% 321_Addendum_FramingAudit.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1 framing audit. Since A296/A299 the thread (A296-A320, about eighteen
% probes) has operationalized the selection rule as an overlap-diagonal functional
% B[n] = <psi_n | c_k pi^k x^k | psi_n> / lambda_n and asked whether B reproduces
% the target (c1,c2,c3) = (2,3,16), normalized (1, 1.5, 8). Across seven arenas the
% result is a stubborn pattern: channels that do not distinguish the three families
% undershoot the mu entry (1.22 to 1.34), channels that give three distinct families
% overshoot (2.3 to 6.28), and mu = 1.5 sits in a gap nothing reaches. This addendum
% re-grounds the framing against the corpus text and finds FRAMING-DRIFT. The target
% (2,3,16) are the Taylor coefficients of the density, c_k = rho^(k)(0)/(pi^k k!)
% (A296). The corpus states the family-to-coefficient relationship as a lepton MASS
% correction law, beta_n = 6 mu1/mu0 - c_n alpha^(1/4)/lambda_1, accurate at 1e-4
% (A287), not as an overlap-diagonal. The overlap-diagonal is precisely A296's "bare
% Taylor probe", which A296 (Third exclusion) and A297 (position moments) BOTH
% excluded; it was revived in A299 as a lead and carried forward unquestioned. The
% corpus-faithful object, the direct coefficient readout, reproduces (1,1.5,8)
% trivially. The magnitude gap is an artifact of matching a functional the corpus
% excluded; the eighteen-probe arc was computing the wrong object.
% Verifier: verify_P321.py.

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

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

\title{Addendum 321: The Framing Audit\\
\large The selection-rule magnitude gap is an artifact of an excluded functional;
the target is the density's own Taylor coefficients}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} FRAMING-DRIFT (framing audit; the eighteen-probe overlap-diagonal
arc was computing a functional the corpus excluded) \quad|\quad
\textbf{Addresses:} OI-287-1 framing; A296 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P321.py} --- all PASS

\begin{abstract}
Since A296 and A299 the selection rule OI-287-1 has been operationalized as an
overlap-diagonal functional, $B[n] = \langle\psi_n\,|\,c_k\pi^k x^k\,|\,\psi_n
\rangle / \lambda_n$, with the question whether $B$ reproduces the target
$(c_1,c_2,c_3) = (2,3,16)$, normalized $(1, 1.5, 8)$. Eighteen probes across seven
arenas have established a sharp pattern: realizations that fail to distinguish the
three families undershoot the $\mu$ entry to between $1.22$ and $1.34$; realizations
that distinguish them overshoot to between $2.3$ and $6.28$; the target $\mu = 1.5$
lies in a gap that no arena reaches. This addendum re-grounds the framing against
the corpus text and finds that the overlap-diagonal is not the object the corpus
names. The target $(2,3,16)$ are the Taylor coefficients of the density,
$c_k = \rho^{(k)}(0)/(\pi^k k!)$, established in A296. The corpus states the
family-to-coefficient relationship as a lepton mass correction law,
$\beta_n = 6\mu_1/\mu_0 - c_n\,\alpha^{1/4}/\lambda_1$, accurate to $10^{-4}$ on both
mass ratios (A287), not as an overlap expectation value. The overlap-diagonal is the
``bare Taylor probe'' that A296 (Third exclusion) and A297 (position moments) both
excluded; A299 revived it as the strongest lead, and the lead was carried forward
without re-examination. The corpus-faithful reading of ``family $n$ draws coefficient
$n$'' is the direct coefficient readout, which equals $(1, 1.5, 8)$ by definition.
The magnitude gap is an artifact of matching a functional the corpus had already set
aside. The verdict is FRAMING-DRIFT, and it redirects the open frontier of OI-287-1.
\end{abstract}

% ----------------------------------------------------------------------
\section{The Origin of the Target and the Mandated Functional}

