% 306_Addendum_AdditiveComposition.tex
% P18-T2, sub-result (ii), the composition question. Additive composition
% (O_hat = sum of sector operators, P18 S4.1) is the unique way to compose
% the named sectors GIVEN the independent-decoupling axiom: each sector
% must contribute a generator whose coupling switches off continuously,
% removing that sector and leaving the others' action unchanged. The
% forced cross-coupling lives at the operator level: d^2 O / dc_i dc_j is
% identically zero for additive composition (O linear in the couplings)
% and nonzero for a multiplicative alternative O = K(I + c_i S_i)(I + c_j S_j),
% which carries the cross-term K S_i S_j. Demonstrated on two sectors of
% P18's radial operator (density alpha*rho, renormalization zeta*r d/dr):
% additive operator mixed partial vanishes to machine precision, the
% multiplicative one matches the analytic cross-term, and the c_j -> 0
% limit recovers the reduced spectrum for additive but distorts the
% surviving sector for multiplicative. This is a Case C partial on
% sub-result (ii) given the decoupling axiom; operadic compositions
% engineered to decouple are not ruled out, the axiom is not derived from
% deeper principles, and A294's re-typing of the whole P18-T2 stands.
% Verifier: verify_P306.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
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\newtheorem{proposition}{Proposition}[section]
\newtheorem{remark}[proposition]{Remark}

\title{Addendum 306: Additive Composition is Forced\\
\large A verified result on P18-T2 sub-result (ii),
given the decoupling axiom}
\author{TOE Corpus}
\date{2026-06-13}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case C (partial: sub-result (ii) advanced given the
decoupling axiom) \quad|\quad
\textbf{Addresses:} P18-T2 sub-result (ii); A294 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P306.py} --- all PASS

\begin{abstract}
P18-T2 (assembly uniqueness) requires three independent sub-results; A294
re-typed the question as pure mathematics and offered no new tools, A302
gave the third sub-result a first-order local result, and A304 forced one
sector of the first. This addendum advances the second sub-result, that
additive composition is the only admissible way to compose the named
sectors, as opposed to multiplicative, operadic, or other compositions.
The advance is conditional on a stated axiom and bounded as such. The
axiom is independent decoupling: each sector contributes a generator whose
coupling switches off continuously, so that $c_j \to 0$ removes sector $j$
and leaves the remaining sectors' action unchanged. This is a
coverage-style requirement, the operator-level reading of the design
requirements (R1)--(R8) of P18 \S1.2, each naming an independent category.
Under it the forcing is exact and checkable at the operator level. The
forced cross-coupling between two sectors is the mixed second partial
$\partial^2 \hat O / \partial c_i\,\partial c_j$. For additive composition
$\hat O$ is linear in the couplings, so this partial is identically the
zero operator; for the multiplicative alternative
$\hat O = K\,(I + c_i S_i)(I + c_j S_j)$ it is the nonzero cross-term
$K S_i S_j$. Demonstrated on two sectors of P18's radial operator, the
density coupling $\alpha\rho(r)$ (R5) and the renormalization generator
$\zeta\,r\partial_r$ (R7), at the canonical couplings: the additive
operator mixed partial has Frobenius norm $0$ to machine precision while
the multiplicative one is $3.06\times10^{6}$, matching the analytic
cross-term to $10^{-6}$ relative; by Hellmann--Feynman the additive
per-sector spectral response $\partial\lambda/\partial\alpha$ is invariant
under the other coupling (drift $1.4\times10^{-7}$). The decoupling limit
confirms it on the spectrum: setting one coupling to zero recovers the
bare reduced operator exactly for additive (spectral residual $0$) but
distorts the surviving sector for multiplicative
($\|\lambda_{\rm mult} - \lambda_{\rm add}\| = 0.72$). Additivity is the
unique composition that meets independent decoupling with no forced
cross-coupling. The result is an honest partial. It addresses sub-result
(ii) only, conditional on the decoupling axiom; it does not rule out
operadic compositions engineered to satisfy decoupling, does not derive
the axiom from deeper principles, and A294's re-typing of the whole
P18-T2 stands.
\end{abstract}

% ----------------------------------------------------------------------
\section{The Forcing}

