% 310_Addendum_LinearCouplingDischarged.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
%
% P18-T2 sub-result (ii), strengthening A308. A308 reduced the second
% sub-result to one remaining assumption: each coupling c_j enters as a
% linear strength c_j*S_j rather than f_j(c_j)*S_j for a nonlinear f_j.
% A308 retained that assumption. This addendum tests whether the linear
% FORM is forced by P18 S4.2's own definition of the couplings as
% dimensional strengths. Two readings: (L1) a strength's per-sector
% operator response is constant, so d^2 O/dc^2 = 0 (the zero operator);
% (L2) a strength is degree-1 homogeneous, f(2c)=2f(c). Over a declared
% family of coupling forms {linear c, quadratic c^2, sqrt c, exp-1} on the
% density sector rho(r) [R5] over fixed K, only the linear form satisfies
% both: linear d^2 O/dc^2 = 8.9e-4 (machine zero) and homogeneity residual
% 0; quadratic d^2 O/dc^2 = 17.18 (matches ||2S||), sqrt 3.47e3, exp-1
% 8.66, all with nonzero homogeneity residual. So the linear form is forced
% by the strength definition. Combined with A308 (IC2 forces additivity
% within the linear class), additive composition is forced with both hinges
% grounded in P18 requirements. Honest boundary: "couplings are strengths"
% is P18 S4.2 made precise, not derived from deeper principles; a finite
% declared family is tested; sub-results (i)=A304, (iii)=A302 unchanged;
% A294 stands; P18-T2 not closed. Case C, strictly stronger than A308.
% Verifier: verify_P310.py.

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

\title{Addendum 310: The Linear-Coupling Hinge Discharged to the
Strength Definition\\
\large A verified result on P18-T2 sub-result (ii):
both hinges now grounded in P18}
\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): linear-coupling hinge discharged
to the dimensional-strength definition; both hinges now grounded) \quad|\quad
\textbf{Addresses:} P18-T2 sub-result (ii); A308; A306; A294 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P310.py} --- all PASS

\begin{abstract}
A308 advanced P18-T2's second sub-result, that additive composition is the
only admissible way to compose the named sectors, by discharging A306's
decoupling axiom to the no-category-bleed coverage requirement IC2: within
the linear-coupling class, no category bleed is equivalent to additivity.
A308 left one hinge in place. The reduction held inside the linear-coupling
class, in which each coupling $c_j$ scales its sector linearly as
$c_j S_j$ rather than entering through a nonlinear $f_j(c_j) S_j$ or through
$S_j$'s internal structure. That linear form was retained, not derived.
This addendum tests whether the linear form is itself forced by P18's
treatment of the couplings. P18 \S4.2 fixes $\alpha, \gamma, \zeta, \beta$
as coupling \emph{strengths}, each pinned by geometric and dimensional
consistency, and \S4.1 writes the density sector as $\alpha\rho(r)$, a
strength multiplying a fixed sector operator. A strength multiplying a
fixed operator is a degree-1 homogeneous linear prefactor by definition,
and that definition has an operator content. Two readings carry it. First,
dimensional homogeneity: a strength's per-sector operator response is
constant in the coupling, $\partial \hat O_j / \partial c_j = S_j$, so the
operator second derivative $\partial^2 \hat O_j / \partial c_j^2$ is the
zero operator; only a degree-1 prefactor has vanishing operator curvature,
and a units rescaling $c_j \to s c_j$ preserves the spectrum structure only
for that form. Second, superposition within a sector: a strength satisfies
$f_j(2c) = 2 f_j(c)$, degree-1 homogeneity. Over a declared family of
coupling forms, the linear $c$, the quadratic $c^2$, the $\sqrt{c}$, and the
$e^{c}-1$, applied to the density sector $\rho(r)$ (R5) over the fixed
kinetic-plus-self-lensing operator $K$ at canonical $\alpha = 1/137.036$,
only the linear form satisfies both readings: its operator second
derivative is $8.9\times10^{-4}$, the zero operator to the finite-difference
floor, and its homogeneity residual is $0$, while the quadratic carries
operator curvature $17.18$ matching $\|2S\|_F$, the $\sqrt{c}$ carries
$3.47\times10^{3}$, and the $e^{c}-1$ carries $8.66$, each with a nonzero
homogeneity residual. The linear form is the unique member of the family on
both criteria. The renormalization sector (R7) under linear coupling shows
the same vanishing curvature, confirming the property is sector
independent. Combined with A308, additive composition is forced with both
hinges grounded in P18's stated requirements: the R1--R8 independent
categories give IC2, and the dimensional-strength definition of the
couplings gives the linear form. Three boundaries remain. The premise that
the couplings are strengths is P18 \S4.2's definition made precise, not a
result derived from deeper principles. The family of coupling forms is
finite and declared, not every nonlinear form. Sub-results (i) and (iii)
are unchanged, with A304's verified result for one sector, A302's local
result for the coefficients, and A294's re-typing standing.
\end{abstract}

