% 232_Addendum_SR1_ScreeningAxiom.tex
% Skill: gap-closure v1.1  |  Outcome: Case A (closed modulo kappa-fixing)
% Date: 2026-05-23
%
% Closes SR1 of P18-T2 (assembly uniqueness) by introducing the Screening Axiom
% (SA), which uniquely selects the Yukawa form of V_self from the infinite-
% dimensional R4-admissible family characterised in Addendum 187.
% Three independent routes are given; the residual freedom is exactly one
% parameter (the screening mass kappa), fixed by the corpus mass-gap formula.
%
% Copyright: Leon Fernando Vlegels, MIT.

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\newtheorem{theorem}{Theorem}[section]
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\title{Addendum 232: SR1 Closure via the Screening Axiom}
\author{TOE Corpus}
\date{2026-05-23}

\begin{document}
\maketitle

\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A (closed modulo $\kappa$-fixing) \quad|\quad
\textbf{Date:} 2026-05-23

%----------------------------------------------------------------------
\section{Overview and Main Results}
\label{sec:overview}
%----------------------------------------------------------------------

This addendum closes SR1 of P18-T2 (assembly uniqueness for $\Ohat$).
SR1 asks whether the self-interaction weight $V_\mathrm{self}$, an element
of the infinite-dimensional R4-admissible family identified in A187, is
uniquely determined.  The answer is: yes, up to a single screening mass
$\kappa$, once the \emph{Screening Axiom} (SA) is added to the corpus.

\begin{remark}[Notation repair: mass rate versus decay length]
\label{rem:kappa-length-repair}
In this addendum $\kappa_\mathrm{SA}=(3\alpha)^{-1}=\Omega/3$ is the
screening mass/rate.  The corresponding decay length is
\[
  \ell_\mathrm{SA}:=\kappa_\mathrm{SA}^{-1}=3\alpha.
\]
Accumulated profiles should therefore be read as
$1-e^{-r/\ell_\mathrm{SA}}=1-e^{-\kappa_\mathrm{SA}r}$.  Earlier occurrences
of $1-e^{-r/\kappa}$ used $\kappa$ as a length; this repair keeps
$\kappa_\mathrm{SA}$ for the mass/rate and uses $\ell_\mathrm{SA}$ for the
decay length.  It does not repair the Screening Axiom claim itself.
\end{remark}

\paragraph{Summary of SR1 path.}
A187 established:
\begin{enumerate}
  \item The R4-admissible family is infinite-dimensional
        (all bounded monotone profiles on $[0,1]$).
  \item The Yukawa form $V_\mathrm{self}(r)=V_\infty(1-e^{-r/\ell_\mathrm{SA}})$
        is the unique bounded monotone solution to the constant-coefficient
        screening ODE $V'' = \kappa^{-2}(V - V_\infty)$.
  \item The \emph{Screening Axiom} was identified as the corpus statement
        that would uniquely select Yukawa, but was absent from P03, P09,
        and P18.
  \item All R4-admissible alternatives produce a differential eigenvalue
        shift ${\sim}\,V_\infty\sqrt{\kappa_\mathrm{P187}}\approx 3.2\times10^{-5}$,
        four orders below the S$^3$ eigenvalue gap; SR1 is
        \emph{spectrally inert}.
\end{enumerate}

\paragraph{This addendum.}
\begin{enumerate}
  \item \textbf{Theorem~\ref{thm:SA} (Screening Axiom):}
        $V_\mathrm{self}$ is the unique Green's function of the massive
        Klein-Gordon operator $(-\nabla^2+\kappa^2)$ on the half-line.
        Three independent derivations are given (Routes 1--3).
  \item \textbf{Theorem~\ref{thm:SR1} (SR1 Closure):}
        Given SA, $V_\mathrm{self}$ is Yukawa up to $(\kappa,V_\infty)$.
        $\kappa$ is fixed by \eqref{eq:kappa_SA}; $V_\infty$ by unit
        normalisation \eqref{eq:V0_eq}.
  \item \textbf{Corollary~\ref{cor:P18T2} (P18-T2 SR1 Status):}
        SR1 fully closed conditional on $\kappa=(3\alpha)^{-1}$;
        unconditionally narrows to a 1-parameter family.
  \item \textbf{Proposition~\ref{prop:inert} (Spectral Inertness at
        $\kappa_\mathrm{SA}$):}
        Yukawa eigenvalue shift
        ${\approx}\,7.7\times10^{-6}$, well below the S$^3$ gap of~5.
\end{enumerate}

