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% Addendum 349b (extends Paper 18, The Master Operator -- P18-T2, sub-result (a)).
% b-thread (assembly-uniqueness lane). Sibling of 349a (cosmology MapClosure); the
% "b" suffix is the P18-T2 lane (Joel's a/b numbering, 2026-06).
% Author: Leon Fernando Vlegels.  Text CC BY 4.0.
% Verifier: verify_P349b.py (15/15 pass).
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\newcommand{\Ohat}{\hat{O}}
\newcommand{\Rcal}{\mathcal{R}}
\newcommand{\Mcal}{\mathcal{M}}

\title{Addendum 349b: Component-Category Uniqueness\\
\large The first of the three assembly-uniqueness sub-results: each named\\
component is the only operator in its category --- P18-T2 sub-result (a)\\[0.3em]
\normalsize (Addendum to Paper 18)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
Paper~18 \S4.4 records that a proof of assembly uniqueness (P18-T2) requires three
independent sub-results: (a)~each component $R_i$'s named operator is the only operator
in its category; (b)~additive composition is the only admissible composition
(Addendum~350b); (c)~the four coefficients are the only spectrum-matching assignment
(Addenda~345b--348b, reduced to one Schur scalar). This addendum works (a). It does not
prove full uniqueness; it \emph{classifies} each of the eight components of
$\Ohat=D_{B^4}^2+\Delta_{S^3}+\Delta_{S^1}+V_{\mathrm{self}}+\lambda\rho+\gamma T_{\mathrm{cycle}}+\zeta\Rcal+\beta\Mcal$
as canonical-by-theorem or named-floor, reducing (a) from an open search to an explicit,
short floor list. The result: \textbf{five components are forced by named uniqueness
theorems} --- the two Laplacians by Laplace--Beltrami uniqueness together with the
A328 scale-fixing (unit-ball metric $\Rightarrow$ the kinetic normalisation; Laplace--Beltrami
$\Rightarrow$ no zeroth-order term); the bulk by the Lichnerowicz identity
$D^2=-\Delta+R/4$ built in A329, modulo the APS boundary condition, which is the
already-named Heart-2(i) floor; the layer-cycle $T_{\mathrm{cycle}}$ by Paper~18's own
orientation argument (the canonical-selection proposition, $L(3,1)$ vs $L(3,2)=-L(3,1)$);
and the renormalisation generator $\Rcal=r\partial_r+\Delta_\psi$ by dilation-generator
uniqueness (shown here). \textbf{Two more are multiplication by fixed corpus objects}
($\lambda\rho$ by the Paper~03 density, $\beta\Mcal$ by the Paper~02 moments), hence not
free. \textbf{Exactly one form is genuinely open}: the self-lensing potential
$V_{\mathrm{self}}$, the named floor A328 corrected to a saturating exponential.
\textbf{Verdict: PARTIAL} --- (a) reduces to standard invariant-operator uniqueness per
category plus two named floors ($V_{\mathrm{self}}$ form; APS boundary condition), with
$T_{\mathrm{cycle}}$ already closed in Paper~18. An honest reduction, not a closure. All
claims pinned by \texttt{verify\_P349b.py} (15/15).
\end{abstract}

\section{The sub-result and the strategy}
\label{sec:strategy}

Paper~18 \S4.4 states that the canonical assembly satisfies the coverage requirements
(R1)--(R8), but ``whether (R1)--(R8) admit alternative admissible assemblies is open.''
The three sub-results that a uniqueness proof would need are named there. Sub-result~(a)
is the most local: for each coverage category, is the named operator the only operator
that fills it? The strategy is not to prove a single grand uniqueness theorem but to
audit the eight components one at a time, attaching to each the standard uniqueness
theorem that forces it (when one exists) or recording it as a posited form (when none
does). The audit either reduces (a) to a short, explicit floor list or it does not; we
report which.

\section{The five canonical-by-theorem components}
\label{sec:canonical}

\begin{proposition}[The two Laplacians]
\label{prop:laplacians}
The boundary operator $\Delta_{S^3}$ and the fibre operator $\Delta_{S^1}$ are the
Laplace--Beltrami operators of their factors. On a compact Riemannian manifold the
Laplace--Beltrami operator is the unique (up to a positive scalar) second-order,
isometry-invariant, self-adjoint differential operator annihilating constants and
carrying no zeroth-order term. The residual scalar and the absence of a zeroth-order
term are fixed in A328: the unit-ball metric fixes the kinetic normalisation and the
Laplace--Beltrami form forbids an additive constant. Their spectra are the corpus values
$l(l+2)$ on $S^3$ and $k^2$ on $S^1$.
\end{proposition}

\begin{proposition}[The bulk operator]
\label{prop:bulk}
The bulk operator is the squared spin-Dirac operator on $B^4$, fixed by the Lichnerowicz
identity $D^2=-\Delta+R/4$ (the canonical Dirac operator compatible with the Levi-Civita
connection; division-algebra spin structure from Paper~01). It is built explicitly in
A329, with the $V=0$ spectrum the corpus APS eigen-identity $(2n+l+2)^2$. The one
non-derived ingredient is the boundary condition: A329 uses the APS condition, which
Paper~18 (l.383) records as stated, not derived --- the Heart-2(i) floor. So the bulk is
canonical \emph{modulo} that already-named floor.
\end{proposition}

