% 322_Addendum_GroundingAudit.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1 (corrected) grounding probe and framing audit. A321 reverted the open
% problem to its A287/A296 form: derive why the lepton mass correction law assigns
% level n the degree-n Taylor coefficient c_n of the density rho, the integers read
% directly. This addendum runs the mandatory A321-style framing audit at thread
% start, before any further probing. Three candidate objects a derivation could use
% are gated against the corpus. O-Taylor-origin (the n-th family's near-origin onset
% samples rho's n-th derivative) is EXCLUDED: A297 already measured the onset and
% found it linear for every family (Dirichlet psi(0)=0 forces leading power one), so
% the eigenstate cannot carry n in its origin jet; and any readout of c_n from psi_n
% is the A296/A297-excluded position-moment overlap, the withdrawn functional in
% disguise. O-moment is not the assignment (the moments give the hierarchy scale,
% not the family-to-degree pairing) and carries no corpus identity to gate a step on.
% O-Z3-grading (families = omega^k eigenspaces of T_cycle, P20) is corpus-mandated
% and distinct from the overlap-diagonal; it gates a first STRUCTURAL step, the
% consistency deg(rho) = #nonzero c_n = #Z_3 eigenspaces = #families = 3 (A301).
% Verdict GROUNDED+STEP, with honest scope: the gated step is the count-and-grading
% consistency, not the full derivation of why degree-k attaches to eigenspace-k.
% The overlap-diagonal stays withdrawn; no new arena search is opened.
% Verifier: verify_P322.py.

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

\title{Addendum 322: The Grounding Audit\\
\large A framing audit of the corrected selection rule, and the one
candidate object the corpus permits}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} GROUNDED$+$STEP (grounding probe and framing audit;
one gated object, a first structural step) \quad|\quad
\textbf{Addresses:} OI-287-1 (corrected); A321; A287 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P322.py} --- all PASS

\begin{abstract}
A321 found that the eighteen-probe overlap-diagonal arc had been
matching a functional the corpus excluded, and reverted OI-287-1 to its
A287 and A296 form: derive why the lepton mass correction law assigns
charged-lepton family $n$ the degree-$n$ Taylor coefficient $c_n$ of the
density $\rho$, with the coefficients read as the integers $(2,3,16)$.
This addendum runs the mandatory framing audit at the start of that
direction, before any further probing. Three candidate objects a
derivation could use are each gated against the corpus text. The
near-origin object, the reading that the $n$-th family's onset
$\sim x^n$ samples $\rho^{(n)}(0)$ and so extracts $c_n$, is excluded:
A297 already measured the onset and found leading power one for every
family, forced by the Dirichlet condition $\psi(0)=0$, so the eigenstate
carries no $n$ in its origin jet; and any readout of $c_n$ from $\psi_n$
is the position-moment overlap A296 and A297 set aside, the withdrawn
functional in another form. The moment object is a distinct use of
$\rho$ that supplies the hierarchy scale, not the family-to-degree
pairing, and carries no corpus identity on which to base a step. The
$\mathbb{Z}_3$-grading object, the three families as the $\omega^k$
eigenspaces of the layer-cycle operator of P20, is corpus-mandated and
distinct from the overlap-diagonal; it gates a first structural step,
the consistency that the degree of $\rho$, the number of nonzero
coefficients, the number of $\mathbb{Z}_3$ eigenspaces, and the number
of families are all three. The verdict is GROUNDED$+$STEP, with the
scope stated plainly: the gated step is the count-and-grading
consistency, not the derivation of why degree $k$ of $\rho$ attaches to
eigenspace $k$, which remains the open core. The overlap-diagonal stays
withdrawn and no new arena search is opened.
\end{abstract}

% ----------------------------------------------------------------------
\section{The Problem and the Audit}

