% 330_Addendum_CoefficientGlobal.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% P18-T2 sub-result (iii), Heart 2 grounding/orientation probe: the GLOBAL and
% HIGHER-ORDER coefficient-uniqueness question. A302 proved the FIRST-ORDER
% LOCAL form (free-weight Jacobian full rank 4 at the canonical point; the
% density alpha and renormalization zeta sectors ~99.86% spectrally collinear,
% the soft direction). This addendum re-grounds A302, resolves the FREE-vs-TIED
% framing, and takes the first concrete steps on global injectivity and second
% order. THE CENTRAL FINDING: the alpha-zeta soft PLANE that carries A302's
% fragility exists ONLY in the free-weight version; the corpus tie zeta=alpha^5/4
% (P18 l.348) collapses it to a stiff 1-D curve and dissolves the fragility. The
% residual-Hessian is positive-definite (strict local min, second order), and
% strengthens with sectors. The FREE-weight global injectivity is the warranted
% fan-out. Global uniqueness stays OPEN; no number changes. Verifier:
% verify_P330.py (13 PASS / 0 FAIL, exit 0).

\documentclass[11pt]{article}
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\newtheorem{proposition}{Proposition}[section]
\newtheorem{remark}[proposition]{Remark}

\title{Addendum 330: Coefficient Uniqueness, Global and Higher Order\\
\large The corpus tie $\zeta=\alpha^{5/4}$ dissolves the soft direction;
the residual is a strict local minimum; free-weight global injectivity
stays open}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} GROUNDED $+$ STEP (Heart 2 grounding probe for sub-result
(iii); A302 reproduced and upgraded to second order; the free-vs-tied framing
resolved, dissolving the $\alpha$--$\zeta$ fragility; one fan-out pre-registered)
\quad|\quad
\textbf{Addresses:} P18-T2 sub-result (iii); A302; A328; A329; A294 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P330.py} --- 13 PASS / 0 FAIL, exit 0

\begin{abstract}
P18-T2 sub-result (iii) asks whether the four coefficients
$(\alpha,\gamma,\zeta,\beta)$ are the only spectrum-matching assignment on
P18's now-forced radial operator (A328/A329). A302 established the first-order
\emph{local} form: treating the four as free assembly weights, the
spectrum-response Jacobian at the canonical point has full rank $4$, so the
coefficients are locally the unique spectrum-matching assignment; but the
density coupling $\alpha$ and the renormalization generator $\zeta$ are
$\sim\!99.86\%$ spectrally collinear, the \emph{soft direction} where uniqueness
is fragile. A302 was explicitly silent on global injectivity and on second
order. This addendum reproduces A302 from scratch and takes the first concrete
steps on the global frontier, and the central step resolves a framing question
that by itself changes the fragility reading. \textbf{Free versus tied.} A302
treated all four coefficients as free weights, the hardest version. P18 also
fixes $\zeta=\alpha^{5/4}$ (lines 348, 358, 579, 608), a functional tie of
$\zeta$ to $\alpha$. We show that the soft direction, in raw coordinates, is
almost exactly ``move $\zeta$ alone'' (the raw near-null direction of the
$\alpha$--$\zeta$ block is $(0.014,\,0.99990)$), and $\zeta$ is spectrally cheap
($\lVert\partial\lambda/\partial\alpha\rVert$ is $71\times$
$\lVert\partial\lambda/\partial\zeta\rVert$). The corpus tie forbids moving
$\zeta$ alone: along $\zeta=\alpha^{5/4}$, $d\zeta/d\alpha=\tfrac54\alpha^{1/4}$,
so any motion in $\zeta$ drags $\alpha$, which carries the large response. The
tie direction is transverse to the flat direction ($\lvert\text{overlap}\rvert
=0.356$), and the tied single-direction spectral response is $\sim\!820\times$
the free flat-direction response. The $\alpha$--$\zeta$ soft \emph{plane} that
carries A302's fragility exists only in the free-weight version; under the corpus
tie it collapses to a stiff $1$-D curve. \textbf{Second order.} The residual
Hessian at the canonical point is positive-definite (eigenvalues $3.20,3.62,54,
2.2\times10^{6}$), so the canonical point is a strict local minimum of the
spectrum-matching residual, not merely a non-degenerate critical point; the
softest curvature strengthens with sector coverage ($0.053\to0.845\to3.20$),
mirroring A302's first order. \textbf{Global.} A coarse second-preimage scan
exhibits the free-weight fragility directly: a near-flat $\zeta$ valley (raising
$\zeta$ to $4\times$ lifts the residual only to $3.6\times10^{-3}$), while the
tied curve is a sharp isolated well ($\alpha$ to $1.5\times/2\times$ lifts it to
$0.14/0.28$). So sub-result (iii) for the corpus-tied family is far less fragile
than A302's free-weight bound suggests; the free-weight global injectivity is
the genuinely hard residual and the warranted fan-out. Global uniqueness stays
open. Verifier \texttt{verify\_P330.py}: 13 PASS / 0 FAIL. No published number
changes.
\end{abstract}

