% 292_Addendum_QuarkScopingDerivation.tex
% Q2 of the closure program. (i) Quark extension test: FAILS decisively
% — required coefficients (-49,-81) up-type, (119,58) down-type; the
% correction law is charged-lepton-specific; scoped. (ii) Derivation
% attempt: universal first-order perturbation EXCLUDED (ratios 1.2-1.8
% vs needed 8.58); the law requires level-selective layer coupling;
% OI-287-1 sharpened to the selection rule. Verifier: verify_P292.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
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\usepackage{booktabs}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}

\title{Addendum 292: Quark Scoping and the Derivation Frontier\\
\large The lepton law is lepton-specific; universal first-order\\
coupling excluded; the selection rule is the remaining step}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B (decisive negative scoping $+$ a route
exclusion; the open core sharpened) \quad|\quad
\textbf{Addresses:} OI-287-1 (sharpened, kept open) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P292.py} --- all PASS

\begin{abstract}
Q2 of the closure program, both halves, both honest.
\emph{(i) Quark extension: fails, decisively.} Inverting the A287 law
against PDG quark masses: the up-type sector would require
coefficients $c = (-49.1, -81.2)$ (MS-bar top: $-79.3$) and the
down-type $c = (119.5, 58.1)$ --- wrong signs (both quark sectors need
\emph{rising} exponents where leptons fall), wrong magnitudes, wrong
ordering (down-type's would have to \emph{decrease} with level), and
no small-integer structure. The verdict is unambiguous: \textbf{the
correction law is a charged-lepton law.} Plausible reasons are noted
without weight (QCD dressing and scheme dependence pollute the bare
comparison; color may enter the operator differently), but no rescue
is attempted: the corpus scopes the law to the sector where it is
parameter-free and $10^{-4}$-exact.
\emph{(ii) Derivation attempt: the universal route is excluded.} If
the law arose from a single perturbation $gW(x)$ of $\hat O$ at first
order, the required relative eigenvalue shifts
($\varepsilon_2 - \varepsilon_1 = -0.0126$,
$\varepsilon_3 - \varepsilon_1 = -0.1084$, ratio $8.58$) would equal
$g\,\Delta_n[\langle\psi|W|\psi\rangle/\lambda]$. Computed for eight
corpus-natural candidates ($\rho$ and each layer component, $x$,
$x^2$, $x(1-x)$, $V_{\rm obs}$): every ratio lies in $1.20$--$1.84$
--- a factor $\geq 4.7$ short. \textbf{No universal first-order
perturbation produces the law.} What survives is the reading the
law's own form suggests: \emph{level-selective} coupling --- family
$n$ couples to \emph{its} layer component $\rho_{\ell(n)}$ with one
universal strength --- which reproduces the form by construction and
leaves exactly two things underived: the selection rule
$\ell(n)$ (edge$\to$boundary$\to$bulk with rising mass) and the
strength $\kappa\Omega/\lambda_1$. OI-287-1 is restated in this final
form. The closure program's Q2 ends: extension scoped negative,
derivation localized to a named missing mechanism.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{Quark Scoping}

\begin{proposition}[Required coefficients]
With the A281 spectrum and PDG masses: up-type $(u, c, t)$ requires
$\beta_2 = 5.630$, $\beta_3 = 6.198$ (pole top; $6.165$ MS-bar) ---
exponents \emph{rising} where the lepton law falls --- equivalent to
$c_2 = -49.1$, $c_3 = -81.2$. Down-type $(d, s, b)$ requires
$\beta_2 = 2.645$, $\beta_3 = 3.732$, i.e.\ $c_2 = 119.5$,
$c_3 = 58.1$ --- positive but inverted in order and an order of
magnitude beyond the density coefficients. No assignment of
$(2, 3, 16)$, no sign convention, and no anchor choice reconciles
either sector. The law does not extend; it is scoped to charged
leptons, where it stands at $10^{-4}$ with zero free parameters
(A287).
\end{proposition}

% ────────────────────────────────────────────────────────────────────────────
\section{The Derivation Frontier}

\begin{proposition}[Universal first-order coupling excluded]
A perturbation $H \to H + gW$ shifts
$\ln(m_n/m_e)$ by $\beta(\varepsilon_n - \varepsilon_1)$ with
$\varepsilon_n = g\langle\psi_n|W|\psi_n\rangle/\lambda_n$. The law
requires the level-3-to-level-2 shift ratio $8.58$. Numerically
(eigenvectors at $N = 2000$): $W = \rho_{\rm edge}$ gives $1.29$;
$\rho_{\rm bnd}$ $1.43$; $\rho_{\rm blk}$ $1.68$; full $\rho$ $1.63$;
$x$ $1.29$; $x^2$ $1.43$; $x(1-x)$ $1.20$; $V_{\rm obs}$ $1.84$.
All excluded by factor $\geq 4.7$. The deficit is structural:
expectation values of smooth positive weights grow too slowly with
level to produce the bulk-coefficient jump.
\end{proposition}

\begin{remark}[OI-287-1, final form]
The surviving route is the one the law's algebra already encodes:
family $n$ couples \emph{only} to its layer component, with the
universal strength $\kappa\Omega/\lambda_1$ and coefficient
$c_{\ell(n)}$ read off $\rho$. Two objects remain underived: the
\emph{selection rule} (why generation index maps to layer depth ---
heaviest to bulk, anchor to edge; consonant with the corpus's
inward-mass intuition and with A289's chord voicing, derived from
neither) and the \emph{strength}. A derivation must produce both;
nothing else is missing. Any future mechanism can be tested in one
line against the pinned shifts $(-0.0126, -0.1084)$.
\end{remark}

\paragraph{Verifier.} \texttt{verify\_P292.py}: quark inversions for
both sectors and both top conventions; the eight-candidate exclusion
table with the $8.58$ target; the pinned shifts; the scoping and
sharpening assertions. All PASS.

\end{document}