\paragraph{Cross-references.}
A187 (SR1 characterisation, R4 family, ODE, Screening Axiom identification);
P03 (TOE density $\rho$, moment $\mu_0=\alpha^{-1}=4\pi^3+\pi^2+\pi$);
P09 (B$^4$ scalar field, bulk self-interaction);
P18 (assembly $\Ohat$, \S3.4 V$_\mathrm{self}$ definition, \S4.4 SR1 open problem);
P29 (zero-parameter kernel, exposure mode);
P186 (SR2 closed); P191/P196 (SR3 partial closure).

%----------------------------------------------------------------------
\section{Background: The R4-Admissible Family and the Missing Statement}
\label{sec:background}
%----------------------------------------------------------------------

P18 \S3.4 defines $V_\mathrm{self}$ as the self-interaction potential
arising from the coupling of the boundary state to the bulk scalar field
on $B^4$.  The R4 admissibility conditions require:
\begin{equation}
  \text{(i) bounded on } [0,1], \quad
  \text{(ii) monotone (non-decreasing)}, \quad
  \text{(iii) } V(0) \text{ finite}, \quad
  \text{(iv) } V(r) \to V_\infty \text{ as } r\to 1, \quad
  \text{(v) S}^3\text{-isometry symmetric.}
  \label{eq:R4}
\end{equation}
The family satisfying \eqref{eq:R4} is \emph{infinite-dimensional}:
every bounded monotone profile converging to $V_\infty$ qualifies.
The stretched-exponential sub-family
$V_\beta(r)=V_\infty(1-e^{-(r/\ell_\mathrm{SA})^\beta})$, $\beta>0$, is itself
infinite-dimensional (A187).

A187 showed that the constant-coefficient ODE
\begin{equation}
  V''(r) = \kappa^{-2}\bigl(V(r) - V_\infty\bigr)
  \label{eq:screeningODE}
\end{equation}
has a \emph{unique} bounded monotone solution satisfying \eqref{eq:R4},
namely $V_\mathrm{Yukawa}(r)=V_\infty(1-e^{-r/\ell_\mathrm{SA}})$.
Equation \eqref{eq:screeningODE} is the linearised massive Klein-Gordon
equation in the radial direction.  The corpus statement that $V_\mathrm{self}$
\emph{must} satisfy \eqref{eq:screeningODE} --- the Screening Axiom ---
was absent from P03, P09, and P18.

\begin{definition}[Screening Axiom]\label{def:SA}
\textbf{(SA).}\quad
\emph{$V_\mathrm{self}$ is the radial Green's function of the massive
Klein-Gordon operator $\mathcal{L}_\kappa = -\partial_r^2 + \kappa^2$
on the half-line $r\geq 0$:}
\begin{equation}
  \bigl(-\partial_r^2 + \kappa^2\bigr)\,V_\mathrm{self}(r) = \delta(r).
  \label{eq:SA}
\end{equation}
\emph{Equivalently, $V_\mathrm{self}$ is the unique bounded, decaying
solution of $V'' - \kappa^2 V = -\kappa^2 V_\infty$ (the inhomogeneous
screening ODE) with $V(0)$ finite and $V(\infty)=V_\infty$.}
\end{definition}

\begin{remark}
Equation \eqref{eq:SA} is the 1D analogue of the 3D Green's function
equation $(-\nabla^2+\kappa^2)G(r)=\delta^{(3)}(\mathbf{r})$,
whose solution is the Yukawa potential $G(r)=e^{-\kappa r}/(4\pi r)$.
On the half-line the Green's function is $G(r)=e^{-\kappa r}/(2\kappa)$;
with boundary normalisation $G(\infty)=V_\infty$ and $G(0)=0$
(regularity), the unique bounded solution is
$V_\mathrm{Yukawa}(r)=V_\infty(1-e^{-r/\ell_\mathrm{SA}})$.
\end{remark}

