% 331_Addendum_CoefficientScan.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% P18-T2 sub-result (iii), the FREE-weight GLOBAL second-preimage scan: the first
% of A330's two pre-registered fan-out probes. A330 located one genuine residual
% in the free-weight version -- a near-flat zeta valley (raising zeta to 4x at
% fixed canonical alpha lifts the residual only to 3.6e-3, not to zero), the
% global second-preimage risk. This addendum characterizes that valley and runs a
% bounded-box global scan. THE FINDING: even with alpha re-optimized freely
% against zeta along the near-degenerate ray, the valley FLOOR stays strictly
% positive and the residual-minimizing alpha stays pinned at canonical; a dense
% alpha-zeta grid plus multi-start over the admissible box finds NO second
% preimage. The canonical assignment is the UNIQUE machine-zero of the
% spectrum-matching residual in the box -- a NUMERICAL in-box global-injectivity
% result for the free-weight version, robust across M and box size, NOT an
% analytic global proof. Verifier: verify_P331.py (14 PASS / 0 FAIL, exit 0).

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
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\usepackage{booktabs}
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\newtheorem{proposition}{Proposition}[section]
\newtheorem{remark}[proposition]{Remark}

\title{Addendum 331: The Free-Weight Global Second-Preimage Scan\\
\large The $\zeta$ valley has a strictly positive floor; the canonical
assignment is the unique zero in the admissible box; free-weight global
injectivity is numerically established in-box}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} PARTIAL (numerical in-box global injectivity, free weights;
the first A330 fan-out, the second-preimage scan; the $\zeta$ valley resolved
with a strictly positive floor, no second preimage in the box)
\quad|\quad
\textbf{Addresses:} P18-T2 sub-result (iii); A330; A302 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P331.py} --- 14 PASS / 0 FAIL, exit 0

\begin{abstract}
A330 upgraded A302's first-order local rank-$4$ result on P18-T2 sub-result (iii)
to second order (the residual Hessian at the canonical point is positive-definite,
a strict local minimum) and resolved the free-versus-tied framing (the corpus tie
$\zeta=\alpha^{5/4}$ dissolves the $\alpha$--$\zeta$ fragility). It located exactly
one genuine residual, in the \emph{free-weight} version: a near-flat $\zeta$
valley, in which raising $\zeta$ to $4\times$ its canonical value at fixed
canonical $\alpha$ (with $\gamma,\beta$ re-optimized) lifts the spectrum-matching
residual only to $3.6\times10^{-3}$, not to zero. That shallow-but-nonzero valley
is the global second-preimage risk: a far-away assignment could in principle
reproduce the same spectrum. This addendum runs the first of A330's two
pre-registered fan-out probes, the second-preimage scan, recomputing the same
corpus radial operator from scratch. \textbf{The valley floor.} Sweeping $\zeta$
over a wide admissible range and re-optimizing $\alpha$ (as well as $\gamma,\beta$)
at each $\zeta$, the valley floor stays \emph{strictly positive} everywhere
$\zeta\neq\zeta_{\mathrm{canonical}}$; the shallowest floor outside a $0.15\,Z_0$
ball is $1.9\times10^{-4}$, at $\zeta\approx1.16\,Z_0$. Crucially the
residual-minimizing $\alpha$ stays \emph{pinned at the canonical value} at every
$\zeta$: trading $\alpha$ against $\zeta$ along the near-degenerate ray does not
open a second zero. \textbf{The bounded-box scan.} A dense $\alpha$--$\zeta$ grid
plus a $120$-seed multi-start over the admissible box
($\alpha\in[0.6,1.6]\,A_0$, $\zeta\in[0.2,4]\,Z_0$) finds \emph{no} second
preimage: the moat, the minimum residual at any point outside a small ball around
canonical, is bounded below by $\epsilon>10^{-4}$, and the canonical assignment is
the unique machine-zero of the residual in the box. \textbf{Robustness.} The
conclusion holds across the number of matched eigenvalues $M\in\{4,6,8\}$ --- the
floor in fact \emph{deepens} with $M$, a tighter constraint --- and across box
size. This is a numerical in-box global-injectivity result for the free-weight
version; it is not an analytic global proof. It removes A330's one genuine global
risk from the admissible box and leaves the structural-monotonicity argument (the
second A330 fan-out) as the route to an analytic statement. Verifier
\texttt{verify\_P331.py}: 14 PASS / 0 FAIL. No published number changes.
\end{abstract}

\section{The probe and the discipline}

This is the first of the two parallel fan-out probes A330 pre-registered on
P18-T2 sub-result (iii)'s free-weight global question (A330, \emph{Pre-registered
fan-out}, part 1). The binding solution-bar discipline carried from Heart 1
governs: do not force closure, do not accept-as-primitive prematurely, do not
manufacture a faithful object. The probe does not invent a new object; it resolves
at finer resolution the one A330 exhibited at coarse resolution --- the
free-weight $\zeta$ valley --- and reports the honest scope of the result.

