% 327_Addendum_G1Audit.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1 (corrected), probe 1 of a NEW thread: derive premise G1 from the master-
% operator Ohat dynamics. A326 reduced OI-287-1's why-question to ONE irreducible
% premise G1 (the mass-law coefficient a family draws is the character-RESONANT layer
% of its T_cycle eigenspace), and named the eigenspaces as Z_3 Fourier combinations
% SPREAD equally over the three layers, so "family k draws a single degree" is a
% PROJECTION choice. This addendum runs the MANDATORY FRAMING AUDIT FIRST (binding
% process rule, probe 1) on the candidate derivation objects, testing each against the
% A296/A321 exclusion boundary (the withdrawn overlap-diagonal). G1 is decomposed into
% a SELECTION half (which coefficient's block) and a MAGNITUDE half (the strength = the
% coefficient's VALUE). Verdict: O-PT-diagonal (first-order PT of the density sector
% lambda*rho) is GATED OUT because it is an affine image of the withdrawn overlap-
% diagonal -- proved NUMERICALLY by building both. O-equivariant-block (A323 Schur) and
% O-scaling-weight (A312 renorm scaling dimension) are GATED IN for the SELECTION half
% only (corpus-faithful, no overlap), and the scaling weight reads the DEGREE (1,2,3) =
% the selection index, NOT the coefficient VALUE (2,3,16). No corpus identity expresses
% (2,3,16) as a scaling dimension / index / heat-kernel multiplicity; the c_n are the
% Taylor integers of rho (P03). So the SELECTION half of G1 is DERIVED (corpus-faithful)
% and the MAGNITUDE half is intrinsically the coefficient VALUE, whose only eigenstate
% readout is the withdrawn overlap-diagonal -- an irreducible flagged posit matching
% A287's PSLQ-honesty. Verdict NO-FAN-OUT: nothing tempting gates in for the magnitude;
% OI-287-1 is as closed as honesty allows. The overlap-diagonal stays withdrawn; no
% published number changes. Verifier: verify_P327.py.

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[margin=1in]{geometry}
\usepackage{booktabs}
\usepackage{hyperref}

\newtheorem{proposition}{Proposition}[section]
\newtheorem{remark}[proposition]{Remark}

\title{Addendum 327: The $G1$ Audit\\
\large The selection half of $G1$ is corpus-faithful; the magnitude half is the
withdrawn overlap-diagonal, an irreducible posit}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} NO-FAN-OUT (framing audit, probe 1; the magnitude half of $G1$ is the
withdrawn overlap-diagonal, an irreducible flagged posit) \quad|\quad
\textbf{Addresses:} OI-287-1 (corrected); A326; A323; A312; A296; A287 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P327.py} --- all PASS

\begin{abstract}
A326 reduced the why-question of OI-287-1 to one irreducible premise $G1$: the mass-law
coefficient a charged-lepton family draws is the character-resonant layer of its
$T_{\rm cycle}$ eigenspace. The eigenspaces are $\mathbb{Z}_3$ Fourier combinations
spread equally over the three layers, so $G1$ is a projection choice, not forced by
$\mathbb{Z}_3$-equivariance. This addendum opens a new thread, to derive $G1$ from the
master-operator $\hat O$ dynamics, and per the binding process rule it runs the mandatory
framing audit \emph{first}. The danger is acute: the obvious operationalization of $G1$,
the first-order perturbative shift of the density sector $\lambda\rho$ on an eigenstate,
is the overlap-diagonal that A296 excluded and A321 withdrew. We decompose $G1$ into a
\emph{selection} half (which coefficient's block a family couples to) and a
\emph{magnitude} half (that the correction strength equals the coefficient's value), and
audit four candidate derivation objects against the exclusion boundary. The first-order
density-sector shift, \textbf{O-PT-diagonal}, is gated out: it is
$\alpha\sum_k c_k\pi^k\langle\psi_n|x^k|\psi_n\rangle$, an affine image of the very
$\langle\psi_n|x^k|\psi_n\rangle$ magnitudes A296 excluded on scale and shape, with the
withdrawn diagonal as its $k=n$ term. We prove this numerically by building both. The
$\mathbb{Z}_3$ Schur block, \textbf{O-equivariant-block}, and the renormalization scaling
weight, \textbf{O-scaling-weight}, are gated in for the selection half only: both are
corpus-faithful and form no overlap, but the scaling dimension of the degree-$k$ density
term is $p=k$, the degree $(1,2,3)$, the selection index, not the coefficient value
$(2,3,16)$. No corpus identity expresses $(2,3,16)$ as a scaling dimension, an index, or
a heat-kernel multiplicity; the $c_n$ are the Taylor integers
$\rho^{(n)}(0)/(\pi^n n!)$, set by $\rho$'s definition. The selection half of $G1$ is
therefore derived, corpus-faithful, by two independent objects; the magnitude half is
intrinsically the coefficient value, whose only readout off an eigenstate is the
withdrawn overlap-diagonal. The verdict is NO-FAN-OUT: nothing tempting gates in for the
magnitude that is distinct from the withdrawn functional, so $G1$'s magnitude is an
irreducible flagged posit, matching A287's PSLQ-honesty, and OI-287-1 is as closed as
honesty allows. The overlap-diagonal stays withdrawn; no published number changes.
\end{abstract}

