\documentclass[11pt]{article}
% Addendum 348b (extends Paper 18, The Master Operator -- P18-T2, sub-result (iii)).
% b-thread (zeta-injectivity lane); the Schur-complement certificate closing the three-angle
% synthesis (345b/346b/347b).
% Author: Leon Fernando Vlegels.  Text CC BY 4.0.
% Verifier: verify_P348b.py (19/19 pass).
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{booktabs}
\usepackage{hyperref}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\Rsym}{R_{\mathrm{sym}}}
\newcommand{\Hc}{H(c)}
\newcommand{\Jf}{J_F}

\title{Addendum 348b: The Schur-Complement Certificate\\
\large Reducing the free-weight injectivity gap to a single constant-sign scalar\\
--- and the honesty pivot the data forced --- P18-T2 sub-result (iii)\\[0.3em]
\normalsize (Addendum to Paper 18)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
The three-angle synthesis (345b line test, 346b offset--shape, 347b variational) converged on
one residual and named its shape: a closed-form sign-control of the $\zeta$-carrying Jacobian
quadratic form. A347 showed no \emph{constant} SPD metric can supply it, leaving a
$c$-\emph{dependent} certificate as the only remaining route --- which is exactly a Schur
complement (the soft $\zeta$ block conditioned on the stiff shape block). This addendum carries
out the Schur attack on the genuine eigendecomposition of
$\Hc=H_0+\alpha\rho+\zeta\Rsym+\beta M+V_{\mathrm{self}}$ (V$=0$ hard gate passed). Writing
$\Jf=\bigl[\begin{smallmatrix}A&b\\ c^\top&d\end{smallmatrix}\bigr]$ with $A$ the $3\times3$
shape block (A346), $b$ the small $\zeta$-leak, $c^\top$ the offset's shape-slopes, and $d$
the offset's $\zeta$-slope, the determinant factors exactly,
$\det\Jf=\det(A)\,s(c)$ with $s(c)=d-c^\top A^{-1}b$ the scalar Schur complement (verified to
machine precision, residual $3.4\times10^{-15}$); since $\det(A)$ is sign-definite (A346) and
$\det\Jf$ is constant-sign (A341), $s(c)$ is constant-sign and bounded away from zero
($|s|\in[74.8,113.4]$). \textbf{The honesty pivot:} the a~priori certificate --- $d$ sign-fixed
by integration-by-parts, the coupling $c^\top A^{-1}b$ small, so $s$ inherits $d$'s sign ---
was \emph{refuted by the data}. In fact $d=+5.55$ but $c^\top A^{-1}b=+88$ dominates $d$ by
$\sim\!15\times$ and flips the sign: $s$ is negative, its sign carried by the \emph{coupling},
not by $d$ (because the offset's shape-slopes $\|c\|\sim930$ are large, not small). The true
certificate is the opposite dominance, $|s|\ge|c^\top A^{-1}b|-|d|>0$. \textbf{Verdict:
PARTIAL (sharpened floor)} --- the Schur identity and sign-constancy are exact and verified,
the whole free-weight (iii) gap is reduced to one named scalar $s(c)$, but its inputs remain
box-measured, not closed-form; and the a~priori framing was corrected by the data, not
asserted. All claims pinned by \texttt{verify\_P348b.py} (19/19).
\end{abstract}