\section{The grounding probe and the discipline}

This is the Heart 2 grounding/orientation probe (A322 pattern) for P18-T2's
third sub-result, the global and higher-order coefficient-uniqueness question.
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 to justify a fan-out. The probe establishes the framing and gates a
fan-out; it need not close anything. The brief-authoring discipline of
re-grounding against the canonical source before fixing intent governs too, and
it earns its keep: the question of whether sub-result (iii) means ``free
weights'' (A302's version) or ``the corpus-tied family'' is exactly the kind of
framing question that, read against P18 rather than the obvious default,
dissolves a fragility that looked irreducible.

The operator the coefficients sit on is itself now pinned. A328 closed the
scale-fixing of both Laplacians (the unit-ball metric forces the unit radius,
Laplace--Beltrami forces the zero shift), 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.

\section{Re-grounding A302}

\begin{proposition}[A302, reproduced]
With the four named sectors carrying free weights on P18's radial operator
(density $\alpha\rho$, layer-cycle $\gamma\,\mathrm{Re}\,\omega^k$,
renormalization $\zeta\,r\partial_r$, moment $\beta\,\mu_n/\mu_0$), the
column-normalized spectrum-response Jacobian at the canonical point
$(\alpha,\gamma,\zeta,\beta)=(1/137.036,\,3/4,\,\alpha^{5/4},\,3\pi/20)$ has
four nonzero singular values with $\sigma_{\min}/\sigma_{\max}=9.3\times10^{-3}$
on $l=0$, rank exactly $4$; the ratio strengthens to
$2.3\times10^{-2}\to3.5\times10^{-2}$ as $l\le1,2$ are added; and the density
and renormalization response columns are collinear to $0.9986$, the soft
direction.
\end{proposition}

The verifier reproduces every number of A302 from scratch (checks 1--3), using
A302's exact operator construction so the spectrum is the corpus one. This is
the re-grounding floor for everything that follows.

\section{The free-versus-tied resolution}

The decisive question the probe must answer is what sub-result (iii) means.
P18 line 376 states that ``the four coefficients $(\alpha,\gamma,\zeta,\beta)$
are pinned by the references cited in the Coefficient Determination
proposition'', each independently: $\alpha$ from $\int\rho=\alpha^{-1}$ (P03),
$\gamma=3/4$ from five-fold convergence (P10), $\zeta=\alpha^{5/4}$ from
dimensional analysis (P16), $\beta=3\pi/20$ from gravity (P17). The strongest
result is therefore global uniqueness of the \emph{free-weight} version, which
is A302's framing and the hardest one. But P18 also fixes $\zeta=\alpha^{5/4}$
as a \emph{functional} tie of $\zeta$ to $\alpha$ (lines 348, 358, 579, 608), and
whether that tie is part of the admissible assembly changes the soft direction.

\begin{proposition}[The corpus tie removes the soft plane]
The soft direction of A302, in raw $(\alpha,\zeta)$ coordinates, is almost
exactly ``move $\zeta$ alone'': the raw near-null right-singular vector of the
$\alpha$--$\zeta$ response block is $(0.014,\,0.99990)$. The renormalization
sector is spectrally cheap,
$\lVert\partial\lambda/\partial\alpha\rVert/
\lVert\partial\lambda/\partial\zeta\rVert=71.3$, so moving $\zeta$ alone barely
perturbs the spectrum, which is the fragility. The corpus tie
$\zeta=\alpha^{5/4}$ forbids this motion: along the tie
$d\zeta/d\alpha=\tfrac54\alpha^{1/4}$, so the admissible direction is
$(1,\tfrac54\alpha^{1/4})$, which is transverse to the soft direction
($\lvert\text{overlap}\rvert=0.356$), and the spectral response along the tied
curve is $\sim\!820\times$ the response along the free flat direction. The
$\alpha$--$\zeta$ soft plane collapses to a stiff $1$-D curve.
\end{proposition}

