% Addendum 187: SR1 — The R4-Admissible Family for V_self and Yukawa Uniqueness
%
%   P18 §1.2 lists R4 (Self-consistency: include self-lensing potential) as a
%   coverage requirement, not an algebraic constraint.  P18 §4.4 identifies, as
%   Sub-result SR1, whether the Yukawa functional form
%     V_self(r) = (E_self/m_0^2)(1 - e^{-r/r_0})
%   is the unique admissible operator in the R4 category.
%
%   Prior session analysis (Documents/Specs/P18-T2-research.md, 2026-05-17)
%   concluded: SR1 is genuinely open; R4 alone admits an infinite family; no
%   Klein-Gordon grounding exists in the current corpus.
%
%   This paper:
%     (1) States R4 precisely and maps its admissible family.
%     (2) Derives the 1D massive-screening ODE whose unique bounded solution
%         with V(0)=0 is exactly the Yukawa form.
%     (3) Documents the corpus gap: the ODE is absent from P03/P09/P18.
%     (4) Characterises the stretched-exponential alternative family V_beta.
%     (5) Proves by first-order perturbation theory and numerical verification
%         that V_self perturbations produce spectral shifts of order V_inf ~ 7e-4,
%         negligible relative to S^3 Laplacian eigenvalues (3, 8, 15, ...).
%     (6) Concludes: the m_mu/m_e failure is NOT attributable to V_self freedom;
%         varying V_self within the R4-admissible family cannot lift the ratio
%         from ~3 to ~207 at leading order.
%
%   Status: "space of alternatives" paper (P18-T2 SR1 characterisation).
%   Numerical companion: Lumen/corpus/addenda/verify/verify_P187.py
%
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\newtheorem{theorem}{Theorem}[section]
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\title{%
  Addendum 187:\\
  SR1 --- The R4-Admissible Family for $V_{\mathrm{self}}$\\
  and Yukawa Uniqueness\\[0.4em]
  \large What R4 Actually Constrains, the Missing Screening ODE,\\
  and the Spectral Inertness of $V_{\mathrm{self}}$ Alternatives
}
\author{L\'eon Fernando Vlegels}
\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
Paper~18's requirement~R4 (``include self-lensing potential'') admits an
infinite-dimensional family of admissible operators.  The canonical choice
$V_{\mathrm{self}}(r) = (E_{\mathrm{self}}/m_0^2)(1-e^{-r/r_0})$ is the
unique bounded monotone solution to the 1D massive screening ODE
$V'' = r_0^{-2}(V - V_\infty)$ with $V(0)=0$, where
$r_0 = \kappa = \alpha^{5/4}$ and $V_\infty = E_{\mathrm{self}}/m_0^2$.
That ODE is \emph{absent from the current TOE corpus}: neither P03 nor P09
states it as a constraint on $V_{\mathrm{self}}$.  Without it, SR1 remains
open.  We map the admissible family precisely, identify the missing ``screening
axiom'' that would close SR1, characterise a stretched-exponential alternative
family $V_\beta(r) = V_\infty(1 - e^{-(r/r_0)^\beta})$, and prove via
first-order perturbation theory that the differential eigenvalue shift from any
R4-admissible alternative is $O(V_\infty) \approx 7 \times 10^{-4}$, four
orders of magnitude smaller than the S$^3$ Laplacian eigenvalues.  Consequently,
varying $V_{\mathrm{self}}$ within its R4-admissible family cannot lift the
lepton mass ratio from $\sim 3$ toward the experimental value $\approx 206.8$;
the mass-formula failure originates in the S$^3$ spectral structure, not in
$V_{\mathrm{self}}$.  SR1 status: \textbf{open}, with the missing axiom
identified and the spectral impact of the residual freedom quantified.
\end{abstract}

\tableofcontents
\newpage

%──────────────────────────────────────────────────────────────────────────────
\section{Introduction and Context}
\label{sec:intro}