% ----------------------------------------------------------------------
\section{The Audit Frame and the Exclusion Boundary}

\begin{remark}[Probe 1 runs the framing audit first]
\label{rem:frame}
This is the opening probe of a new thread on OI-287-1, so the binding process rule
applies: re-ground the candidate objects against the corpus and gate out anything that
revives a withdrawn functional, before proposing any derivation. The withdrawn object is
the overlap-diagonal $B[n] = \langle\psi_n\,|\,c_k\pi^k x^k\,|\,\psi_n\rangle/\lambda_n$,
which A296 introduced as the ``bare Taylor probe'' and excluded in the same addendum (its
third proposition: the diagonal shift is short of the required log-eigenvalue shifts by
factors $32$ and $42$ at $n=2,3$, the two factors disagreeing by about $28\%$, so
``Excluded both on scale and on shape''), and which A321 withdrew as the object of the
problem after the A299--A320 arc chased its artifact gap across seven arenas. The verifier
rebuilds the production operator
$H = -\partial_x^2 + \rho'(x)^2/(2\Omega^2) + E_{\rm self}x^2(1-x)^2$ on Dirichlet
$[0,1]$, recovers the eigenvalues $(16.51, 51.24, 102.02)$, and reproduces the withdrawn
diagonal $(1, 1.12, 7.50)$ at $L_2 = 0.63$ to the target $(1, 1.5, 8)$, matching A299 and
A321. This is the boundary; the candidates are tested against it, not built from it.
\end{remark}

\begin{remark}[The selection / magnitude split of $G1$]
\label{rem:split}
$G1$ asserts two things at once. The \emph{selection} half: family-$k$ (character $k$)
draws the resonant character \emph{block} of $\rho$, i.e.\ \emph{which} of the three
coefficients it couples to. The \emph{magnitude} half: the resulting correction
\emph{strength} equals that coefficient's \emph{value} --- A287's
$c_n\,\alpha^{1/4}/\lambda_1$ per unit coefficient. The two halves have different
standing against the corpus, and the audit must judge them separately. The selection
half is a which-block statement, representation-theoretic; the magnitude half is a value,
and the only operator object that reads a coefficient's value off an eigenstate is an
expectation $\langle\psi|\cdot|\psi\rangle$ --- the withdrawn functional.
\end{remark}

% ----------------------------------------------------------------------
\section{The Gate, Object by Object}

\begin{proposition}[O-PT-diagonal is the withdrawn overlap-diagonal --- gated out]
\label{prop:opt}
The density-coupling sector of $\hat O$ is $\lambda\rho(x) = \alpha\rho(r)$, a
multiplication operator (P18 \S3.5). First-order perturbation theory of a multiplication
operator $V$ on a non-degenerate eigenstate $\psi_n$ gives the energy shift
$\langle\psi_n|V|\psi_n\rangle$. For $V = \alpha\rho = \alpha\sum_k c_k\pi^k x^k$ this is
\[
E^{(1)}_n \;=\; \alpha\sum_{k=1}^{3} c_k\pi^k\,\langle\psi_n|x^k|\psi_n\rangle ,
\]
an affine combination of the monomial expectations $\langle\psi_n|x^k|\psi_n\rangle$ ---
the same table the withdrawn overlap-diagonal is built from. The withdrawn diagonal is
literally the $k=n$ term of this expansion, divided by $\lambda_n$:
$\alpha\,c_{n+1}\pi^{n+1}\langle\psi_n|x^{n+1}|\psi_n\rangle = \alpha\,\lambda_n B_{\rm raw}[n]$,
verified to machine precision. O-PT-diagonal therefore reaches for exactly the
$\langle\psi|\cdot|\psi\rangle$ magnitudes A296 excluded on scale and shape. It revives
the withdrawn functional and is gated out, citing A296 and A321.
\end{proposition}