\begin{proposition}[The target is the density's own coefficients; the rule is a
mass law]
\label{prop:origin}
Write $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$, so that
$\rho'(x) = 48\pi^3 x^2 + 6\pi^2 x + 2\pi$ integrates to $\rho$ with coefficients
$(2, 3, 16)$ in $\pi^k$ units at degrees $1, 2, 3$. These coefficients are the
Taylor data of $\rho$ at the center, $c_k = \rho^{(k)}(0)/(\pi^k\,k!) = (2, 3, 16)$,
which is the reformulation of the selection rule in A296: charged-lepton family $n$
draws the degree-$n$ coefficient of $\rho$. The target $(2,3,16)$, normalized
$(1, 1.5, 8)$, is therefore literally the density's own defining data, not a
quantity to be reconstructed from a separate functional. The corpus states the
family-to-coefficient relationship once, in A287: the lepton mass correction law
$m_n/m_e = (\lambda_n/\lambda_1)^{\beta_n}$ with
$\beta_n = 6\mu_1/\mu_0 - c_n\,\alpha^{1/4}/\lambda_1$, which reproduces
$m_\mu/m_e$ and $m_\tau/m_e$ to $10^{-4}$ with the $c_n$ entering as the integer
coefficients of $\rho$. The relationship is a statement about lepton masses through
the coefficients read as integers, not about an overlap expectation value of any
operator. The moments $\mu_n = \int_0^1 x^n\rho\,dx$, which the corpus does use for
the mass hierarchy, form a monotone-decreasing sequence
$(\mu_0,\mu_1,\mu_2,\mu_3) = (137.04, 108.72, 90.18, 77.06)$ with
$\mu_0 = \alpha^{-1}$; their ratios are not $(2,3,16)$ and were never claimed to be.
The coefficients and the moments are distinct uses of $\rho$, and the target lives
in the former.
\end{proposition}

% ----------------------------------------------------------------------
\section{The Drift}

\begin{remark}[The overlap-diagonal was excluded, then carried forward]
\label{rem:drift}
The functional that the eighteen-probe arc has been testing,
$B[n] = \langle\psi_n\,|\,c_k\pi^k x^k\,|\,\psi_n\rangle / \lambda_n$, is precisely
the diagonal Taylor probe that A296 introduced as a candidate mechanism and
excluded in the same addendum: its third proposition records that the diagonal
coupling of level $n$ to its own monomial falls short of the required shifts by
factors $32$ and $42$, and is excluded on both scale and shape. A297 then excluded
the related position-moment reading, finding the normalized diagonal ratios
$(1, 0.74, 0.59)$ running opposite to the target. A299 reintroduced the
overlap-diagonal as ``the strongest lead the selection rule has'', reporting the
normalized diagonal $(1, 1.12, 7.50)$ at $L_2 = 0.63$, and the subsequent arc
(A300 through A320) inherited that lead as the object to be repaired, asking across
seven arenas what factor lifts the $\mu$ entry from $1.12$ to $1.5$. The present
audit recomputes the overlap-diagonal and confirms $(1, 1.12, 7.50)$ at $L_2 = 0.63$
on the production eigenvalues $(16.51, 51.24, 102.02)$, in agreement with A299. The
corpus-faithful object, by contrast, is the direct coefficient readout
$(c_1, c_2, c_3)/c_1$, which equals $(1, 1.5, 8)$ to machine precision because the
coefficients are the target. The verdict is FRAMING-DRIFT: the magnitude gap that
seven arenas could not close is an artifact of matching an overlap expectation value
the corpus had set aside, not a property of the physics. The redirected frontier of
OI-287-1 is not ``what factor lifts $\mu$ to $1.5$'' but the original A287 and A296
question, why the mass correction law assigns level $n$ the degree-$n$ coefficient
of $\rho$, with the coefficients read as the integers they are. The eighteen-probe
arena search retains value as a thorough exclusion of the overlap-diagonal across
radial, measure, fiber, Dirac, boundary, and coupled realizations, but it is an
exclusion of the wrong object, and the open problem stands where A287 and A296 left
it.
\end{remark}

\paragraph{Status.} OI-287-1 remains open, and its form reverts to the A296
statement: derive why charged-lepton family $n$ draws the degree-$n$ Taylor
coefficient of $\rho$ in the mass correction law of A287. The overlap-diagonal
operationalization of A299 is withdrawn as the object of the open problem; it is
recorded as an excluded functional, consistent with A296 and A297. No published
number changes; the lepton law and its $10^{-4}$ accuracy (A287) are untouched.

\paragraph{Verifier.} \texttt{verify\_P321.py}: recomputes the Taylor coefficients
$(2,3,16)$ and the antiderivative arithmetic; the direct coefficient readout against
$(1,1.5,8)$; the moments $\mu_n$ with $\mu_0 = \alpha^{-1}$ and their monotone-
decreasing ratios; the production eigenvalues and the overlap-diagonal
$(1,1.12,7.50)$ at $L_2 = 0.63$; and the defining FRAMING-DRIFT condition, that the
coefficient readout matches the target at $L_2 = 0$ while the overlap-diagonal
misses it. Eight checks, all PASS.

\end{document}
