\documentclass[11pt]{article}
% Addendum 347b (extends Paper 18, The Master Operator -- P18-T2, sub-result (iii)).
% b-thread (zeta-injectivity lane); Angle 3 of the three-angle attack (345b/346b/347b),
% closed by the Schur certificate 348b.
% Author: Leon Fernando Vlegels.  Text CC BY 4.0.
% Verifier: verify_P347b.py (16/16 pass).
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\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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\newcommand{\Rsym}{R_{\mathrm{sym}}}
\newcommand{\Hc}{H(c)}
\newcommand{\DPhi}{D\Phi}
\newcommand{\lmin}{\lambda_{\min}}

\title{Addendum 347b: The Variational / Monotone-Operator Structure\\
\large $\lambda_{\min}$ is concave, but no constant SPD metric makes the map\\
monotone: the obstruction is a metric-invariant $\zeta$-soft eigenvalue\\
--- P18-T2 sub-result (iii), Angle~3\\[0.3em]
\normalsize (Addendum to Paper 18)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
Angles~1 (345b, line test) and 2 (346b, offset--shape) both reduce the lone non-floor gap of
P18-T2 sub-result~(iii) to the \emph{same} missing lever: a closed-form sign-control of the
$\zeta$-carrying Jacobian quadratic form. Angle~3 attacks that lever directly through the
variational structure of the eigenvalues of
$\Hc=H_0+\sum_i c_i S_i$ ($S_i=\rho,\,\text{character }I,\,\Rsym,\,M$ fixed; $c$ the scalar
weights). Four findings on the genuine eigendecomposition (V$=0$ hard gate passed).
\textbf{(1)} $\lmin(c)=\inf_\psi\langle\psi|\Hc|\psi\rangle$ is an infimum of affine functions,
hence \textbf{concave} in $c$ (verified, $0/200$ midpoint violations); higher eigenvalues are
concave-minus-concave, so no global-convexity argument forces injectivity directly.
\textbf{(2) The prize attempt:} a $C^1$ map is globally injective on a convex set if it is
\emph{monotone} in some inner product, i.e.\ the symmetrised Jacobian $D\,\DPhi+\DPhi^\top D$
is sign-definite for a single constant SPD metric $D$. On the true operator the Euclidean
symmetrised Jacobian is \emph{indefinite} (one persistent eigenvalue $\sim\![-0.032,-0.028]$),
and the best well-conditioned SPD metric ($\mathrm{cond}\le10^3$) drives the box-minimum
symmetrised eigenvalue only to $\sim\!-10^{-3}$, never positive; a degenerate metric reaches a
spurious $\sim\!-10^{-55}$, a $\mathrm{cond}\!\to\!\infty$ artifact, not a certificate.
\textbf{(3)} The $\gamma$ column of $\DPhi$ is exactly $c$-independent ($\sim\!10^{-15}$) ---
the variational form of A346's character block, the one closed-form sector. \textbf{(4)} The
residual $R(c)=\|\Phi(c)-\Phi^*\|^2$ is \emph{non-convex} (a zero-eigenvalue $\zeta$ valley;
Hessian not positive-definite at $63/86$ box points). \textbf{Verdict: PARTIAL} with a genuine
new structural fact --- the monotone-operator obstruction is a single metric-invariant
$\zeta$-soft eigenvalue --- and the prize honestly not delivered. The variational route lands
on the same $\zeta$-soft boundary as Angles~1 and~2 and explains it. All claims pinned by
\texttt{verify\_P347b.py} (16/16).
\end{abstract}

\section{The shared lever}
\label{sec:lever}

Both prior angles converge on one residual. A345's line test is met numerically but its clean
analytic sufficient condition ($v^\top J v$ sign-constancy) fails on $\sim\!1\%$ of chords;
A346's shape block closes but the $\zeta$-coupling does not vanish. In each case the missing
piece is a closed-form statement that the Jacobian quadratic form keeps a definite sign in the
$\zeta$ direction. The variational structure is the natural place to seek it: the operator is
\emph{affine} in the weights,
\begin{equation}
  \Hc \;=\; H_0 + \sum_i c_i\,S_i,
  \qquad S_i \in \{\rho,\ \text{character }I,\ \Rsym,\ M\}\ \text{fixed},
  \label{eq:affine}
\end{equation}
so its eigenvalues inherit variational convexity properties in $c$. We test whether those
properties deliver the lever.

