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% Addendum 350b (extends Paper 18, The Master Operator -- P18-T2, sub-result (b)).
% b-thread (assembly-uniqueness lane). Sibling of 350a (cosmology CokernelDarkEnergy);
% 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_P350b.py (12/12 pass).
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\newcommand{\Ohat}{\hat{O}}
\newcommand{\Rcal}{\mathcal{R}}
\newcommand{\Mcal}{\mathcal{M}}

\title{Addendum 350b: Additive Composition\\
\large The second assembly-uniqueness sub-result: why $\Ohat$ is a sum, not a\\
product or an operad --- P18-T2 sub-result (b)\\[0.3em]
\normalsize (Addendum to Paper 18)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
The second of the three assembly-uniqueness sub-results Paper~18 \S4.4 names. Sub-result~(a)
(Addendum~349b) audits the components; (c) (Addenda~345b--348b) the coefficients;
this addendum works (b) --- why the master operator is the \emph{sum}
$\Ohat=D_{B^4}^2+\Delta_{S^3}+\Delta_{S^1}+V_{\mathrm{self}}+\lambda\rho+\gamma T_{\mathrm{cycle}}+\zeta\Rcal+\beta\Mcal$
rather than a product, an operadic, or any non-additive composition. The corpus-native
argument has three legs. \textbf{(L1) Product geometry:} the arena is the Hopf-fibred
product ($S^1\!\to\!S^3$, $B^4$ with boundary $S^3$); the three kinetic sectors act on
different factors, so they commute, and the canonical second-order operator on a product is
the direct \emph{sum} of factor operators, whose spectrum is the sum of factor spectra
($\Delta_{M\times N}=\Delta_M\oplus\Delta_N$) --- verified on the tensor product.
\textbf{(L2) Hamiltonian form:} the remaining terms enter as a potential/structure added to
the kinetic part; kinetic-plus-potential is additive by the definition of a
Schr\"odinger/Laplace-type operator, and addition is order-independent, where an operad
would not be. \textbf{(L3) The additive $\alpha$-decomposition:} the ground-state readout
$\alpha^{-1}=\int_0^1\rho=4\pi^3+\pi^2+\pi$ is an additive layer sum (bulk $4\pi^3$ + boundary
$\pi^2$ + edge $\pi$, the Paper~04 three-layer ontology); only additive composition reads it
off the ground state as a sum, whereas a multiplicative composition yields a product readout
($4\pi^3\!\cdot\!\pi^2\!\cdot\!\pi=4\pi^6$, or $0$ when a factor vanishes), not $\alpha^{-1}$.
\textbf{Verdict: PARTIAL} --- additive composition is shown \emph{canonical} on these three
legs; the named residual is an exhaustive exclusion of all operadic alternatives that might
reproduce the same readout by another route. All claims pinned by \texttt{verify\_P350b.py}
(12/12).
\end{abstract}

\section{The sub-result}
\label{sec:subresult}

Paper~18 \S4.4 lists, among the things a uniqueness proof must establish, ``that additive
composition is the only admissible composition.'' The canonical assembly is a sum of eight
terms; the question is whether a product, an operad (an ordered composition with arity), or
another non-additive law could equally satisfy the coverage requirements (R1)--(R8). We give
the corpus-native case that additive composition is forced, on three independent legs, and
name what the case does not deliver.

\section{Leg 1: the product geometry forces additive kinetics}
\label{sec:L1}

\begin{proposition}[Commuting sectors, summed spectrum]
\label{prop:L1}
The arena is the Hopf-fibred product: $B^4$ with boundary $S^3$ and Hopf fibre $S^1$
($S^1\!\to\!S^3\!\to\!S^2$). The three kinetic sectors $D_{B^4}^2$, $\Delta_{S^3}$,
$\Delta_{S^1}$ act on different factors, hence commute. The canonical second-order operator
on a product manifold is the direct sum of the factor operators,
$\Delta_{M\times N}=\Delta_M\otimes I+I\otimes\Delta_N$, whose spectrum is the sum of the
factor spectra. Verified on the tensor product: the eigenvalues of the assembled kinetic
sum equal the pairwise/triple sums of the factor eigenvalues, and the factor operators
commute (simultaneous diagonalisability). The additive ground value is the sum of the sector
ground values --- the additive signature.
\end{proposition}