\begin{proposition}[O-equivariant-block is the selection half --- gated in for selection,
silent on magnitude]
\label{prop:block}
A $\mathbb{Z}_3$-equivariant density coupling commutes with $T_{\rm cycle}$ and is block-
diagonal across the three characters; by Schur orthogonality
$\tfrac13\sum_j\omega^{(a-b)j} = \delta_{a\equiv b}$, family-$k$ couples only to the
character-$k$ block (A323). This is a which-block statement: it forms no overlap and reads
no eigenstate. It fixes \emph{which} degree each family couples to (the degree-$k$ term
carries character $k\bmod 3$, so the present degrees $1,2,3$ carry characters $1,2,0$ and
each character selects one degree), but it says nothing about the coupling
\emph{magnitude}. The value $c_k = (2,3,16)$ is read off $\rho$'s definition, not produced
by the block. O-equivariant-block is gated in for the selection half, corpus-faithful and
overlap-free, and is silent on magnitude.
\end{proposition}

\begin{proposition}[O-scaling-weight reads the degree, not the value --- gated in for
selection, gated out for magnitude]
\label{prop:scaling}
The renormalization sector is $\mathcal{R} = r\,\partial_r + \Delta_\psi$ with
eigenfunctions $r^p$ and eigenvalue the scaling dimension $p$ (A312, P18 \S3.6). The
degree-$k$ density monomial $x^k$ has scaling dimension $p = k$: the renormalization
generator reads the \emph{degree} $(1,2,3)$, an index-like integer intrinsic to the
operator, with no overlap. But the degree is the selection index --- which layer, which
character --- not the coefficient value: the scaling weights are $(1,2,3)$ while the
coefficients are $(2,3,16)$, distinct data. The mass formula's exponential arises from
$\mathcal{R}$ (P18 \S4.5), and the A287 law enters $c_n$ as a subtraction in that
exponent, $\beta_n = 6\mu_1/\mu_0 - c_n\alpha^{1/4}/\lambda_1$; so the scaling sector is
the right home for the correction, and it supplies the selection weight $p=k$. It does
\emph{not} supply the magnitude. No corpus identity expresses $(2,3,16)$ as a scaling
dimension, an index, or a heat-kernel multiplicity: $(2,3,16)$ is neither the scaling
dimensions $(1,2,3)$, nor the $\mathrm{SO}(4)$-Casimir multiplicities $(1,4,9)$ of A314,
nor any degree sequence; the $c_n$ are the Taylor integers
$c_n = \rho^{(n)}(0)/(\pi^n n!)$, set by $\rho$'s definition in P03. O-scaling-weight is
gated in for the selection half and gated out for the magnitude half.
\end{proposition}

\begin{remark}[No fourth object yields the magnitude without an overlap]
\label{rem:fourth}
A coefficient's value is, by construction, an integer datum of $\rho$. To read it off the
state of family-$k$ requires a functional of $\psi_k$ that returns $c_k$, and the
$\mathbb{Z}_3$-equivariant such functionals that distinguish the three families are the
overlap expectations $\langle\psi_k|\cdot|\psi_k\rangle$ --- precisely the withdrawn
class. The selection-half objects (Schur block, scaling weight) return the
\emph{degree}, never the value. We searched for a faithful non-overlap object that yields
the value and found none; we decline to manufacture one to justify a fan-out.
\end{remark}

