% 308_Addendum_DecouplingDischarged.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
%
% P18-T2 sub-result (ii), strengthening A306. A306 forced additive
% composition GIVEN the independent-decoupling axiom, which it assumed.
% This addendum tests whether that axiom follows from P18's R1-R8
% coverage requirements, replacing it with the no-category-bleed
% condition IC2 -- the operator reading of "R1-R8 name independent
% categories". The operator-level bleed measure is B_ij =
% ||d^2 O / dc_i dc_j||_F. Over a declared linear-coupling family
% {additive, multiplicative, affine-with-product, operadic} on two
% sectors of P18's radial operator (density alpha*rho [R5],
% renormalization zeta*r d/dr [R7]), B_ij = 0 (no bleed) holds IFF the
% composition is additive: additive B_ij = 0, multiplicative 3.06e6,
% affine-product and operadic 5.47. So IC2 forces additivity with no
% separate decoupling axiom. Honest boundary: linear-strength couplings
% still assumed (P18's own treatment), a finite declared family is
% tested (not every operad), sub-results (i),(iii) unchanged, A294
% stands. Case C, strictly stronger than A306.
% Verifier: verify_P308.py.

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
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\usepackage{hyperref}

\newtheorem{proposition}{Proposition}[section]
\newtheorem{remark}[proposition]{Remark}

\title{Addendum 308: The Decoupling Axiom Discharged to Coverage\\
\large A verified result on P18-T2 sub-result (ii):
no category bleed forces additivity}
\author{TOE Corpus}
\date{2026-06-13}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case C (sub-result (ii) strengthened: decoupling axiom
discharged to the IC2 coverage requirement) \quad|\quad
\textbf{Addresses:} P18-T2 sub-result (ii); A306; A294 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P308.py} --- all PASS

\begin{abstract}
A306 advanced P18-T2's second sub-result, that additive composition is
the only admissible way to compose the named sectors, conditional on one
stated assumption: the independent-decoupling axiom, under which each
$c_j \to 0$ removes sector $j$ and leaves the other sectors' action
unchanged. That axiom was assumed. This addendum tests whether it follows
from P18's design requirements (R1)--(R8) of \S1.2, which P18 states are
categorical coverage requirements, each naming an independent category of
structure, not algebraic constraints. The operator reading of
``independent category'' is no category bleed: the contribution of sector
$j$ to the operator is a function of $c_j$ and $S_j$ alone, not of the
other couplings $c_i$ (IC2), together with coverage surviving removal,
that setting $c_i \to 0$ does not distort the remaining categories (IC1).
The operator-level measure of category bleed between two sectors is the
mixed second partial of the operator itself,
$B_{ij} = \|\partial^2 \hat O / \partial c_i\,\partial c_j\|_F$, the same
operator quantity A306 used as its discriminator. Over a declared,
finite, linear-coupling family of four compositions 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
$\alpha = 1/137.036$, $\zeta = \alpha^{5/4}$, no bleed holds if and only
if the composition is additive: the additive $B_{ij}$ is the zero
operator to machine precision, the multiplicative $B_{ij}$ is
$3.06\times10^{6}$, and the affine-with-product and operadic forms are
each $5.47$, matching their analytic cross-terms. Within the
linear-coupling class IC2 is therefore equivalent to additivity, so the
no-category-bleed reading of (R1)--(R8) forces additive composition
without a separate decoupling axiom. The advance is strictly stronger
than A306 because the assumption is now grounded in P18's coverage
requirements rather than asserted. Three boundaries bound it: couplings
are still assumed to act as linear strengths, which is P18's own
treatment; the family tested is finite and declared, not every
conceivable operad; and sub-results (i) and (iii) are unchanged with
A294's re-typing standing.
\end{abstract}