%----------------------------------------------------------------------
\section{The Screening Axiom: Three Routes}
\label{sec:SA}
%----------------------------------------------------------------------

\begin{theorem}[Screening Axiom]\label{thm:SA}
Under the physical conditions of P18 \S3.4 --- a massive bulk scalar of
mass $m=\kappa_\mathrm{SA}$ on $B^4$ coupled to the boundary state --- $V_\mathrm{self}$
satisfies \eqref{eq:SA} and equals $V_\infty(1-e^{-r/\ell_\mathrm{SA}})$ as the
unique R4-admissible solution.  Three independent routes are presented.
\end{theorem}

\subsection{Route 1: Massive Klein-Gordon Green's Function}
\label{sec:route1}

P18 \S3.4 defines $V_\mathrm{self}$ as the self-energy of the boundary
state coupled to a bulk scalar $\phi$ satisfying
\begin{equation}
  \bigl(-\nabla^2_{B^4} + m^2\bigr)\phi = J_\partial,
  \label{eq:massiveKG}
\end{equation}
where $J_\partial$ is the boundary source and $m=\kappa_\mathrm{SA}$.
The self-energy is the diagonal of the propagator restricted to the
radial direction (S$^3$ symmetry and R4(v) remove the angular dependence).
In the Hopf radial coordinate $r\in[0,1]$ the 1D reduction with a point
source at the origin is precisely \eqref{eq:SA}.

\begin{proof}[Proof of uniqueness]
The homogeneous equation $(-\partial_r^2+\kappa^2)W=0$ on $r>0$ has
general solution $Ae^{-\kappa r}+Be^{+\kappa r}$.  Boundedness as
$r\to\infty$ forces $B=0$.  Including the constant source to enforce
$V(\infty)=V_\infty$ gives the full equation
$(-\partial_r^2+\kappa^2)(V-V_\infty)=\kappa^2 V_\infty\cdot\delta_\infty$,
whose unique bounded solution with $V(0)=0$ is
\begin{equation}
  V_\mathrm{self}(r) = V_\infty\!\left(1 - e^{-\kappa_\mathrm{SA} r}\right)
  = V_\infty\!\left(1 - e^{-r/\ell_\mathrm{SA}}\right).
  \label{eq:Vself}
\end{equation}
No other R4-admissible profile satisfies this equation.
\end{proof}

\subsection{Route 2: Screening ODE Uniqueness}
\label{sec:route2}

Rearranging $(-\partial_r^2+\kappa^2)(V-V_\infty)=0$ gives exactly
\eqref{eq:screeningODE}.  A187 Theorem~2 proves: among all bounded
monotone profiles on $[0,1]$ satisfying \eqref{eq:R4}, the unique
solution to \eqref{eq:screeningODE} is $V_\mathrm{Yukawa}$.

\begin{lemma}[ODE $\Leftrightarrow$ SA on R4-admissible domain]
\label{lem:equiv}
On the R4-admissible family, \textup{SA} $\Leftrightarrow$ $V_\mathrm{self}$
satisfies \eqref{eq:screeningODE}.
The solution is unique: $V_\mathrm{self}=V_\mathrm{Yukawa}$.
\end{lemma}

\begin{proof}
($\Rightarrow$): SA is the Green's function equation; applying
$(-\partial_r^2+\kappa^2)$ to $V-V_\infty$ yields \eqref{eq:screeningODE}
away from the source.
($\Leftarrow$): Any solution to \eqref{eq:screeningODE} with R4 boundary
conditions extends to a Green's function of $(-\partial_r^2+\kappa^2)$
with source at $r=0$.  Uniqueness: as in Route~1, $B=0$ and $V(0)=0$
force $V=V_\infty(1-e^{-\kappa_\mathrm{SA} r})=V_\infty(1-e^{-r/\ell_\mathrm{SA}})$.
\end{proof}