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

\begin{proposition}[The linear coupling form is forced by the
dimensional-strength definition]
Write the per-sector operator $\hat O_j(c) = K + f_j(c)\, S_j$ over the
fixed kinetic-plus-self-lensing part $K$ and the bare density sector
$S = \rho(r)$ (R5), normalized, at canonical $\alpha = 1/137.036$, with the
renormalization generator $S_2 = r\partial_r$ (R7) at $\zeta = \alpha^{5/4}$
cross-checked. P18 \S4.2 treats the coupling as a \emph{strength}: a fixed
prefactor multiplying a fixed sector operator, pinned by dimensional and
geometric consistency. Read this at the operator level two ways.
\begin{enumerate}
\item[\textnormal{(L1)}] Dimensional homogeneity. A strength's per-sector
operator response is constant in the coupling,
$\partial \hat O_j / \partial c = S$, so the operator second derivative
\[
\Big\| \frac{\partial^2 \hat O_j}{\partial c^2} \Big\|_F
\]
is the zero operator. Under a units rescaling $c \to s c$, only the
degree-1 prefactor preserves this constancy; a nonlinear $f_j$ acquires
$c$-dependence in the response and breaks the dimensional reading. Across
the declared family $\{c,\ c^2,\ \sqrt{c},\ e^{c}-1\}$ the linear form has
$\|\partial^2 \hat O / \partial c^2\|_F = 8.9\times10^{-4}$, the zero
operator to the finite-difference floor, with linear
$\partial \hat O / \partial c$ matching the constant operator
$\|S\|_F = 8.59$. Every nonlinear form carries genuine operator curvature:
the quadratic $\|\partial^2 \hat O / \partial c^2\|_F = 17.18$, matching the
analytic $\|2S\|_F$; the $\sqrt{c}$ form $3.47\times10^{3}$, matching
$\tfrac14 c^{-3/2}\|S\|_F$ to the finite-difference residual; the $e^{c}-1$
form $8.66$, matching $e^{c}\|S\|_F$.
\item[\textnormal{(L2)}] Superposition within a sector. A strength means
doubling the coupling doubles the sector's contribution,
$f_j(2c) = 2 f_j(c)$, degree-1 homogeneity. Only the linear form satisfies
it: its residual $\max_c |f(2c) - 2f(c)|/|2f(c)|$ is $0$, while the
quadratic gives $1$, the $\sqrt{c}$ gives $0.29$, and the $e^{c}-1$ gives
$1.9\times10^{-2}$.
\item[\textnormal{(L3)}] The linear form is the unique member of the family
that is both curvature-free (L1) and degree-1 homogeneous (L2). The
renormalization sector under linear coupling has the same vanishing
curvature, $6.2\times10^{-4}$, so the property is sector independent. A
coupling strength, as P18 \S4.2 defines it, is therefore the linear form.
\end{enumerate}
Within ``coupling is a dimensional strength'' the linear form is forced.
\end{proposition}

\begin{remark}[The assembled chain, and what remains open]
The advance relative to A308 is in the assumption, not the numbers. A308
forced additivity from IC2 within the linear-coupling class and retained
the linear form. Here the linear form is replaced by P18 \S4.2's own
definition of the couplings as dimensional strengths, made precise at the
operator level: a strength's per-sector response is constant (L1) and
doubling the coupling doubles the sector (L2), and only the linear form
meets both. The chain assembles. The R1--R8 statement that the named
categories are independent gives IC2, the no-category-bleed condition, which
A308 showed is equivalent to additivity within the linear class. The \S4.2
statement that the couplings are dimensional strengths gives the linear
form, which this addendum shows is forced within the declared family. So
additive composition with $c_j S_j$ is forced given (R1--R8 independent
categories $\Rightarrow$ IC2) and (couplings are dimensional strengths
$\Rightarrow$ linear), with no remaining free assumption beyond the R1--R8
coverage requirements and the definition of a coupling as a strength. Both
hinges of sub-result (ii) are now grounded in P18's stated requirements
rather than asserted. Three boundaries bound the claim. The premise that
the couplings are strengths is P18 \S4.2's definition, made precise here,
not derived from something deeper that the couplings must be strengths; a
treatment in which a coupling enters nonlinearly is outside the definition,
not contradicted by it. The family of coupling forms is finite and
declared, four forms spanning the natural alternatives, not every
conceivable nonlinear $f_j$. 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.
\end{remark}

\paragraph{Status.} P18-T2 remains open. Its second sub-result now rests on
two hinges, both grounded in P18's stated requirements rather than assumed
ad hoc: additive composition is forced by the no-category-bleed reading of
R1--R8 (IC2, A308) together with the linear form of the couplings, which
follows from P18 \S4.2's definition of $\alpha, \gamma, \zeta, \beta$ as
dimensional strengths. The forcing of the linear form is exact within the
declared family, where the linear form is the unique curvature-free,
degree-1 homogeneous member. The premise that the couplings are strengths
is P18's definition rather than a derived result, the family of forms is
finite, and sub-results (i) and (iii) are unchanged. No published number
changes.

\paragraph{Verifier.} \texttt{verify\_P310.py}: rebuilds the density sector
of P18's radial operator and the four-member family of coupling forms from
scratch, asserts the linear operator second derivative vanishes
($<10^{-2}$, the finite-difference floor set by the radial stencil) while
the quadratic and $e^{c}-1$ forms match their analytic curvature to
$10^{-4}$ relative and the $\sqrt{c}$ form to $10^{-2}$ relative, asserts
that only the linear form is degree-1 homogeneous, asserts that the linear
form is the unique member of the family on both criteria, and confirms the
same vanishing curvature on the renormalization sector. Eight checks, all
PASS.

\end{document}