\section{Concavity of $\lambda_{\min}$ --- clean but insufficient}
\label{sec:concave}

\begin{proposition}[$\lmin$ is concave; the differences are not]
\label{prop:concave}
Since $\Hc$ is affine in $c$, $\lmin(c)=\inf_{\|\psi\|=1}\langle\psi|\Hc|\psi\rangle$ is an
infimum of functions affine in $c$, hence concave in $c$. This is verified numerically
($0/200$ midpoint-concavity violations). By Courant--Fischer the higher eigenvalues are
min--max values, hence differences of concave functions; the matched difference-functionals
$F_i=\lambda_{m_i}-\lambda_{(1,2)}$ are concave-minus-concave, with no definite curvature.
\end{proposition}

\begin{remark}[Why concavity does not close it]
\label{rmk:concave}
A globally concave (or convex) coordinate map need not be injective, and in any case the
relevant objects are the \emph{differences} $F_i$, which carry no definite curvature. So no
global-convexity argument forces injectivity directly --- precisely the structural reason the
gap is hard. Concavity of $\lmin$ is real and clean, but it is not the lever.
\end{remark}

\section{The monotone-operator test}
\label{sec:monotone}

\begin{theorem}[Monotone-operator criterion]
\label{thm:monotone}
A $C^1$ map $\Phi$ is globally injective on a convex set if it is monotone in some inner
product: $\langle D(\Phi(c)-\Phi(c')),\,c-c'\rangle$ has one sign for a constant SPD metric
$D$. By the mean-value theorem this is equivalent to sign-definiteness of the symmetrised
Jacobian $D\,\DPhi(c)+\DPhi(c)^\top D$ for all $c$ --- a $v^\top(D\,\DPhi)_{\mathrm{sym}}v$
sign-control, exactly the closed-form lever Angles~1 and~2 need.
\end{theorem}

We test this on the genuine eigendecomposition, with the gradient supplied in closed form by
Hellmann--Feynman, $\partial\lambda_n/\partial c_i=\langle\psi_n|S_i|\psi_n\rangle$ on the
true mixed eigenstate.

\begin{proposition}[The obstruction is metric-invariant]
\label{prop:obstruction}
\emph{Euclidean ($D=I$):} the symmetrised Jacobian is indefinite --- one persistent negative
eigenvalue (normalised coordinates $\sim\![-0.032,-0.028]$) and three positive; the negative
eigenvector mixes $\gamma,\zeta,\beta$ nearly equally, with the soft direction the $\zeta$
sector. \emph{Best well-conditioned metric ($\mathrm{cond}\le10^3$):} the box-minimum
symmetrised eigenvalue is driven up to $\sim\!-10^{-3}$ but cannot cross zero; the map is at
best semidefinite-monotone (on the boundary), never strictly monotone in any non-degenerate
metric. \emph{Degenerate metric (collapsing the $\beta$ weight):} reaches a spurious
$\sim\!-10^{-55}$ --- a $\mathrm{cond}\!\to\!\infty$ numerical artifact, flagged and excluded,
not a certificate. The obstruction is therefore a single near-zero symmetrised eigenvalue in
the $\zeta$ direction that no constant SPD metric can lift.
\end{proposition}

\begin{remark}[The mechanism]
\label{rmk:mechanism}
The offset-functional construction couples the soft $\zeta$ sector into every difference row,
so the symmetrised Jacobian inherits one near-zero eigenvalue regardless of the metric. This
is the same $\zeta$-soft floor named in A330/A332/A341/A345 --- now seen as a
\emph{metric-invariant} property of the monotone-operator form, which is the new structural
fact of Angle~3.
\end{remark}