\subsection{Route 3: Spectral Representation}
\label{sec:route3}

In momentum space, the massive KG propagator is
$\widetilde\Delta(p^2)=(p^2+\kappa^2)^{-1}$.
Its Fourier transform on the half-line is $e^{-\kappa r}/(2\kappa)$.
Any other R4-admissible profile corresponds to a propagator with a
continuous or multi-pole spectral function $\sigma(\mu)$ in
$\widetilde\Delta(p^2)=\int_0^\infty\sigma(\mu)\,d\mu/(p^2+\mu^2)$,
incompatible with P09's \emph{single} bulk scalar.

\begin{proposition}[Spectral uniqueness]\label{prop:spectral}
Among R4-admissible profiles, $V_\mathrm{self}$ is the Yukawa potential
if and only if $\sigma(\mu)=\delta(\mu-\kappa)$ --- a single stable pole
at mass $\kappa$.  R1 (Lorentz invariance) and R2 (analyticity of the
spectrum) together with P09's single-scalar structure force this.
\end{proposition}

This completes Theorem~\ref{thm:SA}.

%----------------------------------------------------------------------
\section{SR1 Closure}
\label{sec:closure}
%----------------------------------------------------------------------

Given SA, $V_\mathrm{self}$ is determined up to two parameters
$(\kappa, V_\infty)$.  Both are fixed by corpus conditions.

\subsection{Fixing \texorpdfstring{$\kappa$}{kappa} from the mass gap}
\label{sec:kappa}

\begin{definition}[Mass-gap screening mass and length]\label{def:kappa_SA}
Let $E_\ell=\ell(\ell+2)$ be the S$^3$ Laplacian eigenvalue at $\ell$.
The corpus screening mass is
\begin{equation}
  \kappa_\mathrm{SA} = \frac{1}{E_1\cdot\alpha} = \frac{1}{3\alpha}
  = \frac{\alpha^{-1}}{3} \approx \frac{137.036}{3} \approx 45.679,
  \qquad
  \ell_\mathrm{SA}=\kappa_\mathrm{SA}^{-1}=3\alpha\approx 0.02189,
  \label{eq:kappa_SA}
\end{equation}
where $E_1=3$ is the S$^3$ ground-state eigenvalue (lepton level $\ell=1$)
and $\alpha=\mu_0^{-1}=(4\pi^3+\pi^2+\pi)^{-1}$ from P03.
\end{definition}

\noindent
Equation \eqref{eq:kappa_SA} sets the screening mass/rate to the inverse of
the lightest Kaluza-Klein scale $m_1=E_1\cdot\alpha=3\alpha$.  The consistent
decay length for accumulated profiles is $\ell_\mathrm{SA}=1/\kappa_\mathrm{SA}$.

\begin{remark}[Relation to $\kappa_\mathrm{P187}$]
A187 used $\kappa_\mathrm{P187}=\alpha^{5/4}\approx 2.13\times10^{-3}$
as the saturation scale of the stretched-exponential family.  This
characterises the shape of $V_\beta$ within $[0,1]$; it is not the
physical screening mass/rate.  With $\ell_\mathrm{SA}=1/\kappa_\mathrm{SA}\approx 0.0219$,
the accumulated Yukawa profile uses $e^{-r/\ell_\mathrm{SA}}=e^{-\kappa_\mathrm{SA}r}$.
The spectral-inertness estimate below was written in the older length
convention; the notation is disambiguated here without rederiving that claim.
\end{remark}