\begin{proposition}[The integer assignment is the density's own jet; one
candidate object is corpus-mandated and distinct from the withdrawn
functional]
\label{prop:audit}
The coefficients of $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ are its
Taylor data at the center,
$c_n = \rho^{(n)}(0)/(\pi^n\,n!) = (2,3,16)$, normalized
$(1, \tfrac32, 8)$, computed here from
$\rho'(0) = 2\pi$, $\rho''(0) = 6\pi^2$, $\rho'''(0) = 96\pi^3$. The
corrected OI-287-1 asks why the mass correction law of A287 assigns
family $n$ the integer $c_n$ read directly. Three objects a derivation
could use are gated against the corpus:
\begin{itemize}
\item[(i)] \emph{The near-origin object.} The reading that the $n$-th
family's onset $\sim x^n$ samples $\rho^{(n)}(0)$ and so extracts $c_n$.
This is not corpus-mandated and is excluded. A297 measured the onset of
the lowest three eigenstates of the production operator and found leading
powers $(0.999, 0.997, 0.995)$, recomputed here, all one: the Dirichlet
condition $\psi(0)=0$ forces every family linear at the center, so the
origin jet carries no $n$. Further, any readout of $c_n$ from $\psi_n$ is
an overlap $\langle\psi_n|x^k|\psi_n\rangle$, the position-moment probe
A296 and A297 excluded and A321 withdrew. The object is the withdrawn
functional in another form.
\item[(ii)] \emph{The moment object.} The reading that an identity links
the moments $\mu_n = \int_0^1 x^n\rho\,dx$ to $c_n$. The moments are
$(\mu_0,\dots,\mu_3) = (137.04, 108.72, 90.18, 77.06)$ with
$\mu_0 = \alpha^{-1}$, monotone decreasing, with ratios at $L_2 = 7.3$
from $(1,\tfrac32,8)$; the corpus uses them for the mass hierarchy, not
for the family-to-degree pairing. The object is distinct from the
overlap-diagonal but empty: no corpus identity derives the assignment
from the moments.
\item[(iii)] \emph{The $\mathbb{Z}_3$-grading object.} The three families
as the $\omega^k$ eigenspaces of the layer-cycle operator,
$T_{\rm cycle}\psi^{(k)} = \omega^k\psi^{(k)}$, $k=0,1,2$, enforced by
$\pi_1(L(3,1)) = \mathbb{Z}_3$ (P20). This is corpus-mandated and
distinct from the overlap-diagonal: a representation-theoretic grading is
not an expectation value of $\psi_n$ against a monomial. It is the one
object that passes the gate.
\end{itemize}
\end{proposition}

% ----------------------------------------------------------------------
\section{The Gated Step and Its Scope}

\begin{remark}[A structural first step, and what it does not yet reach]
\label{rem:step}
On the gated object the first checkable step is the structural
consistency already recorded in A301, restated here as the grading
object's own statement: the degree of $\rho$, the number of nonzero
coefficients $c_n$, the number of $\mathbb{Z}_3$ eigenspaces, and the
number of families are all equal to three. The verifier confirms this.
The scope is stated plainly. The step establishes that the coupling-side
grading and the topological grading agree on the count; it does not
derive why degree $k$ of $\rho$ attaches to eigenspace $k$, which is the
open core of the corrected problem. A316 found that the real reduction of
the layer-cycle phase, $\mathrm{Re}(\omega^k)$, takes equal values on
$k=1$ and $k=2$ and folds families two and three, so the genuine pairing
requires the complex $\omega^k$ phase; the structural step does not
supply that pairing. The fan-out, to be entered only after this grounding
confirms the framing, is then: test whether the genuine complex
$\omega^k$ eigenspace $k$ pairs with the degree-$k$ graded piece of
$\rho$ by representation theory, at the structural and grading level
rather than as a magnitude functional; ask whether the identity
$c_n = \rho^{(n)}(0)/(\pi^n n!)$ is itself a reading of the
$\mathbb{Z}_3$-graded structure of $\rho$ on the lens space, independent
of any wavefunction overlap; and only after a corpus-faithful pairing is
confirmed, test whether it reproduces the mass-law assignment. The
overlap-diagonal of A299 through A320 remains withdrawn and is not
revived by any of these; no new arena search is opened.
\end{remark}

\paragraph{Status.} OI-287-1 remains open in its corrected A287 and A296
form. The framing audit gates one corpus-faithful object, the
$\mathbb{Z}_3$ grading of P20, and excludes the near-origin reading as
the withdrawn functional in another form. The first step is structural
and consistent; the derivation of why degree $k$ attaches to eigenspace
$k$ is the standing open core. No published number changes; the A287
lepton law and its $10^{-4}$ accuracy are untouched.

\paragraph{Verifier.} \texttt{verify\_P322.py}: recomputes
$c_n = \rho^{(n)}(0)/(\pi^n n!) = (2,3,16)$ and the direct readout
$(1,\tfrac32,8)$ at $L_2 = 0$; the moments with $\mu_0 = \alpha^{-1}$,
monotone decreasing and at $L_2 = 7.3$ from the target; the production
eigenvalues $(16.51, 51.24, 102.02)$ with node counts $(0,1,2)$ and the
near-origin onset powers $(0.999, 0.997, 0.995)$, all linear; the
exclusion of the near-origin object as the withdrawn functional; and the
GROUNDED$+$STEP verdict with the structural consistency
$\deg\rho = 3$. Eight checks, all PASS.

\end{document}