\section{The frozen $\gamma$ column and the non-convex residual}
\label{sec:gammares}

\begin{proposition}[The one closed-form sector]
\label{prop:gamma}
The $\gamma$ column of $\DPhi$ is exactly $c$-independent (maximum deviation $\sim\!10^{-15}$
over the box): the $\gamma$ sector is a scalar multiple of $I$, so
$\partial\lambda_n/\partial\gamma=\operatorname{Re}\omega^l$ is frozen. This is the
variational form of A346's exact character block --- the one sector the variational structure
pins in closed form. It is necessary but not sufficient: with $\gamma$ frozen, the residual
sign-control still requires the $\zeta$-soft $(\alpha,\zeta,\beta)$ $3\times3$, the boundary
case of Proposition~\ref{prop:obstruction}.
\end{proposition}

\begin{proposition}[The residual is non-convex]
\label{prop:residual}
The residual $R(c)=\|\Phi(c)-\Phi^*\|^2$ has a Hessian with a zero eigenvalue at canonical
(the $\zeta$-flat valley of A330/A332) and is not positive-definite at the majority of sampled
box points ($63/86$). So the ``unique critical point from a positive-definite Hessian'' route
fails: the $\zeta$ valley makes $R$ only quasi-flat along $\zeta$, not strictly convex.
(A330's second-order strict local minimum at canonical is a statement about $R$ restricted to
the residual manifold; the full $4\times4$ $c$-Hessian is rank-deficient along $\zeta$ ---
consistent, not contradictory.)
\end{proposition}

\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), the variational / monotone-operator
structure yields: (i)~$\lmin(c)$ is concave (infimum of affine), and the matched
difference-functionals are concave-minus-concave, so no global-convexity argument forces
injectivity directly; (ii)~the chosen-functional map is not a monotone operator in the
Euclidean metric (symmetrised Jacobian indefinite, one persistent $\sim\!-0.03$ eigenvalue),
and no well-conditioned constant SPD metric makes it strictly monotone (best
$\mathrm{cond}\le10^3$ box-minimum eigenvalue $\sim\!-10^{-3}$, never positive; the only
``monotone'' metric is the degenerate $\mathrm{cond}\!\to\!\infty$ limit, a numerical artifact,
not a certificate); (iii)~the $\gamma$ column of $\DPhi$ is exactly $c$-independent --- the one
closed-form sector; (iv)~the residual $R(c)$ is non-convex (zero-eigenvalue $\zeta$ valley,
Hessian not positive-definite at $63/86$ box points), so the residual-convex-program route
fails. The key prize --- a closed-form sign-control of the $\zeta$-carrying Jacobian quadratic
form --- is \textbf{not delivered}: the variational route lands on the same $\zeta$-soft
boundary as Angles~1 and~2 and characterises it sharply (the symmetrised Jacobian is
semidefinite-at-best on the $\zeta$ direction in every non-degenerate metric, because the
offset-functional construction couples the soft $\zeta$ sector into every difference row).
This is an honest \textbf{PARTIAL} with a genuine new structural fact (the monotone-operator
obstruction is the $\zeta$-soft eigenvalue, metric-invariant) and the prize honestly not
delivered --- no fourth overclaim on this thread (after A333 Dirichlet-as-corpus, A340
shift-model, A342 total-positivity dead-end). The residual across all three angles is the same
named floor: a closed-form sign-definite control of the $\zeta$-soft direction; the corpus tie
$\zeta=\alpha^{5/4}$ (P18~l.348) removes $\zeta$ and remains the cleanly-closed
corpus-posed-family case. A294's re-typing of P18-T2 stands; (i)$=$A328/A329, (ii)$=$A310,
(iii) A335 in-box injectivity $+$ A341 constant-sign determinant $+$ A345 line test $+$ A346
shape block all unchanged; no canon \texttt{.tex} edited and no published number changes.
Verifier: \texttt{verify\_P347b.py}, 16/16 pass.

\paragraph{Relation to the other angles.}
A347's negative result --- no \emph{constant} metric works --- forces the remaining hope to a
\emph{$c$-dependent} certificate, which is exactly a Schur complement (the soft $\zeta$ block
conditioned on the stiff shape block). That is carried out in 348b.

\end{document}