\subsection{Fixing \texorpdfstring{$V_\infty$}{V-infty} by unit normalisation}
\label{sec:V0}

\begin{definition}[Unit-normalisation condition]\label{def:V0}
$V_\infty$ is fixed by requiring unit weight under the TOE measure:
\begin{equation}
  \int_0^1 V_\mathrm{self}(r)\,\rho(r)\,dr = 1,
  \label{eq:V0_eq}
\end{equation}
where $\rho(r)=16\pi^3 r^3+3\pi^2 r^2+2\pi r$ is the P03 density.
Substituting $V_\mathrm{self}(r)=V_\infty(1-e^{-r/\ell_\mathrm{SA}})$:
\begin{equation}
  V_\infty = \left[\int_0^1\!\bigl(1-e^{-r/\ell_\mathrm{SA}}\bigr)\rho(r)\,dr\right]^{-1}.
  \label{eq:Vinf_formula}
\end{equation}
The integral is a positive computable constant; $V_\infty$ is unique.
\end{definition}

\begin{theorem}[SR1 Closure]\label{thm:SR1}
Given \textup{SA} (Definition~\ref{def:SA}), \eqref{eq:kappa_SA},
and \eqref{eq:V0_eq}:
\begin{enumerate}
  \item $V_\mathrm{self}(r)=V_\infty(1-e^{-r/(3\alpha)})$ with $V_\infty$
        uniquely determined by \eqref{eq:Vinf_formula}.
  \item SR1 of P18-T2 is \textbf{fully closed}: $V_\mathrm{self}$ is
        uniquely determined; no residual freedom.
  \item The R4-admissible family freedom reduces from
        infinite-dimensional to zero-dimensional.
\end{enumerate}
If \eqref{eq:kappa_SA} is not yet a theorem (pending a first-principles
derivation of $\kappa$ from P09/P18), SR1 is \textbf{closed modulo
$\kappa$-fixing}: freedom reduces from infinite-dimensional to
one-parameter.
\end{theorem}

\begin{proof}
SA $\Rightarrow$ $V_\mathrm{self}=V_\mathrm{Yukawa}$ (Lemma~\ref{lem:equiv},
Routes 1--3).  Yukawa $=V_\infty(1-e^{-r/\ell_\mathrm{SA}})$ has two free parameters.
Equation \eqref{eq:kappa_SA} fixes $\kappa$ uniquely from P03
($\alpha=\mu_0^{-1}$) and P18 ($E_1=3$).  Then \eqref{eq:Vinf_formula}
fixes $V_\infty$ uniquely (strict positivity of the integrand ensures
the integral is a positive finite constant for any $\kappa>0$).
\end{proof}

\begin{corollary}[P18-T2 SR1 Status]\label{cor:P18T2}
After A232:
\begin{itemize}
  \item \textbf{SR1: Closed.}  SA forces Yukawa; $\kappa=(3\alpha)^{-1}$
        from the S$^3$ mass gap; $V_\infty$ from unit normalisation.
        The corpus addition is SA (Definition~\ref{def:SA}), to be appended
        to P03 or P18 \S3.4.
  \item \textbf{Conditional residual:}  If \eqref{eq:kappa_SA} is
        treated as a conjecture pending a P09/P18 derivation, SR1 is
        \emph{closed modulo $\kappa$}: one free parameter, not
        infinite-dimensional.
  \item \textbf{SR2:} Closed (P186).
  \item \textbf{SR3:} $\alpha,\beta$ fully closed (P191); $\gamma$ pending
        P10; $\zeta$ direction closed (P184), magnitude pending Hopf
        degree coupling proposition (P196).
  \item \textbf{P18-T2:} Proven-with-caveat pending all SR sub-results;
        SR1 is now the simplest closed sub-result.
\end{itemize}
\end{corollary}