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

\begin{proposition}[No category bleed is equivalent to additivity, within
the linear-coupling class]
Write P18's radial operator over the fixed kinetic-plus-self-lensing part
$K$ and the bare sector operators $S_1, S_2$ carrying linear coupling
strengths $c_1, c_2$, restricting to two sectors for a clean
demonstration: the density coupling $S_1 = \rho(r)$ (R5) with weight
$c_1 = \alpha$ and the renormalization generator $S_2 = r\partial_r$ (R7)
with weight $c_2 = \zeta$, at the canonical $\alpha = 1/137.036$,
$\zeta = \alpha^{5/4}$. Define the operator-level category-bleed measure
\[
B_{12}(\hat O)
  = \Big\| \frac{\partial^2 \hat O}{\partial c_1\,\partial c_2} \Big\|_F .
\]
The no-category-bleed condition IC2, the operator reading of ``R1 and R7
name independent categories'', is $B_{12} = 0$. The coverage-on-removal
condition IC1 is that $\hat O(c_1, 0)$ leaves category $1$ equal to the
bare reduced operator $K + c_1 S_1$. Take the declared linear-coupling
family in which each $c_j$ scales its sector linearly, per P18's treatment
of $(\alpha, \gamma, \zeta, \beta)$ as multiplicative strengths:
\[
\begin{aligned}
\hat O_{\rm add}  &= K + c_1 S_1 + c_2 S_2, \\
\hat O_{\rm mult} &= K\,(I + c_1 S_1)(I + c_2 S_2), \\
\hat O_{\rm affp} &= K + c_1 S_1 + c_2 S_2 + c_1 c_2\, S_1 S_2, \\
\hat O_{\rm oper} &= K + c_1 S_1 + c_2 S_2
                     + \tfrac{1}{2} c_1 c_2\,(S_1 S_2 + S_2 S_1).
\end{aligned}
\]
Then no category bleed holds if and only if the composition is additive:
\begin{enumerate}
\item[\textnormal{(i)}] The additive composition is affine in
$(c_1, c_2)$, so $B_{12}(\hat O_{\rm add})$ is the zero operator
identically, $0$ to machine precision; and IC1 holds, the $c_2 \to 0$
limit recovers $K + \alpha S_1$ with spectral residual $0$.
\item[\textnormal{(ii)}] Every non-additive member of the family carries a
nonzero $c_1 c_2$ cross-structure, so its $B_{12}$ is a forced bleed:
$B_{12}(\hat O_{\rm mult}) = 3.06\times10^{6}$, matching the analytic
$\|K S_1 S_2\|_F$ to $10^{-6}$ relative, while
$B_{12}(\hat O_{\rm affp}) = B_{12}(\hat O_{\rm oper}) = 5.47$, matching
$\|S_1 S_2\|_F$ and $\|\tfrac{1}{2}(S_1 S_2 + S_2 S_1)\|_F$ to the
finite-difference residual.
\item[\textnormal{(iii)}] Across the family, $B_{12} = 0$ selects the
additive composition alone. The multiplicative form additionally fails
IC1, distorting the surviving category on removal (spectral residual
$0.72$); the affine-product and operadic forms preserve IC1 but are
excluded by their nonzero bleed.
\end{enumerate}
Within the linear-coupling class IC2 is therefore equivalent to
additivity. The no-category-bleed reading of (R1)--(R8) forces additive
composition.
\end{proposition}

\begin{remark}[What the result rests on, what remains open]
The advance relative to A306 is in the assumption, not the numbers. A306
forced additivity given the independent-decoupling axiom and assumed that
axiom. Here the axiom is replaced by IC2, the operator reading of P18
\S1.2's statement that (R1)--(R8) name independent categories of
structure. IC2 is not a new postulate; it is what ``the categories are
independent'' says at the operator level, namely that turning on one
sector does not enter another sector's contribution. The forcing is then
the family-level equivalence above: no bleed selects the additive form
alone. Three boundaries remain. The couplings are assumed to act as linear
strengths, which is P18's own treatment of $(\alpha, \gamma, \zeta,
\beta)$ but is an assumption nonetheless; a composition in which a coupling
acts nonlinearly is outside the class tested. The family is finite and
declared, four compositions chosen to span the natural alternatives;
exotic operads engineered so that their cross-terms cancel on the relevant
sectors are not enumerated, and the full operadic sweep stays open. The
result addresses sub-result (ii) alone: the per-category operator
uniqueness (i) carries A304's verified result for one sector and the
coefficient uniqueness (iii) carries A302's local result, both unchanged.
A294's re-typing of the whole P18-T2 stands. The standing of the question
is that sub-result (ii) is now forced by a coverage requirement rather
than by a separate axiom, a strictly weaker hypothesis than A306 carried.
\end{remark}

\paragraph{Status.} P18-T2 remains open. Its second sub-result now rests
on a weaker hypothesis than before: additive composition is forced by the
no-category-bleed reading of (R1)--(R8), a P18 coverage requirement,
rather than by a separately assumed decoupling axiom. The forcing is exact
within the declared linear-coupling family, where no bleed is equivalent
to additivity. The assumption that couplings act as linear strengths
remains, the full operadic sweep remains open, and sub-results (i) and
(iii) are unchanged. No published number changes.

\paragraph{Verifier.} \texttt{verify\_P308.py}: rebuilds the two sectors
of P18's radial operator and the four-member linear-coupling family from
scratch, asserts the additive operator mixed partial $B_{12}$ vanishes
($<10^{-4}$) as an exact identity while each non-additive member is a
forced bleed ($>10^{-2}$) matching its analytic cross-term, asserts the
$c_2 \to 0$ coverage check on the additive form, and asserts that within
the family no bleed is equivalent to additivity. Eight checks, all PASS.

\end{document}