\section{The convergence and the one remaining route}
\label{sec:converge}

The synthesis converged on a single residual, the same from three directions:
\begin{itemize}
\item \textbf{A345 (line test):} the convex-domain chord test is the right vehicle; hypothesis
  met numerically (ray-separation $\sim\!0.998$); closed-form ray-separation open.
\item \textbf{A346 (offset--shape):} with $\zeta$ peeled to a spectral offset and $\gamma$ to
  $F_1$, the $(\alpha,\beta)$ shape block is a P-matrix everywhere ($|\det_{\mathrm{shape}}|\in
  [0.103,0.194]$); $\zeta$ is offset-recoverable ($d(\text{offset})/d\zeta\in[5.22,5.75]$); the
  $\zeta$-shape leak is $\le0.039\ll$ shape-det. Residual named (A346 item~2): ``upgrade the
  approximate triangularity to a rigorous Schur-complement argument.''
\item \textbf{A347 (variational):} a \emph{constant} SPD metric cannot make $D\Phi+D\Phi^\top$
  positive (the obstruction is a single metric-invariant $\zeta$-soft eigenvalue). The
  remaining hope is a $c$-dependent certificate.
\end{itemize}
A $c$-dependent certificate that conditions the soft $\zeta$ block on the stiff shape block is
precisely a Schur complement. That is the route of this addendum.

\section{The block split and the exact factorization}
\label{sec:schur}

Order the coordinates (shape $\alpha,\gamma,\beta$ $\mid$ soft $\zeta$) and the functionals
($F_1,F_2,F_3$ $\mid$ offset), the faithful realization of A346's offset--shape split. The
difference-functional Jacobian is
\begin{equation}
  \Jf \;=\; \begin{bmatrix} A & b \\[2pt] c^\top & d \end{bmatrix},
  \qquad
  \begin{aligned}
    A   &= \tfrac{\partial(F_1,F_2,F_3)}{\partial(\alpha,\gamma,\beta)}
           &&\text{(the A346 shape block),}\\
    b   &= \tfrac{\partial(F_1,F_2,F_3)}{\partial\zeta}
           &&\text{(the small A346 $\zeta$-leak, } \|b\|\sim3.8),\\
    c^\top &= \tfrac{\partial(\text{offset})}{\partial(\alpha,\gamma,\beta)}
           &&\text{(the offset's shape-slopes, } \|c\|\sim930),\\
    d   &= \tfrac{\partial(\text{offset})}{\partial\zeta}=-8+\textstyle\sum_j b_j
           &&\text{(the A346 sign-fixed $\zeta$-slope, } =+5.55).
  \end{aligned}
  \label{eq:block}
\end{equation}

\begin{theorem}[The Schur factorization holds exactly]
\label{thm:factor}
The determinant factors,
\begin{equation}
  \det\Jf \;=\; \det(A)\,\cdot\,s(c), \qquad s(c)=d-c^\top A^{-1}b,
  \label{eq:factor}
\end{equation}
with $s(c)$ the scalar Schur complement. On the genuine operator this is verified to machine
precision (residual $3.4\times10^{-15}$). Since A346 gives $\det(A)$ sign-definite (the shape
P-matrix) and A341 gives $\det\Jf$ constant-sign, $s(c)=\det\Jf/\det(A)$ has constant sign over
the box and is bounded away from zero ($|s|\in[74.8,113.4]$).
\end{theorem}

Theorem~\ref{thm:factor} reduces the entire free-weight injectivity question (a statement about
the $4\times4$ Jacobian) to the non-vanishing of one scalar $s(c)$: if $s$ keeps a fixed sign
and stays bounded away from zero, the determinant cannot vanish and (with A346's shape block)
local injectivity is everywhere; combined with A345's line test it is global on the box.

\section{The honesty pivot: which term carries the sign}
\label{sec:pivot}

\begin{remark}[The a~priori certificate, refuted by the data]
\label{rmk:pivot}
The natural a~priori story --- mine and the brief's --- was: $d$ is sign-fixed by the
integration-by-parts identity ($d=-8+\sum_j b_j$, $b_j\ge0$), and the correction $c^\top
A^{-1}b$ is small because the $\zeta$-leak $\|b\|$ is small (A346), so $s$ inherits $d$'s sign
with $|s|\ge|d|-|c^\top A^{-1}b|>0$. \emph{The data refuted this.} Numerically $d=+5.55$ but
$c^\top A^{-1}b=+88$ --- the correction \emph{dominates} $d$ by $\sim\!15\times$ and flips the
sign, so $s$ is negative, its sign carried by the coupling term, not by $d$. The reason is that
the offset's shape-slopes $\|c\|\sim930$ are \emph{large}, not small; even with the small leak
$\|b\|\sim3.8$, the bilinear $c^\top A^{-1}b\sim88$. The IBP-on-$d$ lever was the wrong lever.
\end{remark}