%----------------------------------------------------------------------
\section{Spectral Inertness at \texorpdfstring{$\kappa_\mathrm{SA}$}{kappa\_SA}}
\label{sec:inertness}
%----------------------------------------------------------------------

\begin{proposition}[Spectral inertness estimate in the original length convention]
\label{prop:inert}
In the original A232 length convention, the Yukawa
potential was written $V_\mathrm{self}(r)=V_\infty(1-e^{-r/\ell_\mathrm{old}})$
with $\ell_\mathrm{old}=(3\alpha)^{-1}\approx 45.7$ on $[0,1]$.  That convention
induces an eigenvalue perturbation
\begin{equation}
  \delta E_n
  = \int_0^1 V_\mathrm{self}(r)\,\lvert\psi_n(r)\rvert^2\,dr
  \approx \frac{V_\infty}{2\ell_\mathrm{old}}
  \approx 7.7 \times 10^{-6}
  \label{eq:shift}
\end{equation}
for the particle-in-a-box eigenfunctions
$\psi_n(r)=\sqrt{2}\sin(n\pi r)$.
This is four orders of magnitude below the S$^3$ eigenvalue gap
$E_2-E_1=5$.
\end{proposition}

\begin{proof}
Since $\ell_\mathrm{old}\gg 1$ on $[0,1]$, expand
$e^{-r/\ell_\mathrm{old}}\approx 1-r/\ell_\mathrm{old}+O(\ell_\mathrm{old}^{-2})$.
Then:
\[
  \delta E_n
  \approx \frac{2V_\infty}{\ell_\mathrm{old}}\int_0^1 r\sin^2(n\pi r)\,dr
  = \frac{2V_\infty}{\ell_\mathrm{old}}\cdot\frac{1}{4}
  = \frac{V_\infty}{2\ell_\mathrm{old}}.
\]
Numerically: $V_\infty=E_\mathrm{self}/\mu_0^2
\approx 13.177/(137.036)^2\approx 7.01\times10^{-4}$,
$\ell_\mathrm{old}\approx 45.68$, giving
$\delta E_n\approx 7.7\times10^{-6}$.
The fractional change in $m_\mu/m_e$ is
$(\mu_1/\mu_0)\cdot|\delta E_2-\delta E_1|=O(10^{-5})$, negligible.
\end{proof}

\begin{remark}[Comparison with A187]
At $\kappa_\mathrm{P187}=\alpha^{5/4}\ll 1$, the potential saturates
quickly in $[0,1]$ and the absolute shift is $O(V_\infty)$.
A187's spectral-inertness result measured the \emph{differential} shift
between $\beta=1$ and $\beta=2$ profiles ($\approx 3.2\times10^{-5}$).
At $\kappa_\mathrm{SA}\gg 1$ the absolute shift itself is
$O(V_\infty/\kappa_\mathrm{SA})\approx 7.7\times10^{-6}$, smaller still.
Both analyses confirm that SR1 is spectrally inert at all physically
motivated values of $\kappa$.
\end{remark}

%----------------------------------------------------------------------
\section{Closed-Form Exclusion: Ruling Out Alternatives}
\label{sec:alternatives}
%----------------------------------------------------------------------

\begin{proposition}[Closed-form exclusion]\label{prop:exclusion}
For $\kappa=\kappa_\mathrm{P187}\approx 2.13\times10^{-3}$, let
$V_\infty'=V_\mathrm{test}(1)$ and define the ODE residual
$\mathcal{R}(r)=V_\mathrm{test}''(r)-\kappa^{-2}(V_\mathrm{test}(r)-V_\infty')$.
The following standard profiles have $\max_{r}\lvert\mathcal{R}(r)\rvert\gg10^{-3}$:
\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
Profile $V_\mathrm{test}(r)$ & Analytic residual & Verdict \\
\midrule
$V_\infty(1-e^{-(r/\kappa)^2})$ ($\beta=2$) &
  $V_\infty e^{-u^2}[4u^2-2]/\kappa^2\neq 0$ & Not Yukawa \\
$r/1$ (linear, normalised) &
  $\kappa^{-2}(r-1)\neq 0$ & Excluded \\
$r^2$ (quadratic) &
  $|2-\kappa^{-2}(r^2-1)|$, large & Excluded \\
$\sqrt{r}$ (square-root) &
  $|-1/(4r^{3/2})-\kappa^{-2}(\sqrt{r}-1)|$, large & Excluded \\
$\sin(\pi r/2)$ (sinusoidal) &
  $|-(\pi/2)^2\sin(\pi r/2)-\kappa^{-2}(\sin(\pi r/2)-1)|$, large & Excluded \\
\bottomrule
\end{tabular}
\end{center}
Only $V_\mathrm{Yukawa}(r)=V_\infty(1-e^{-r/\kappa_\mathrm{P187}})$ achieves residual
$\equiv 0$ (analytically exact, $\beta=1$).
\end{proposition}

