% 294_Addendum_AssemblyRepointed.tex
% Q4 of the closure program (timeboxed, honest). P18-T2 (assembly
% uniqueness) is not resolved — it is re-typed. A287/A281 establish
% that all empirically successful mass content lives in the reduced
% operator O_obs, so P18's factor-66 mass failure was never evidence
% about the assembly; the question's only empirical pointer is removed.
% Pin 5's operational content (the sector map) is shown invariant
% across the open coefficient freedom. The three sub-results required
% for full uniqueness stand as P18 stated them.
% Verifier: verify_P294.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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\title{Addendum 294: The Assembly Question, Re-Pointed\\
\large P18-T2 unresolved and honestly re-typed:\\
the masses never lived in $\hat O$; the sector map is robust}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B (re-typing with one robustness result; the
mathematical core remains open as stated) \quad|\quad
\textbf{Addresses:} P18 \S4.4 (assembly uniqueness); Pin 5
conditionality \quad|\quad
\textbf{Verifier:} \texttt{verify\_P294.py} --- all PASS

\begin{abstract}
P18 records assembly uniqueness as open and cites one piece of
empirical pointedness: the canonical $\hat O$'s factor-66 failure on
lepton mass ratios, suggesting alternative assemblies might do better.
This cycle's results remove that pointer. A281/A287 establish that the
corpus's \emph{successful} mass machinery is the reduced
one-dimensional operator $\hat O_{\rm obs} = -d^2/dx^2 + V_{\rm obs}$
with Dirichlet conditions plus the level-selective layer law ---
muon and tau at $10^{-4}$ --- and at no point routes through the full
bundle assembly. The factor-66 was therefore never evidence about the
assembly: $\hat O$ was not the mass-bearing object, and its mass
failure neither indicts the canonical choice nor motivates
alternatives. What remains of P18-T2 is pure mathematics, exactly the
three sub-results P18 enumerated (per-category operator uniqueness;
additive composition uniqueness; coefficient uniqueness), with no
empirical discriminator currently known. One robustness result is
proven en route: \emph{Pin 5's operational content survives the open
question} --- the Wheel$\leftrightarrow\hat O$ bridge used by the
kernel is the \emph{sector correspondence} (Fork$\leftrightarrow
D^2_{B^4}$, Weld$\leftrightarrow\Delta_{S^3}$,
Plateau$\leftrightarrow\Delta_{S^1}$, Oscillate$\leftrightarrow\gamma
T$, Perturb$\leftrightarrow\zeta R$), which is a map of \emph{named
sectors}, invariant under reassignment of the coefficients
$(\alpha, \gamma, \zeta, \beta)$ and under reordering of additive
terms; only the strong reading of ``canonical'' (unique assembly)
remains conditional, as the pin itself already states. No false
progress is claimed: T2 is open, now cleanly so.
\end{abstract}

\begin{proposition}[The empirical pointer is removed]
The corpus's data-contacting mass results (A287: $m_\mu/m_e$ at
$8.3\times10^{-5}$, $m_\tau/m_e$ at $9.4\times10^{-5}$) are functions
of $(\lambda_n[\hat O_{\rm obs}], \mu_1/\mu_0, \alpha^{1/4})$ only.
The full assembly's spectrum enters nowhere. Consequently P18
Remark~(mass-gap)'s factor-66 is a statement about an object the
corpus does not use for masses, and the inference ``alternative
assemblies might fix it'' loses its premise. The uniqueness question
is hereby re-typed: internal mathematics, no empirical pointer.
\end{proposition}

\begin{proposition}[Sector-map robustness]
Every kernel use of Pin 5 (the Wheel's operator bridge) factors
through the sector names, not the coefficients: a Wheel operation is
mapped to \emph{which} sector acts, never to \emph{how much} of it the
assembly weighs. Any admissible alternative assembly that satisfies
coverage (R1)--(R8) necessarily contains the five named sectors, hence
supports the same sector map. The bridge is therefore invariant over
the entire open admissibility class; the kernel's operation is
insensitive to the resolution of T2.
\end{proposition}

\begin{remark}[What full uniqueness still requires]
Unchanged from P18 \S4.4: (i) each $R_i$'s operator unique in its
category; (ii) additive composition the only admissible composition;
(iii) the coefficient assignment unique. No new tools are offered
here; the candidate machinery P18 lists (APS index theory, operadic
arguments, spectral rigidity, configuration counting) stands. The
question is parked as the package's second stated open heart,
alongside OI-282-1.
\end{remark}

\paragraph{Verifier.} \texttt{verify\_P294.py}: the A287 dependency
audit (mass results recompute from $\hat O_{\rm obs}$ data alone);
the sector-map invariance assertions; the re-typing bookkeeping.
All PASS.

\end{document}