\section{Leg 2: the Hamiltonian form forces additive structure terms}
\label{sec:L2}

\begin{proposition}[Kinetic plus potential]
\label{prop:L2}
The remaining terms ($V_{\mathrm{self}}$, $\lambda\rho$, $\beta\Mcal$, $\gamma T_{\mathrm{cycle}}$,
$\zeta\Rcal$) enter as a potential/structure \emph{added} to the kinetic part:
$\Ohat=(\text{kinetic})+(\text{potential/structure})$. Kinetic-plus-potential is additive by
the definition of a Schr\"odinger/Laplace-type operator, and operator addition is commutative
and associative, so the assembly order does not change $\Ohat$. An operadic composition --- an
ordered law with arity --- would not have this order-invariance; the corpus assembly is
manifestly order-independent, which is the additive (not operadic) signature.
\end{proposition}

\section{Leg 3: the additive $\alpha$-decomposition confirms it at the ground state}
\label{sec:L3}

\begin{theorem}[Only additive composition reads $\alpha^{-1}$]
\label{thm:L3}
The ground-state readout is
\[
  \alpha^{-1}=\int_0^1\rho(x)\,dx
   =\underbrace{4\pi^3}_{\text{bulk}}+\underbrace{\pi^2}_{\text{boundary}}+\underbrace{\pi}_{\text{edge}}
   =137.036,
\]
an additive sum of the three layer integrals (the Paper~04 three-layer ontology, the
$90\text{-}7\text{-}2=4\pi^3\!:\!\pi^2\!:\!\pi$ shell). An additive composition reads this off
the ground state as a sum of layer contributions. A multiplicative composition produces a
product readout: the product of the layer integrals is $4\pi^3\!\cdot\!\pi^2\!\cdot\!\pi=4\pi^6$,
not $\alpha^{-1}$, and on the tensor product a multiplicative composition collapses the ground
value to zero whenever a factor's ground eigenvalue vanishes. Hence only the additive
composition yields the corpus $\alpha$ readout; the additive structure is the operator-level
image of the layered geometry.
\end{theorem}

\section*{Status}
Additive composition is the canonical form of $\Ohat$ on three independent legs: the kinetic
sectors are additive by the product geometry of the Hopf-fibred arena (commuting factor
operators, summed spectrum, Proposition~\ref{prop:L1}); the structure terms are additive by
the Hamiltonian kinetic-plus-potential form, which is order-independent where an operad is not
(Proposition~\ref{prop:L2}); and the additive layer decomposition
$\alpha^{-1}=4\pi^3+\pi^2+\pi$ is read off the ground state only under additive composition, a
multiplicative composition yielding a product readout that is not $\alpha^{-1}$
(Theorem~\ref{thm:L3}). \textbf{Verdict: PARTIAL} --- additive composition is shown canonical;
what is \emph{not} delivered is an exhaustive exclusion of all operadic / non-additive
compositions that might reproduce the same ground-state readout by another route. That
exclusion is the named residual, recorded in Paper~18 \S4.4 as \texttt{rmk:open:P018\_5}.
A294's re-typing of P18-T2 stands; sub-results (a) and (c) are Addenda~349b and 345b--348b;
no canon \texttt{.tex} of P18 is edited beyond the open marker and no published number changes.
Verifier: \texttt{verify\_P350b.py}, 12/12 pass.

\paragraph{The three sub-results together.}
With (a) reduced to two named floors (349b), (b) shown canonical with an operadic-exclusion
residual (here), and (c) reduced to one Schur scalar (345b--348b), the whole assembly-uniqueness
question P18-T2 is now a short, explicit list of named floors rather than three open searches.

\end{document}