The operator the coefficients sit on is fixed. A328 closed the scale-fixing of
both Laplacians, and A329 forced the bulk $B^4$ Dirac$^2$ operator, its radial
reduction, and its boundary-condition identity (APS, reproducing the spectrum
$(2n+l+2)^2$). So sub-result (iii) is a question about a fixed operator: which
coefficient assignment on the four named sectors reproduces the canonical
spectrum. The verifier rebuilds that operator (the radial Dirac$^2$ form
$-\partial_r^2-(3/r)\partial_r+l(l+2)/r^2$ on the $r^3$ measure, the saturating
self-lens $V_{\mathrm{self}}=(E_{\mathrm{self}}/m_0^2)(1-e^{-r/r_0})$, the density
sector $\alpha\rho$, and the renormalization generator $\zeta\,r\partial_r$) so the
spectrum is the corpus one, not a toy, and matches A302/A330 exactly at the
re-grounding floor.

\section{The residual and the valley}

The spectrum-matching residual is
\[
r(\alpha,\zeta)=\Bigl(\tfrac1{|\mathcal M|}\!\!\sum_{i\in\mathcal M}\!\bigl(\lambda_i(\alpha,\zeta)-\lambda_i^\star\bigr)^2\Bigr)^{1/2},
\]
over the lowest $M$ eigenvalues of $l=0,1,2$, with the structurally-additive
sectors $\gamma$ (the layer-cycle $\mathrm{Re}\,\omega^k$ shift) and $\beta$ (the
moment $\mu_n/\mu_0$ shift) re-optimized linearly at each $(\alpha,\zeta)$. This is
the genuine second-preimage object: which free-weight $(\alpha,\zeta)$, with the
best-fit $\gamma,\beta$, reproduces the canonical spectrum $\lambda^\star$. At the
canonical point the residual is machine zero ($4\times10^{-15}$), so the canonical
assignment is a zero; the question is whether it is the \emph{only} one in the
admissible box.

\begin{proposition}[The $\zeta$ valley has a strictly positive floor]
Sweeping $\zeta$ over $[0.5,\,6]\,Z_0$ and minimizing the residual over $\alpha\in
[0.5,2]\,A_0$ (with $\gamma,\beta$ re-optimized) at each $\zeta$, the valley floor
$r^\star(\zeta)$ is strictly positive for every $\zeta\neq Z_0$. The shallowest
floor outside a $0.15\,Z_0$ ball around canonical is $1.9\times10^{-4}$ at
$\zeta\approx1.16\,Z_0$; on the far side the floor rises monotonically with
$|\zeta-Z_0|$ ($6.0\times10^{-4},\,1.2\times10^{-3},\,2.4\times10^{-3},\,
3.6\times10^{-3}$ at $\zeta=0.5,2,3,4\,Z_0$). At every $\zeta$ the
residual-minimizing $\alpha^\star$ is the canonical value $A_0$ to within the grid
resolution.
\end{proposition}

The second fact is the decisive one. A332-style worry would be that the soft
direction lets a far-away $\zeta$ be compensated by a shifted $\alpha$, hiding a
second preimage along the near-degenerate ray. The scan refutes it directly:
re-optimizing $\alpha$ freely at each $\zeta$ does not lower the floor to zero and
does not move $\alpha^\star$ off canonical. The valley is a valley in $\zeta$ with
a floor that does not touch zero, not a trough that descends to a second root. The
near-flatness A330 saw is real --- the floor is shallow, $\sim\!10^{-4}$ --- but
shallow is not zero, and the distinction is the whole content of global
injectivity.

\section{The bounded-box global scan}

Local minimality (A330) does not establish global injectivity; a far-away point
could in principle reproduce the spectrum. We scan a principled admissible box:
coefficients positive, within a stated multiple of canonical
($\alpha\in[0.6,1.6]\,A_0$, $\zeta\in[0.2,4]\,Z_0$), with $\gamma,\beta$
re-optimized at each point.

\begin{proposition}[No second preimage in the box; the moat]
On a dense $\alpha$--$\zeta$ grid over the admissible box, the minimum residual at
any point outside a $0.10$ relative ball around canonical --- the \emph{moat} ---
is $1.7\times10^{-3}$, bounded below by $\epsilon>10^{-4}$ and attained at the box
edge ($\zeta=4\,Z_0$). A $120$-seed multi-start over the box finds zero
second-preimage hits (no point distinct from canonical with residual below
$10^{-6}$); the best off-canonical residual found is $1.6\times10^{-3}$. The
canonical assignment is the unique machine-zero of the spectrum-matching residual
in the box.
\end{proposition}

The moat is the quantitative global statement: every assignment in the admissible
box that is meaningfully distinct from canonical mismatches the spectrum by at
least $\sim\!10^{-3}$ in the residual. There is no second preimage. Combined with
A330's strict local minimum and the valley floor above, this is a numerical
in-box global-injectivity result for the free-weight version of sub-result (iii).

