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% Addendum 346b (extends Paper 18, The Master Operator -- P18-T2, sub-result (iii)).
% b-thread (zeta-injectivity lane); Angle 2 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_P346b.py (19/19 pass).
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\newcommand{\Rsym}{R_{\mathrm{sym}}}
\newcommand{\Hc}{H(c)}
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\title{Addendum 346b: The Offset--Shape Separation\\
\large Peeling $\zeta$ to a spectral offset and $\gamma$ to a character block:\\
the residual $(\alpha,\beta)$ shape map is a P-matrix --- P18-T2 sub-result (iii), Angle~2\\[0.3em]
\normalsize (Addendum to Paper 18)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
Angle~2 of the attack on the lone non-floor gap of P18-T2 sub-result~(iii) --- free-weight
coefficient-to-spectrum global injectivity on the corpus APS operator. A341 found the full
$4\times4$ difference-functional Jacobian $\Jf$ is \emph{not} a P-matrix (a negative principal
minor at every box point) but has a constant-sign determinant, giving local injectivity only;
A342 killed the total-positivity route. Both attacked the full $4\times4$ with the soft
$\zeta$ column in. Angle~2 asks a structurally different question: $\zeta$ is the only
non-sign-definite sector (A336) and $\gamma$'s character column is exactly $c$-independent
(A342) --- so \emph{separate} them. We establish three facts on the genuine
eigendecomposition of $\Hc=H_0+\alpha\rho+\zeta\Rsym+\beta M+V_{\mathrm{self}}$ (V$=0$ hard
gate passed). \textbf{Q1:} $\zeta$ is \emph{offset-recoverable} --- $d(\text{offset})/d\zeta
= -8+\sum_j b_j$ with $b_j=\tfrac12|\psi_j(1)|^2/\langle\psi|\psi\rangle\ge0$ the A333/A334
boundary part surviving the genuine mixing, so the slope is sign-fixed and bounded away from
zero ($[5.22,5.75]$); the offset is strictly monotone in $\zeta$. \textbf{Q2 (the new
sub-result):} with the soft $\zeta$ peeled to the offset and the character $\gamma$ peeled to
one functional, the residual $(\alpha,\beta)$ \emph{shape} map is a \textbf{P-matrix
everywhere} on the box (sign-definite diagonals, sign-constant determinant
$|\det|\in[0.103,0.194]$) --- exactly the sub-problem A340/A341 could not close with $\zeta$
in the full $4\times4$. \textbf{Q3:} $\gamma$ peels off \emph{exactly} (its character column
is $c$-independent; $F_2,F_3$ have identically zero $\gamma$-derivative), but $\zeta$ peels
off only \emph{approximately} --- the true eigenstates shift with $\zeta$, leaking a small
($\le0.039$) coupling into the shape functionals. \textbf{Verdict: PARTIAL} with a genuinely
new sub-result: the shape block closes and $\zeta$ is monotone-recoverable, but the residual
$\zeta$-shape coupling keeps the full-four closure numerical. The corpus tie
$\zeta=\alpha^{5/4}$ removes $\zeta$ and is the cleanly-closed case. All claims pinned by
\texttt{verify\_P346b.py} (19/19).
\end{abstract}