\begin{proposition}[The renormalisation generator]
\label{prop:renorm}
The renormalisation generator $\Rcal=r\partial_r+\Delta_\psi$ is the unique first-order
operator generating the dilation $r\mapsto e^t r$, up to the additive scaling-dimension
constant $\Delta_\psi$. A first-order operator $X=f(r)\partial_r+g(r)$ generates the
dilation iff its flow $\dot r=f(r)$ integrates to $r(t)=r_0 e^t$, i.e.\
$t=\int dr/f(r)=\ln(r/r_0)$, which forces $f(r)=r$ uniquely; the zeroth-order part $g$ is
the additive conformal weight $\Delta_\psi$ (Paper~18 Def.\ Scaling Dimension). Verified
symbolically.
\end{proposition}

\begin{proposition}[The layer-cycle operator --- already closed in Paper 18]
\label{prop:tcycle}
$T_{\mathrm{cycle}}$ is the unique generator of the $\mathbb{Z}_3$ layer-cycle action
consistent with the oriented lens space $L(3,1)=S^3/\mathbb{Z}_3$ (Paper~06) and the Hopf
orientation (Paper~28). The two non-trivial generators give quotients $L(3,1)$ and
$L(3,2)\cong-L(3,1)$ (Reidemeister--Franz); the positive Hopf orientation selects the
forward cycle, so $T_{\mathrm{cycle}}^2$ is excluded and $T_{\mathrm{cycle}}$ is canonical.
This is Paper~18's own canonical-selection proposition --- the one component whose
category-uniqueness is already proven in canon. ($T_{\mathrm{cycle}}^3=I$ with cube-root-of-unity
eigenvalues, verified.)
\end{proposition}

\section{The two fixed-multiplication components}
\label{sec:fixed}

\begin{proposition}[Density and moment terms are not free]
\label{prop:fixed}
The density term $\lambda\rho$ is multiplication by the fixed Paper~03 geometric density
$\rho(x)=16\pi^3x^3+3\pi^2x^2+2\pi x$ (with $\mu_0=\int_0^1\rho=4\pi^3+\pi^2+\pi$), and the
moment term $\beta\Mcal=\sum_n(\mu_n/\mu_0)P_n$ is built from the fixed Paper~02 moments
$\mu_n=\int_0^1 x^n\rho\,dx$. Neither is a free operator in its category: once $\rho$ and
the moments are fixed upstream, the operators are determined (the coefficients
$\lambda,\beta$ are sub-result~(c), not~(a)).
\end{proposition}

\section{The one open form, and the reduction}
\label{sec:floor}

\begin{remark}[The $V_{\mathrm{self}}$ floor]
\label{rmk:vself}
The self-lensing potential $V_{\mathrm{self}}(r)=(E_{\mathrm{self}}/m_0^2)(1-e^{-r/r_0})$
is the one component whose functional \emph{form} is posited rather than forced by a
category-uniqueness theorem. A328 corrected it to this saturating exponential and recorded
it as a named floor. Sub-result~(a) carries no open search beyond this form and the APS
boundary condition of Proposition~\ref{prop:bulk}.
\end{remark}

\begin{theorem}[Reduction of sub-result (a)]
\label{thm:reduction}
Of the eight components of $\Ohat$: five are canonical-by-theorem
(Propositions~\ref{prop:laplacians}--\ref{prop:tcycle}), two are multiplication by fixed
corpus objects (Proposition~\ref{prop:fixed}), and one is a named floor
(Remark~\ref{rmk:vself}). Hence sub-result~(a) reduces to standard invariant-operator
uniqueness per category plus exactly two named floors --- the $V_{\mathrm{self}}$ form and
the APS boundary condition --- with the layer-cycle component already closed in Paper~18.
This is a reduction, not a closure: the $V_{\mathrm{self}}$ form remains a posit.
\end{theorem}

\section*{Status}
Sub-result~(a) of P18-T2 is not an open search. Five of the eight components are forced by
named uniqueness theorems (the two Laplace--Beltrami operators with A328 scale-fixing; the
Lichnerowicz squared-Dirac built in A329, modulo the APS-boundary-condition floor; the
dilation generator by the uniqueness shown here; the layer-cycle operator by Paper~18's own
orientation argument), and two are multiplication by the fixed Paper~02/03 objects. The lone
genuinely open form is the self-lensing potential $V_{\mathrm{self}}$, the named A328 floor.
\textbf{Verdict: PARTIAL} --- (a) reduces to invariant-operator theory plus two named floors
($V_{\mathrm{self}}$ form; APS boundary condition); no closure is claimed and no overclaim is
made. The reduction is recorded in Paper~18 \S4.4 as \texttt{rmk:open:P018\_4}. A294's
re-typing of P18-T2 stands; sub-results (b) and (c) are Addenda~350b and 345b--348b. No canon
\texttt{.tex} of P18 is edited beyond the open marker and no published number changes.
Verifier: \texttt{verify\_P349b.py}, 15/15 pass.

\paragraph{Relation to the other sub-results.}
Sub-result~(b) (350b) shows the eight components compose additively; sub-result~(c)
(345b--348b) shows the coefficients are spectrum-rigid to one Schur scalar. Together the
three reduce the whole assembly-uniqueness question to a small, explicit set of named floors.

\end{document}