This is the central finding, and it is a clean resolution rather than a
manufactured one. A302's $99.86\%$ collinearity is a statement about the
\emph{shape} of the two normalized response columns; in raw magnitude the two
sectors are wildly different, $\alpha$ stiff and $\zeta$ soft. The fragility
lives entirely in the freedom to vary $\zeta$ independently of $\alpha$. The
corpus tie removes exactly that freedom. So the honest reading of A302's soft
direction is: it is the worst case of the \emph{free-weight} problem, and the
corpus does not pose the free-weight problem for $(\alpha,\zeta)$ --- it ties
them. For the corpus-tied family the $\alpha$--$\zeta$ fragility is dissolved
(checks 4--7).

\section{Second order: a strict local minimum}

A302 was first order, silent on second-order coincidences. We extend to the
Hessian. The spectrum-matching residual is
$R(c)=\sum_i(\lambda_i(c)-\lambda_i(c^\star))^2$; at the canonical point $c^\star$ the
residual and its gradient vanish, and the Hessian is exactly $2J^\top J$
(Gauss--Newton, exact at a zero-residual root). Positive-definiteness upgrades
A302's non-degenerate critical point to a strict local minimum.

\begin{proposition}[Second-order rigidity]
The free-weight residual Hessian $2J^\top J$ at the canonical point is
positive-definite, with eigenvalues $3.20,\,3.62,\,54,\,2.19\times10^{6}$; the
canonical assignment is a strict local minimum of the spectrum-matching
residual. The softest curvature strengthens with sector coverage,
$0.053\to0.845\to3.20$ for $l\le0,1,2$, mirroring A302's first-order
strengthening. The tied residual Hessian (coordinates $\alpha,\gamma,\beta$, with
$\zeta$ slaved) is also positive-definite, and the $\alpha$--$\zeta$ soft plane
is absent because $\zeta$ is no longer an independent coordinate.
\end{proposition}

So sub-result (iii) is upgraded from first-order local to second-order local: the
canonical point is a genuine strict minimum, and the strengthening-with-coverage
structure A302 found at first order persists at second (checks 8--10).

\section{Global: the free-weight valley and the tied well}

Local and second-order rigidity do not establish global injectivity: a far-away
point could in principle reproduce the same spectrum. The risk is concentrated
along the soft $\zeta$ direction. A coarse second-preimage scan over a bounded
admissible box, with $\gamma,\beta$ re-optimized at each $(\alpha,\zeta)$,
exhibits the free-weight fragility directly and the tie's cure.

\begin{proposition}[Free-weight valley, tied well]
At the canonical point the residual is machine zero ($8\times10^{-16}$), so the
canonical assignment is the global minimum. In the free-weight problem the
$\zeta$ direction is a near-flat valley: raising $\zeta$ to $4\times$ its
canonical value (at fixed canonical $\alpha$, $\gamma,\beta$ re-optimized) lifts
the residual only to $3.6\times10^{-3}$. Along the tied curve
$\zeta=\alpha^{5/4}$ the same problem is a sharp isolated well: raising $\alpha$
to $1.5\times/2\times$ lifts the residual to $0.14/0.28$, more than $10\times$
the free $\zeta$ valley. The global second-preimage risk lives in the free-weight
$\zeta$ valley, and the corpus tie removes it.
\end{proposition}

This is the global statement matching the local one. The free-weight problem has
a shallow valley along $\zeta$ where a far-away $\zeta$ with re-optimized
$\gamma,\beta$ nearly reproduces the spectrum --- the genuine global fragility,
and the reason a clean global-injectivity proof of the \emph{free-weight} version
is non-trivial. The corpus-tied problem replaces the valley with a sharp well
(checks 11--13). We do not claim global injectivity is closed; we exhibit
exactly where the free-weight risk lives and that the tie removes it.