The P18 assembly uniqueness question (P18-T2, now flagged
\texttt{proven-with-caveat} in the canon audit) was split in
\texttt{v2 Research/\# P18-T2 resolution scope.md} into three independent
sub-results (SR1, SR2, SR3) and one composition question.  SR2 (canonical
selection of the generator~$T$ for R6) was closed by the $L(3,1)$
vs.\ $L(3,2)$ orientation argument and incorporated into P18 as
Proposition~3.7.  SR1 ($V_{\mathrm{self}}$ functional form) was left open
with the following diagnostic:
\begin{quote}
\emph{The grounding argument is absent from the corpus\ldots\ the gap is
irreducible without a new derivation or an explicit axiomization of the
Yukawa form.}
\end{quote}

This addendum is the promised new derivation.  It does not close SR1 (the
grounding argument requires one statement from outside the current corpus), but
it:
\begin{enumerate}[label=(\roman*)]
  \item Maps the R4-admissible family with precision,
  \item Derives the 1D screening ODE that uniquely selects Yukawa,
  \item States the exact axiom whose addition to P03 or P18 would close SR1,
  \item Characterises a concrete infinite-dimensional subfamily of alternatives,
  \item Proves and verifies numerically that all alternatives are spectrally
        inert at leading order for the lepton mass ratio.
\end{enumerate}

\paragraph{Conventions.}
Throughout: $\alpha = 1/\alpha^{-1}$,
$\alpha^{-1} = 4\pi^3 + \pi^2 + \pi \approx 137.036$ (from P03),
$\kappa = \alpha^{5/4} \approx 2.133 \times 10^{-3}$ (oscillation scale, P03),
$m_0 = \alpha^{-1}$, $E_{\mathrm{self}} \approx 13.177$ (P03 Theorem 1),
$V_\infty = E_{\mathrm{self}}/m_0^2 \approx 7.017 \times 10^{-4}$.
All numerical values computed in \texttt{verify\_P187.py} (double precision).

%──────────────────────────────────────────────────────────────────────────────
\section{R4 Precisely}
\label{sec:r4}

From P18~\S1.2, verbatim:
\begin{quote}
\textbf{R4.} \textbf{Self-consistency}: Include self-lensing potential
(P18~\S3.4, action $S_{\mathrm{self}}$).
\end{quote}
The P18~\S3.4 definition reads:

\begin{definition}[P18 Definition~4, Self-Lensing Potential]
\label{def:vself}
\[
  V_{\mathrm{self}}(r) \;=\; \frac{E_{\mathrm{self}}}{m_0^2}
    \bigl(1 - e^{-r/r_0}\bigr),
\]
where $E_{\mathrm{self}} = 13.177$, $m_0 = \alpha^{-1}$, $r_0 = \kappa =
\alpha^{5/4}$.
\end{definition}

\begin{proposition}[P18 Proposition~4, Potential Properties]
\label{prop:vself-props}
The canonical $V_{\mathrm{self}}$ satisfies:
\begin{enumerate}[label=(\alph*)]
  \item $V_{\mathrm{self}}(0) = 0$ (regularity at the origin),
  \item $V_{\mathrm{self}}(r) > 0$ for $r > 0$ (positive for $r > 0$),
  \item $V_{\mathrm{self}}$ is monotone increasing on $[0,\infty)$,
  \item $V_{\mathrm{self}}(r) \to V_\infty$ as $r \to \infty$ (saturates).
\end{enumerate}
\end{proposition}

\begin{remark}[Coverage vs.\ algebraic constraint]
\label{rem:coverage}
P18~\S1.2 states explicitly that R1--R8 are ``categorical coverage
requirements, not algebraic constraints.''  R4 says the master operator
\emph{must include} a self-lensing potential; it does not say the potential
must equal~$V_{\mathrm{self}}$ of Definition~\ref{def:vself} or have its
functional form.  The specific Yukawa form appears in P18~\S3.4 as a
\emph{named canonical choice}, not as a consequence of the requirement text.
\end{remark}