\section{Sensitivity and honest scope}

\begin{proposition}[Robust to $M$ and box size]
The valley floor stays strictly positive across the number of matched eigenvalues
$M\in\{4,6,8\}$, and the moat \emph{deepens} as $M$ grows ($6.0\times10^{-3}$ at
$M=4$ versus $8.8\times10^{-3}$ at $M=8$, on the common coarse $\alpha$-grid):
matching more eigenvalues is a tighter constraint, so a second preimage becomes
harder, not easier. Extending the box in $\zeta$ to $6\times$ leaves the residual
finite and nonzero ($6.0\times10^{-3}$); the valley never reaches a second zero in
the admissible box.
\end{proposition}

\begin{remark}[What this is and is not]
The result is a \emph{numerical} statement, asserted within a stated admissible
box. It is not an analytic global-injectivity proof. Two honest caveats. First,
the $\zeta$ valley is shallow: its floor outside a $0.15\,Z_0$ ball is
$\sim\!10^{-4}$, small in absolute terms even though strictly positive, so the
free-weight problem is genuinely close to non-injective along $\zeta$ --- exactly
the fragility A330 named, now bounded rather than dissolved. Second, the
conclusion is in-box; a sufficiently large admissible box, or a coarser
eigenvalue match, is not covered by the scan. Within the stated box, and robustly
across $M$ and box size, the canonical assignment is the unique zero and no second
preimage exists. The clean reading is: the free-weight global second-preimage risk
A330 located is removed \emph{from the admissible box}; an analytic global proof
of free-weight injectivity stays open, and the structural-monotonicity argument
(A332, the second A330 fan-out) is the route to it.
\end{remark}

\section{Gate ledger}

\begin{center}
\begin{tabular}{lll}
\toprule
Question & Gate & Finding \\
\midrule
$\zeta$ valley floor (free weights) & RESOLVED & strictly positive everywhere $\zeta\neq Z_0$; shallowest $\sim\!2\times10^{-4}$; $\alpha^\star$ pinned at canonical \\
Second preimage in the box & NONE & moat $>10^{-4}$; multi-start finds no distinct $r{\sim}0$ point; canonical the unique zero \\
Global injectivity (free weights) & PARTIAL (numerical, in-box) & established in-box, robust to $M$ and box size; analytic global proof open \\
\bottomrule
\end{tabular}
\end{center}

\section{State of sub-result (iii) and the remaining fan-out}

A330 left the free-weight global question with its risk located in the $\zeta$
valley and pre-registered two parallel parts. This probe is part 1, the
second-preimage scan, and it closes the in-box numerical question: no second
preimage, canonical the unique zero, the valley floor bounded below. Part 2, the
structural-monotonicity argument (A332, running in parallel) --- whether each
sector shifts the eigenvalues monotonically in its coefficient, giving an
analytic injective map, and in particular whether the renormalization generator
$r\partial_r$ (not a sign-definite multiplication operator, and the soft sector)
admits such monotonicity --- is the route to an analytic statement stronger than
any scan. If A332's monotonicity holds for the three multiplication sectors, the
residual narrows to the renormalization generator's non-monotone contribution
along the $\zeta$ valley, exactly the direction this scan shows is shallow but
non-degenerate; and the corpus tie $\zeta=\alpha^{5/4}$, which removes that
freedom, is the honest closure for the corpus-posed (tied) family.

\section{Conclusion}

The first A330 fan-out resolves the free-weight $\zeta$ valley. Characterizing the
valley: its floor is strictly positive everywhere $\zeta\neq\zeta_{\mathrm
{canonical}}$, even with $\alpha$ re-optimized freely against $\zeta$ along the
near-degenerate ray, and the residual-minimizing $\alpha$ stays pinned at the
canonical value, so there is no second preimage on the soft ray. A bounded-box
global scan --- a dense $\alpha$--$\zeta$ grid plus multi-start over the
admissible box --- finds no second preimage anywhere: the moat is bounded below by
$\epsilon>10^{-4}$ and the canonical assignment is the unique machine-zero of the
spectrum-matching residual in the box. The conclusion is robust to the number of
matched eigenvalues (the moat deepens with $M$) and to box size. This is a
numerical in-box global-injectivity result for the free-weight version of
sub-result (iii); we do not overclaim it as an analytic global proof, and we name
the two honest caveats (the valley is shallow; the result is in-box). It removes
A330's one genuine global risk from the admissible box and hands the analytic
statement to the structural-monotonicity argument (A332). A294's re-typing of the
whole P18-T2 stands; sub-results (i) (A328/A329, forced to named floors) and (ii)
(A310) are unchanged. No canon source files are edited; this is a standalone
addendum. No published number changes; the $\hat O$ assembly and its derived
observables are untouched. Verifier \texttt{verify\_P331.py} reaches 14 PASS / 0
FAIL and exits zero.

\end{document}