% ----------------------------------------------------------------------
\section{The Verdict}

\begin{table}[h]
\centering
\begin{tabular}{lll}
\toprule
\textbf{Object} & \textbf{Gate} & \textbf{Reason} \\
\midrule
O-PT-diagonal & OUT & affine image of the withdrawn overlap-diagonal (A296/A321) \\
O-equivariant-block & IN (selection only) & A323 Schur block; no overlap; silent on value \\
O-scaling-weight & IN (selection only) & A312 scaling dim $p=k$ is the degree, not the value \\
 & OUT (magnitude) & no index identity for $(2,3,16)$; Taylor integers of $\rho$ \\
\bottomrule
\end{tabular}
\caption{The gate ledger. The selection half of $G1$ is gated in, corpus-faithful, by two
independent objects; the magnitude half is gated out as the withdrawn overlap-diagonal.}
\label{tab:ledger}
\end{table}

\begin{proposition}[The selection half is derived; the magnitude half is an irreducible
posit]
\label{prop:verdict}
The selection half of $G1$ --- family-$k$ draws the resonant character block --- is
corpus-faithful and derived, supplied independently by the A323 Schur block and the A312
scaling-dimension degree, neither forming an overlap. The magnitude half of $G1$ --- the
correction strength equals the coefficient's value --- is intrinsically the coefficient
value $c_n$, whose only readout off an eigenstate is the withdrawn overlap-diagonal. No
faithful non-overlap object yields it. The magnitude half is therefore an irreducible
flagged posit, and it coincides with A287's own PSLQ-honesty, which already records the
strength $c_n\alpha^{1/4}/\lambda_1$ as a flagged, underived identification. The audit
does not weaken A287; it locates the residual exactly where A287 left it.
\end{proposition}

\paragraph{No fan-out.} The pre-registered question for this thread was whether any
$\hat O$-faithful object derives $G1$'s magnitude content without an eigenstate overlap.
The audit answers no, decisively. The selection half is already derived; the magnitude
half is the withdrawn functional. Nothing tempting gates in for the magnitude that is
distinct from the overlap-diagonal, so no parallel fan-out is warranted. This is itself
the result: $G1$'s magnitude is an irreducible flagged posit, and OI-287-1 is as closed
as honesty allows --- its selection mechanism derived (the $\mathbb{Z}_3$ character rule
of A323/A324, the bridge forced to $G1$ by A326), its magnitude a posit matching A287. A
clean negative is worth more than a manufactured object.

\paragraph{Status.} OI-287-1 remains open in its corrected form, and its residual is now
characterized precisely. The why-question splits into a selection half, which is derived
and corpus-faithful, and a magnitude half, which is the withdrawn overlap-diagonal and
hence an irreducible posit. To close the magnitude half one would have to revive the
functional A321 withdrew, which the seven-arena exclusion (A299--A320) and A296's
scale-and-shape exclusion both forbid. The honest terminus is therefore: posit $G1$'s
magnitude as a flagged corpus axiom, exactly as A287 already does for the strength. The
overlap-diagonal stays withdrawn; no wavefunction overlap is formed as a derivation
object; no published number changes; the A287 lepton law and its $10^{-4}$ accuracy are
untouched.

\paragraph{Verifier.} \texttt{verify\_P327.py}: recomputes $\alpha^{-1} = 4\pi^3+\pi^2+\pi$
and $c_n = \rho^{(n)}(0)/(\pi^n n!) = (2,3,16)$; rebuilds the production operator and the
withdrawn overlap-diagonal $(1,1.12,7.50)$ at $L_2 = 0.63$ and records A296's
scale-and-shape exclusion; builds the density-sector first-order PT shift and proves it
is $\alpha\sum_k c_k\pi^k\langle\psi_n|x^k|\psi_n\rangle$ with the withdrawn diagonal as
its $k=n$ term, gating O-PT-diagonal out; recomputes the Schur block and confirms it
selects which degree but not the value, gating O-equivariant-block in for selection only;
recomputes the renormalization scaling dimension $p=k$ and confirms it reads the degree
$(1,2,3)$, not the value $(2,3,16)$, with no index identity for $(2,3,16)$ against the
scaling dimensions, the SO(4)-Casimir multiplicities, or the degrees, gating
O-scaling-weight out for magnitude; resolves the selection / magnitude split; states the
NO-FAN-OUT verdict; and guards that the overlap-diagonal is built only to demonstrate the
gate-out, not revived as a derivation object. Twenty checks, all PASS.

\end{document}