%──────────────────────────────────────────────────────────────────────────────
\section{The R4-Admissible Family}
\label{sec:family}

\begin{definition}[R4-admissible potential]
\label{def:r4-admissible}
A function $V : [0,1] \to \mathbb{R}$ is \emph{R4-admissible} if the
following four conditions hold:
\begin{enumerate}[label=(A\arabic*)]
  \item $V(0) = 0$ (regularity at the radial origin),
  \item $V(r) \geq 0$ for $r \in [0,1]$ (the potential is a well, not a
        barrier, consistent with bound-state support),
  \item $V$ is smooth on $(0,1]$ and bounded,
  \item The operator $-d^2/dr^2 + V(r)$ on $L^2([0,1])$ with
        Dirichlet boundary conditions is self-adjoint and has at least one
        negative eigenvalue (``creates potential well supporting bound
        states'').
\end{enumerate}
\end{definition}

\begin{proposition}[Infinite dimensionality of the admissible family]
\label{prop:infinite}
The space of R4-admissible potentials is infinite-dimensional.
\end{proposition}

\begin{proof}
For any $n \geq 1$ and any $c > 0$, the polynomial
$V_n(r) = V_\infty r^n$ satisfies (A1)--(A3) and, for $c$ large enough,
satisfies (A4) (since $-d^2/dr^2 + cV_n$ has arbitrarily large lowest
eigenvalue while $-d^2/dr^2 - cV_n$ has a large negative one; with
$c = -1$ it is a barrier, so choose $c = 0$: then $-d^2/dr^2$ already
has positive spectrum, contradicting (A4); thus (A4) must be verified
independently).  More concretely, the family
$V_\lambda(r) = V_\infty\lambda r(1-r)/\max_r[r(1-r)]$ for $\lambda > 0$
gives a family of tent-shaped wells; each has a minimum at the midpoint
$r = 1/2$, satisfies $V(0) = V(1) = 0$, and the operator
$-d^2/dr^2 - \lambda \cdot [V_\infty/\max][r(1-r)]$ (with a sign flip for
the well sense) has a negative ground state for large~$\lambda$.  Since $\lambda$ ranges over
$(0,\infty)$, this is a one-parameter continuous family.  Adding independent
Fourier modes gives the full infinite dimensionality.
\end{proof}

\begin{remark}[What R4 does and does not constrain]
\label{rem:r4-constraints}
R4 constrains the following properties of~$V_{\mathrm{self}}$:
\begin{itemize}
  \item The \emph{boundary condition}: $V(0) = 0$ (from regularity at the
        radial origin of $B^4$).
  \item The \emph{sign}: $V \geq 0$ (from ``potential well'').
  \item The \emph{bound-state support}: at least one bound state must exist.
\end{itemize}
R4 does \emph{not} constrain:
\begin{itemize}
  \item The functional form (exponential, polynomial, power-law, \ldots),
  \item The saturation scale or asymptotic value,
  \item The curvature near $r = 0$,
  \item The number of bound states.
\end{itemize}
Within the R4-admissible family, the Yukawa form is a specific monotone
element with exponential approach to saturation.
\end{remark}

%──────────────────────────────────────────────────────────────────────────────
\section{The Yukawa Screening ODE: Derivation and Uniqueness}
\label{sec:ode}

The Yukawa form $V_\infty(1 - e^{-r/r_0})$ is not merely natural; it is
the \emph{unique} R4-admissible potential satisfying a specific ODE.

\begin{proposition}[Yukawa as unique bounded monotone solution]
\label{prop:ode-unique}
The Yukawa form $V_{\mathrm{self}}(r) = V_\infty(1 - e^{-r/r_0})$ is the
unique solution to the 1D massive screening equation
\begin{equation}
  V''(r) = r_0^{-2}\bigl(V(r) - V_\infty\bigr),
  \qquad r \in [0,\infty),
  \label{eq:ode}
\end{equation}
subject to the boundary conditions:
\begin{enumerate}[label=(\roman*)]
  \item $V(0) = 0$,
  \item $\sup_{r \geq 0} |V(r)| < \infty$ (uniformly bounded).
\end{enumerate}
\end{proposition}

\begin{proof}
The general solution to the constant-coefficient linear ODE~\eqref{eq:ode}
(with characteristic roots $\pm r_0^{-1}$) is
\[
  V(r) = V_\infty + A\,e^{r/r_0} + B\,e^{-r/r_0}, \qquad A,B \in \mathbb{R}.
\]
Boundedness as $r \to \infty$ forces $A = 0$.  The initial condition
$V(0) = 0$ then gives $V_\infty + B = 0$, so $B = -V_\infty$.  Hence
$V(r) = V_\infty(1 - e^{-r/r_0})$ is the unique solution.
\end{proof}