\section{Gate ledger}

\begin{center}
\begin{tabular}{lll}
\toprule
Approach & Gate & Finding \\
\midrule
Free-vs-tied (the soft direction) & RESOLVED & tie $\zeta=\alpha^{5/4}$ removes the $\alpha$--$\zeta$ soft plane; fragility dissolved for the tied family \\
Second-order (Hessian) rigidity & DERIVED & residual Hessian positive-definite (strict local min); strengthens with sectors \\
Global injectivity (free weights) & PARTIAL / fan-out & free-weight $\zeta$ valley is the genuine global risk; second-preimage scan / monotonicity warranted \\
\bottomrule
\end{tabular}
\end{center}

\begin{remark}[Gate per approach]
\emph{Free-vs-tied} --- RESOLVED, and machine-derivable: the corpus tie is a P18
identity, and imposing it dissolves the $\alpha$--$\zeta$ fragility (the soft
plane becomes a stiff curve, tied response $\sim\!820\times$ the free flat
response). This is the high-value finding of the probe. \emph{Second order} ---
DERIVED: the residual Hessian is positive-definite, a strict local minimum,
strengthening with coverage; A302 is upgraded from first to second order, no
fan-out needed. \emph{Global injectivity} --- PARTIAL: local and second-order
rigidity plus the dissolved tie make the global question far less fragile than
A302's free-weight bound, but global injectivity of the free-weight version is
not established; the free-weight $\zeta$ valley is the residual to control, and
it warrants a fan-out.
\end{remark}

\section{Pre-registered fan-out}

One fan-out is warranted, on the free-weight global question, with two
parallel parts that test the same residual from opposite sides:

\begin{enumerate}
\item \textbf{Global second-preimage scan, free weights.} A finer bounded-box
scan focused on the $\alpha$--$\zeta$ near-degenerate ray with $\gamma,\beta$
re-optimized, matching the lowest $M$ eigenvalues, testing for a second
$(\alpha,\gamma,\zeta,\beta)$ that reproduces the target spectrum. A clean ``no
second preimage in the admissible box'' is a real (numerical) global result; the
$\zeta$ valley exhibited here at coarse resolution is the object to resolve.
\item \textbf{Structural monotonicity argument.} Each sector enters the operator
as a positive multiplication or first-order generator; if each shifts the
eigenvalues monotonically in its coefficient, the coefficient-to-spectrum map is
monotone hence injective, a structural global result stronger than any scan.
Assess whether the $\zeta$ sector (the renormalization generator $r\partial_r$,
which is not a sign-definite multiplication operator) admits such a monotonicity,
since that is precisely the soft sector.
\end{enumerate}

If the scan finds no second preimage and the monotonicity holds for the three
multiplication sectors, the residual narrows to the renormalization generator's
non-monotone contribution along the $\zeta$ valley --- and the corpus tie, which
removes that freedom, is the honest closure for the corpus-posed (tied) family.
If neither closes the free-weight version, sub-result (iii) for free weights
stays open with its risk precisely located, which is itself a faithful result.

\section{Conclusion}

The grounding probe reproduces A302's first-order local rank-4 result from
scratch and advances the third sub-result on three fronts. The free-versus-tied
framing is resolved: the $\alpha$--$\zeta$ soft plane that carries A302's
fragility exists only in the free-weight version, and the corpus tie
$\zeta=\alpha^{5/4}$ (P18 line 348) collapses it to a stiff curve, dissolving the
fragility for the corpus-posed family. The second-order question is answered: the
residual Hessian is positive-definite, so the canonical point is a strict local
minimum, and the rigidity strengthens with sector coverage, upgrading A302 from
first to second order. The global question is framed and its risk located: the
free-weight problem has a near-flat $\zeta$ valley, the genuine global fragility,
while the tied curve is a sharp well. We pre-register one fan-out on the
free-weight global injectivity (a finer second-preimage scan and a structural
monotonicity argument), and we do not manufacture more. Global uniqueness of the
free-weight version stays open with its risk precisely named; for the
corpus-tied family the fragility is dissolved. 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\_P330.py} reaches 13 PASS / 0
FAIL and exits zero.

\end{document}