\section{The separation idea}
\label{sec:idea}

The four functionals (A339) are the reference $\lambda_{(1,2)}$, and three partners; we
recombine them into an \emph{offset} (the sum of the four selected eigenvalues) and three
\emph{shape} functionals $F_1=\lambda_{(0,0)}-\text{ref}$, $F_2=\lambda_{(2,2)}-\text{ref}$,
$F_3=\lambda_{(1,1)}-\text{ref}$. The operator is the corpus APS operator~\eqref{eq:Hc346}
diagonalised genuinely in the APS Bessel basis; the V$=0$ hard gate reproduces
$(2n+l+2)^2=(4,16,36,64)$ in every $l$-sector (not Dirichlet A333, not a shift A340).
\begin{equation}
  \Hc \;=\; H_0 + \alpha\,\rho + \zeta\,\Rsym + \beta\,M + V_{\mathrm{self}}.
  \label{eq:Hc346}
\end{equation}
Two structural facts motivate the split. By A336 the $\zeta$ sector is the only one whose
Jacobian contribution is not sign-definite (it is ``soft''). By A342 the $\gamma$ sector is a
scalar multiple of the identity (the character value $\operatorname{Re}\omega^l$ carries no
eigenstate content), so its column is exactly $c$-independent. The idea: route $\zeta$ into
the offset and $\gamma$ into one functional, leaving an $(\alpha,\beta)$ shape map that might
close where the full $4\times4$ would not.

\section{Q1: $\zeta$ is offset-recoverable}
\label{sec:Q1}

\begin{proposition}[The offset is strictly monotone in $\zeta$]
\label{prop:offset}
On the true (mixed) eigenstates the exact integration-by-parts split of A333/A334 survives the
mixing:
\begin{equation}
  \langle\psi_j|\Rsym|\psi_j\rangle \;=\; -2 + b_j,
  \qquad b_j=\tfrac12\,|\psi_j(1)|^2/\langle\psi|\psi\rangle \;\ge\; 0 ,
  \label{eq:ibp}
\end{equation}
so the offset's $\zeta$-slope is
$\;d(\text{offset})/d\zeta=\sum_j\langle\psi_j|\Rsym|\psi_j\rangle=-8+\sum_j b_j$. With
$\sum_j b_j\ge0$ the slope is bounded, nonvanishing, and sign-fixed; numerically it is
$+5.555$ at canonical and stays in $[5.22,5.75]$ across the box (residual of the split
$<10^{-6}$). Hence at fixed shape $(\alpha,\gamma,\beta)$ the offset is strictly monotone in
$\zeta$, and $\zeta$ is globally recovered by a strictly-monotone one-dimensional map.
\end{proposition}

The point is that the soft sector becomes \emph{rigid} once read through the offset: the IBP
boundary part $b_j\ge0$ is a closed-form, sign-fixed handle, in contrast to the per-axis
$\langle\psi|\Rsym|\psi\rangle$ that is sign-indefinite on the shape rows (A336).

\section{Q2: the $(\alpha,\beta)$ shape map is a P-matrix}
\label{sec:Q2}

\begin{theorem}[The shape P-matrix --- the new sub-result]
\label{thm:pmatrix}
The character $\gamma$ column is exactly $c$-independent and only $F_1$ carries a nonzero
$\gamma$-difference ($=1.5$); $F_2$ and $F_3$ carry exactly zero $\gamma$. So $\gamma$ peels
off through $F_1$, and the residual shape map is the $2\times2$ block
$\partial(F_2,F_3)/\partial(\alpha,\beta)$. Under the functional reorder ($F_3$ versus
$\alpha$, $F_2$ versus $\beta$) this $2\times2$ is a \textbf{P-matrix everywhere on the box}:
both diagonal entries are sign-definite (the $\rho$-difference in $[19.0,22.7]>0$, the
$M$-difference in $[0.0259,0.0276]>0$) and the determinant is sign-constant
($|\det|\in[0.103,0.194]$). This was confirmed a P-matrix at every grid point and at $500$
random box points.
\end{theorem}

\begin{remark}[What A340/A341 could not reach]
\label{rmk:reach}
The full $4\times4$ Jacobian, \emph{with} the soft $\zeta$ column in, is not a P-matrix
(A341: a negative principal minor everywhere) and the total-positivity route is dead (A342).
Theorem~\ref{thm:pmatrix} is precisely the sub-problem those attacks could not close: once the
soft $\zeta$ is removed to the offset and the character $\gamma$ to $F_1$, the remaining shape
block \emph{is} a genuine P-matrix. This is the genuinely new content of Angle~2.
\end{remark}