\begin{proposition}[Additive composition is the unique forced-cross-coupling-free
composition, given independent decoupling]
Write P18's radial operator over a fixed kinetic-plus-self-lensing part
$K$ and the bare sector operators $S_j$ carrying free couplings $c_j$,
restricting to two sectors for a clean demonstration: the density coupling
$S_\alpha = \rho(r)$ (R5) with weight $\alpha$ and the renormalization
generator $S_\zeta = r\partial_r$ (R7) with weight $\zeta$, at the
canonical $\alpha = 1/137.036$, $\zeta = \alpha^{5/4}$. Compose the
sectors two ways,
\[
\hat O_{\rm add}(\alpha,\zeta) = K + \alpha S_\alpha + \zeta S_\zeta,
\qquad
\hat O_{\rm mult}(\alpha,\zeta) = K\,(I + \alpha S_\alpha)(I + \zeta S_\zeta).
\]
Impose independent decoupling: each $c_j \to 0$ must remove sector $j$ and
leave the others' action unchanged. Then:
\begin{enumerate}
\item[\textnormal{(i)}] The forced cross-coupling is the operator mixed
partial $\partial^2 \hat O / \partial\alpha\,\partial\zeta$. For the
additive composition $\hat O_{\rm add}$ is affine in $(\alpha,\zeta)$, so
this partial is the zero operator identically:
$\|\partial^2 \hat O_{\rm add}/\partial\alpha\,\partial\zeta\|_F = 0$ to
machine precision. For the multiplicative composition it is the nonzero
cross-term $K S_\alpha S_\zeta$, with Frobenius norm $3.06\times10^{6}$
matching $\|K S_\alpha S_\zeta\|_F$ to $10^{-6}$ relative.
\item[\textnormal{(ii)}] By Hellmann--Feynman the additive per-sector
spectral response $\partial\lambda/\partial\alpha$ is independent of the
other coupling $\zeta$: its drift between $\zeta = 0$ and $\zeta = \zeta_0$
is $1.4\times10^{-7}$, while the multiplicative response drifts more. The
eigenvalue mixed partial $\partial^2\lambda/\partial\alpha\,\partial\zeta$
is not used as the discriminator; second-order perturbation theory makes
it nonzero even for additive composition through the
$\langle 0|S_\alpha|m\rangle\langle m|S_\zeta|0\rangle$ overlap, so it
measures the sectors' shared spectral content rather than the
composition's forced coupling. The operator partial in (i) isolates the
latter.
\item[\textnormal{(iii)}] The decoupling limit holds for additive and
fails for multiplicative. Setting $\zeta \to 0$ at canonical $\alpha$, the
additive operator recovers the bare reduced operator $K + \alpha S_\alpha$
exactly (spectral residual $0$), whereas the multiplicative operator
becomes $K(I + \alpha S_\alpha)$, which multiplies the kinetic part itself
by the surviving sector and so distorts it:
$\|\lambda[\hat O_{\rm mult}(\alpha,0)] - \lambda[\hat O_{\rm add}(\alpha,0)]\|
= 0.72$.
\end{enumerate}
Additive composition is therefore the unique composition of the named
sectors that meets independent decoupling with no forced cross-coupling.
\end{proposition}

\begin{remark}[What is forced, what remains open]
The forcing is exact at the operator level: additivity is the only
composition whose mixed partial in any two couplings is the zero operator,
equivalently the only composition under which ``turn off sector $j$'' is
independent of the other sectors' strengths. The two-sector demonstration
is representative; the operator-mixed-partial criterion and the decoupling
limit are pairwise and extend term by term to the full set of named
sectors, since the additive form is the unique multilinear-free composition
in every coupling pair. The scope is stated as such, and three boundaries
bound the claim. It is \emph{conditional} on the independent-decoupling
axiom, the operator-level reading of P18's coverage requirements
(R1)--(R8) as independent categories; the axiom is assumed here, not
derived from deeper principles. It does \emph{not} rule out exotic
operadic compositions engineered to satisfy decoupling, those whose
cross-terms cancel on the relevant sectors while remaining non-additive
elsewhere; the result excludes only the natural multiplicative and
factorized-coupling alternatives, and the full categorical and operadic
sweep is open. It addresses sub-result (ii) \emph{alone}; the per-category
operator uniqueness (i) carries A304's verified result for one sector and
A302's local result stands for the coefficients (iii). A294's re-typing of
the whole P18-T2 stands.
\end{remark}

\paragraph{Status.} P18-T2 remains open. Its second sub-result now has a
verified result: additive composition is the unique composition of the
named sectors with no forced cross-coupling, given the
independent-decoupling axiom, with the natural multiplicative alternative
excluded by a nonzero operator cross-term and a distorted decoupling
limit. The operadic and full-categorical sweep is open, and the decoupling
axiom is assumed rather than derived. Sub-results (i) and (iii) are
unchanged. No published number changes.

\paragraph{Verifier.} \texttt{verify\_P306.py}: rebuilds the two sectors
of P18's radial operator and both compositions from scratch, asserts the
additive operator mixed partial vanishes ($<10^{-4}$) as an exact identity
while the multiplicative one is a forced cross-coupling ($>10^{-2}$)
matching the analytic $K S_\alpha S_\zeta$ to $10^{-6}$ relative, asserts
the Hellmann--Feynman per-sector response invariance for additive, and
asserts the $\zeta \to 0$ limit recovers the reduced spectrum for additive
($0$) but distorts the surviving sector for multiplicative ($0.72$). Eight
checks, all PASS.

\end{document}