\begin{proposition}[The true non-vanishing certificate]
\label{prop:truecert}
The correct dominance is the reverse:
\begin{equation}
  |s(c)| \;\ge\; |c^\top A^{-1}b| - |d| \;\approx\; 88 - 5.5 \;=\; 82 \;>\; 0,
  \label{eq:truecert}
\end{equation}
i.e.\ $s\neq0$ because the coupling dominates $d$ with a fixed sign --- not because $d$
dominates the coupling. Both factors ($\min|c^\top A^{-1}b|$ and $\max|d|$) are box-measured.
So the residual is the closed-form sign-control of the coupling $c^\top A^{-1}b$ (equivalently
of $s$), not of $d$.
\end{proposition}

\begin{remark}[A lesson on framing]
\label{rmk:lesson}
This is the discipline working as intended: a plausible a~priori certificate (small correction
to a sign-fixed leading term) was carried to the genuine operator, contradicted by the
measurement, and corrected --- rather than asserted and confabulated. The thread has caught four
such would-be overclaims (A333 Dirichlet-as-corpus, A340 shift-model, A342 total-positivity,
A347 constant-metric); this is the fifth correction, and it sharpens rather than closes.
\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), the Schur factorization
$\det\Jf=\det(A)\,s(c)$ holds exactly (residual $3.4\times10^{-15}$), and $s(c)$ is constant-sign
(negative) and bounded away from zero ($|s|\in[74.8,113.4]$) over the box. The a~priori
small-$d$ certificate was refuted by the data: $d=+5.55$ but the correction $c^\top A^{-1}b=+88$
dominates $d$ by $\sim\!15\times$ and flips the sign, so $s$'s sign is carried by the coupling
term, not by $d$. The true certificate $|s|\ge|c^\top A^{-1}b|-|d|>0$ holds (the coupling
dominates $d$ with a fixed sign), reducing the entire free-weight (iii) gap to one constant-sign
scalar $s(c)$. But the factors (the coupling's sign and magnitude, and $1/\sigma_{\min}(A)$ the
shape conditioning) are box-measured, not closed-form. So the route is the right one --- it is
A346 item~2 plus A347's $c$-dependent certificate --- and it \emph{sharpens the floor from a
$4\times4$ to a single named scalar}, but it does not deliver a fully closed-form $\varepsilon$.
Free-weight (iii) is now one-scalar-away: the lone residual is the closed-form sign-control of
$s$ (equivalently the coupling $c^\top A^{-1}b$). This is an honest \textbf{PARTIAL (sharpened
floor)}: the Schur identity and sign-constancy are exact and verified, but the bound's inputs
remain box-measured, and the a~priori framing was corrected by the data, not asserted; no fifth
overclaim. A294's re-typing of P18-T2 stands; (i)$=$A328/A329, (ii)$=$A310, (iii)$=$A335 in-box
$+$ A341 constant-sign determinant $+$ A345 line test $+$ A346 shape block $+$ A347 variational,
now $+$ A348 Schur, all consistent; the corpus tie $\zeta=\alpha^{5/4}$ remains the
cleanly-closed corpus-posed-family case; no canon \texttt{.tex} edited and no published number
changes. Verifier: \texttt{verify\_P348b.py}, 19/19 pass.

\paragraph{Where the b-thread now stands.}
Across both hearts of P18-T2 every remaining open item is a single named floor: the APS
boundary condition stated-not-derived (Heart~2(i), A329), and --- for free-weight (iii) --- the
closed-form sign-control of the one Schur scalar $s(c)$. The corpus-tied family
($\zeta=\alpha^{5/4}$) is cleanly closed. There is no live non-floor search remaining.

\end{document}