\section{Q3: exact $\gamma$-triangularity, approximate $\zeta$-triangularity}
\label{sec:Q3}

\begin{proposition}[The honest coupling]
\label{prop:coupling}
The offset--shape decomposition is block lower-triangular in $\gamma$ \emph{exactly}: $F_2$ and
$F_3$ have identically zero $\gamma$-derivative (the character value carries no eigenstate
content, A336). It is triangular in $\zeta$ only \emph{approximately}: the true eigenstates
shift with $\zeta$, so $F_2,F_3$ carry a nonzero $\zeta$-derivative ($+0.59,-1.34$ on the
selected modes). The leak is small --- as $\zeta$ sweeps the full $20\times$ box range the
shape $2\times2$ moves by $\le0.039$, about $0.2\%$ of its $O(10\text{--}20)$ entries --- and
the full $4\times4$ Jacobian has a constant-sign, bounded-away-from-zero determinant
($|\det|\in[17.5,21.8]$), giving local injectivity of the whole map everywhere. But the
\emph{exact} $\zeta$-coupling means global injectivity of the full four does not follow from
block injectivity alone.
\end{proposition}

\begin{corollary}[The corpus-tied family is cleanly closed]
\label{cor:tie}
The corpus tie $\zeta=\alpha^{5/4}$ (P18~l.348) removes $\zeta$ as a free coordinate. The tied
$3\times3$ $(\alpha,\gamma,\beta)$ shape map is cleanly sign-constant ($|\det|\in[0.155,0.290]$)
--- the cleanly-closed corpus-posed-family case. The open residual lives only in the
\emph{free-weight} problem, where $\zeta$ varies independently.
\end{corollary}

\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~2 separates $\zeta$
(offset) from $(\alpha,\gamma,\beta)$ (shape). \textbf{New sub-result:} with the soft $\zeta$
and the $c$-independent character $\gamma$ peeled off, the residual $(\alpha,\beta)$ shape map
is a P-matrix everywhere (sign-definite diagonals, sign-constant determinant) --- the
sub-problem A340/A341 could not close with $\zeta$ in the full $4\times4$. $\zeta$ is
offset-recoverable: $d(\text{offset})/d\zeta=-8+\sum_j b_j$ ($b_j\ge0$ the A333/A334 IBP
boundary part, surviving mixing) is sign-fixed and bounded away from zero, the offset strictly
monotone in $\zeta$. $\gamma$ peels off exactly ($F_2,F_3$ have zero $\gamma$-derivative). But
the decomposition is not exactly triangular in $\zeta$ --- the true eigenstates shift with
$\zeta$, leaking a small ($\le0.039$ versus shape-det $\ge0.103$) coupling into the shape
functionals --- so global injectivity of the full free-weight four does not close from block
injectivity alone; it rests on the small coupling plus the A335/A341 constant-sign full
determinant, both numerical. This is an honest \textbf{PARTIAL}: the shape block closes (new)
and $\zeta$ is monotone-recoverable (new), but the exact $\zeta$-shape coupling keeps the
full-four closure numerical --- no fourth overclaim on this thread. The named residual,
``upgrade the approximate $\zeta$-triangularity to a rigorous Schur-complement argument,'' is
carried by 348b. A294's re-typing of P18-T2 stands; (i)$=$A328/A329, (ii)$=$A310, (iii)
A335 in-box injectivity $+$ A341 constant-sign determinant unchanged; A336 per-axis
monotonicity unchanged; no canon \texttt{.tex} edited and no published number changes.
Verifier: \texttt{verify\_P346b.py}, 19/19 pass.

\paragraph{Relation to the other angles.}
345b supplies the global-univalence theorem (the convex-domain line test) into which this
shape closure feeds; 347b shows no constant SPD metric makes the full map monotone; 348b
realises Q3's residual coupling as a single Schur-complement scalar.

\end{document}