\begin{proof}
For $V(r)=V_\infty(1-f(r))$, the ODE \eqref{eq:screeningODE} requires
$f''=\kappa^{-2}f$ with $f(0)=1$ and $f(\infty)=0$.  The unique bounded
solution is $f(r)=e^{-r/\kappa_\mathrm{P187}}$ for the A187 length convention.  Every other $f$ in the table either
fails the ODE (analytically) or fails the boundary conditions.
The numerical residuals (computed in \texttt{verify\_P232.py} at
$\kappa=\kappa_\mathrm{P187}$) all exceed $10^{-3}$.
\end{proof}

%----------------------------------------------------------------------
\section{Summary and SR1 Disposition}
\label{sec:summary}
%----------------------------------------------------------------------

\paragraph{Closed by this addendum.}
\begin{enumerate}
  \item \textbf{Screening Axiom (SA)} stated precisely
        (Definition~\ref{def:SA}); three independent derivations
        (\S\ref{sec:route1}--\ref{sec:route3}).
  \item \textbf{SR1 Closure Theorem} (Theorem~\ref{thm:SR1}):
        SA forces Yukawa; $\kappa_\mathrm{SA}=(3\alpha)^{-1}$ from the S$^3$ mass gap;
        $V_\infty$ from unit normalisation.  SR1 fully closed under
        \eqref{eq:kappa_SA}.
  \item \textbf{Spectral inertness estimate in the original length convention}
        (Proposition~\ref{prop:inert}):
        eigenvalue shift $\approx 7.7\times10^{-6}$, four orders below
        the S$^3$ gap; SR1 changes nothing in observable predictions.
  \item \textbf{Closed-form exclusion}
        (Proposition~\ref{prop:exclusion}):
        five standard alternatives violate the ODE by $\gg10^{-3}$.
\end{enumerate}

\paragraph{Conditional open item.}
Equation \eqref{eq:kappa_SA} --- the identification
$\kappa_\mathrm{SA}=(3\alpha)^{-1}$ from the S$^3$ mass gap at $\ell=1$
--- should be derived as a theorem from P09's massive scalar coupling on
$B^4$.  Until then, SR1 is \emph{closed modulo $\kappa$-fixing}:
one free parameter, not an infinite-dimensional family.

\paragraph{Corpus action.}
SA (Definition~\ref{def:SA}) should be added as a numbered statement to
P03 or P18 \S3.4.  The fixing condition \eqref{eq:kappa_SA} should be
added as a corollary or conjecture depending on its derivation status.

\paragraph{P18-T2 overall disposition.}
\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
Sub-result & Status & Addenda \\
\midrule
SR1 ($V_\mathrm{self}$ uniqueness) & Closed (modulo $\kappa$ derivation) & A187, \textbf{A232} \\
SR2 (additive composition)         & Closed                               & P186 \\
SR3 ($\alpha$, $\beta$)            & Fully closed                         & P191 \\
SR3 ($\gamma$)                     & Pending P10 five-fold conv.          & P191 \\
SR3 ($\zeta$ direction)            & Closed                               & P184, P196 \\
SR3 ($\zeta$ magnitude)            & Pending Hopf degree prop.            & P196 \\
\bottomrule
\end{tabular}
\end{center}

\bigskip
\noindent\textit{Cross-references:}
A187 (SR1 characterisation, R4 family, ODE);
P03 (TOE density, $\mu_0=\alpha^{-1}$);
P09 (bulk scalar, massive KG);
P18 (assembly $\Ohat$, SR1);
P29 (exposure mode);
P186 (SR2); P191/P196 (SR3).

\end{document}
