future re-attempt of the
small-variation lever.
\end{remark}

\paragraph{1b --- boundary injectivity and degree theory: valid, floor unchanged.}
The degree theorem is valid: a $C^1$ local diffeomorphism with constant-sign determinant on a
compact convex set, injective on the \emph{boundary}, is globally injective. It reduces
global injectivity to injectivity on the boundary of the box. But the reduction does not give
an easier sub-problem.

\begin{proposition}[The boundary inherits the $\zeta$-soft floor]
\label{prop:1b}
Over the box, varying $\zeta$ across its full admissible range moves the image by only
$\sim\!0.037$, while varying $\alpha$ moves it $\sim\!1.63$ --- a $\sim\!45\times$
anisotropy. The global minimum image-distance between distinct boundary points is
$\sim\!2\times10^{-3}$, about $10^3$ times smaller than the image diameter ($\sim\!2.56$).
Boundary injectivity therefore holds numerically, but with a margin set by the \emph{same}
$\zeta$-soft floor as the interior problem. Degree theory lowers the dimension of the
question, not the floor.
\end{proposition}

\section{Angle 1c: the convex-domain line test}
\label{sec:1c}

\begin{theorem}[Convex-domain line test]
\label{thm:linetest}
Let $F\in C^1$ on a convex domain $D$. If $F$ is injective on every line segment contained in
$D$, then $F$ is injective on $D$. A sufficient per-segment condition, in
separating-hyperplane form: for each chord $t\mapsto c+tv$, $t\in[0,1]$, there is a fixed
covector $u$ with
\begin{equation}
  u^\top F'(t) \;=\; u^\top \Jf(c+tv)\,v \;>\; 0 \qquad \text{for all } t\in[0,1].
  \label{eq:hyperplane}
\end{equation}
Then $t\mapsto u^\top F(c+tv)$ is strictly increasing, so $F(c+v)\neq F(c)$; the same holds on
every sub-segment, giving injectivity along the whole chord. Equivalently, $0$ does not lie in
the convex hull of the unit velocity vectors $\{\Jf(c+tv)\,v/\|\cdot\|: t\in[0,1]\}$.
\end{theorem}

The hyperplane form~\eqref{eq:hyperplane} is the mechanism the constant-sign determinant
alone cannot supply, and it is met robustly on the true operator.

\begin{proposition}[Ray separation on the true operator]
\label{prop:raystep}
Along every sampled box chord, the within-segment velocity \emph{direction} spread is tiny
(median $\sim\!1.5^\circ$, maximum $<10^\circ$, every chord $<90^\circ$), even though the
velocity \emph{magnitude} swings by $\sim\!10^3$. Consequently the image is a near-monotone
curve along every ray, and $0$ sits outside the convex hull of the unit velocities with
hull-distance margin $\sim\!0.998$ on every chord. This is the genuine local-to-global
mechanism: not small Jacobian variation (refuted, \S\ref{sec:1a1b}) but near-fixed velocity
\emph{direction}.
\end{proposition}

\begin{remark}[Reconciliation with 1a]
\label{rmk:reconcile}
Proposition~\ref{prop:raystep} and Remark~\ref{rmk:1a} are consistent: the velocity magnitude
$\|\Jf v\|$ swings $\sim\!10^3\times$ (so $J$ is far from constant, 1a fails) while the
velocity direction is nearly fixed (so the hyperplane test passes). It is the direction, not
the magnitude, that controls injectivity along a ray.
\end{remark}

\section{The analytic handle and its honest limit}
\label{sec:handle}

The cleanest \emph{rigorous} sufficient condition specialising
Theorem~\ref{thm:linetest} is directional convexity: $v^\top \Jf(c+tv)\,v$ keeps a constant
sign along the chord (then $v^\top(F(c+v)-F(c))=\int_0^1 v^\top\Jf v\,dt\neq0$). This is the
closed-form lever; if it held on every chord, the line test would close in closed form.

\begin{remark}[The $\sim\!1\%$ violation: why the lever is numerical]
\label{rmk:limit}
On the true operator the sign-constancy $v^\top\Jf(c+tv)\,v$ holds on $\sim\!296/300$ sampled
chords --- not all. A $\sim\!1\%$ violation rate forbids asserting it as a theorem hypothesis.
The robust statement that \emph{does} hold on every chord is the half-space / direction-spread
form of Proposition~\ref{prop:raystep}, which is established numerically. Moreover
$v^\top\Jf v$ and $\det\Jf$ are the \emph{same} object: eigenvector-weighted Galerkin moment
ratios, with no elementary closed form --- the identical $\zeta$-soft floor named in A341 and
A342. The line test thus relocates the gap precisely, but does not dissolve it.
\end{remark}

\section*{Status}
On the genuine eigendecomposed corpus APS operator-with-potential (V$=0$ hard gate passed,
$(2n+l+2)^2$ every $l$-sector, not Dirichlet, not a shift), Angle~1 upgrades the status of
sub-result~(iii) from ``constant-sign $\det$ $\Rightarrow$ local injectivity only'' (A341) to
``constant-sign $\det$ $+$ ray separation ($0$ outside the velocity hull on every chord,
direction spread $<90^\circ$) $\Rightarrow$ the convex-domain line test is satisfied
$\Rightarrow$ global injectivity on the box.'' The small-variation route (1a) is refuted; the
degree-theory route (1b) is valid but does not lower the $\zeta$-soft floor. The line-test
hypothesis is met \emph{numerically} with a large margin; the clean closed-form lever
($v^\top\Jf v$ sign-constancy) fails on $\sim\!1\%$ of chords, so a fully rigorous
closed-form proof remains \textbf{open} at the same $\zeta$-soft moment-ratio floor as the
determinant. This is an honest \textbf{PARTIAL}: the correct global-univalence theorem and the
missing local-to-global bridge, met numerically --- not a closed-form HIT. The corpus tie
$\zeta=\alpha^{5/4}$ (P18~l.348) removes $\zeta$ and remains the cleanly-closed
corpus-posed-family case. No canon \texttt{.tex} of P18 is edited and no published number
changes; A335/A336/A341 status is unchanged. Verifier: \texttt{verify\_P345b.py}, 16/16 pass.

\paragraph{Relation to the other angles.}
Angle~2 (346b) peels $\zeta$ off as a spectral offset and closes the residual $(\alpha,\beta)$
shape block as a P-matrix; Angle~3 (347b) shows no constant SPD metric makes the map
monotone; 348b realises the residual as a single Schur-complement scalar. All four converge on
the same named floor: a closed-form sign-control of the $\zeta$-soft direction.

\end{document}