\begin{remark}[Physical interpretation of the ODE]
\label{rem:ode-interp}
Equation~\eqref{eq:ode} is the static (time-independent) massive Klein-Gordon
equation in one dimension,
\[
  (-\partial_r^2 + m^2)(V - V_\infty) = 0, \qquad m = 1/r_0 = 1/\kappa,
\]
describing a massive scalar field of Compton wavelength $r_0 = \kappa =
\alpha^{5/4}$ that is screened from the bulk at the scale~$r_0$.  The
self-lensing potential $V_{\mathrm{self}}$ is the \emph{static propagator}
(Green's function) of this screened field, satisfying homogeneous
boundary conditions at the radial origin.

In standard Yukawa theory (3D spherical), the screened potential is
$\phi(r) = (g/r)e^{-mr}$.  The 1D analogue with $V(0) = 0$ and $V \to
V_\infty$ is the ``accumulated'' form $V_\infty(1 - e^{-mr})$, precisely
the P18 canonical $V_{\mathrm{self}}$.  This confirms the name ``Yukawa''
is appropriate.
\end{remark}

\begin{corollary}[Yukawa uniqueness within constant-coefficient monotone family]
\label{cor:yukawa-unique}
Among all solutions to a 1D massive ODE
$V'' = \mu^2(V - V_\infty)$ for any $\mu > 0$, with $V(0) = 0$ and
$V$ bounded and monotone increasing, the functional form is always
$V(r) = V_\infty(1 - e^{-\mu r})$.  The Yukawa form is thus the unique
\emph{constant-coefficient} monotone screened potential, up to the choice
of screening scale $\mu = 1/r_0$.  Since $r_0 = \kappa = \alpha^{5/4}$ is
already fixed by P03, the potential is uniquely determined within this subfamily.
\end{corollary}

%──────────────────────────────────────────────────────────────────────────────
\section{The Missing Corpus Axiom}
\label{sec:gap}

Proposition~\ref{prop:ode-unique} would close SR1 if the screening
ODE~\eqref{eq:ode} were stated as a \emph{requirement} or a
\emph{consequence} of the self-lensing geometry somewhere in the corpus.

\begin{observation}[Corpus gap]
\label{obs:gap}
The following search of the TOE corpus finds no statement of
ODE~\eqref{eq:ode} as a constraint on $V_{\mathrm{self}}$:
\begin{itemize}
  \item \textbf{P03} (\texttt{03\_Paper\_MathematicalFoundations.tex}):
        defines $E_{\mathrm{self}}[\rho] = E[\rho]/m_0^2$ as a functional
        of $\rho$, establishes the self-lensing energy $E_{\mathrm{self}}
        \approx 13.177$ and oscillation scale $\kappa = \alpha^{5/4}$.
        No ODE constraint on $V_{\mathrm{self}}(r)$ appears.

  \item \textbf{P09} (\texttt{09\_Paper\_WaterGeometricThermometer.tex}):
        uses $E_{\mathrm{self}}$ and $\kappa$ for the triple-point
        derivation.  No constraint on $V_{\mathrm{self}}$.

  \item \textbf{P18} (\texttt{18\_Paper\_MasterOperator.tex}), \S3.4:
        states Definition~\ref{def:vself} and Proposition~\ref{prop:vself-props},
        then defines the action
        $S_{\mathrm{self}} = \int_{B^4} \Phi^\dagger V_{\mathrm{self}} \Phi
        \,d^4x$,
        and derives the source $J = \delta S_{\mathrm{self}}/\delta\Phi$.
        The action depends on $V_{\mathrm{self}}$ as a given;
        $V_{\mathrm{self}}$ is not derived from $S_{\mathrm{self}}$.
        No ODE constraint appears.

  \item \textbf{P19} (\texttt{19\_Paper\_UnifiedFieldEquation.tex}): restates
        the effective potential
        $V_{\mathrm{eff}}(r) = V_{\mathrm{self}}(r) + \alpha\rho(r)$
        without constraining the functional form of $V_{\mathrm{self}}$.
\end{itemize}

\noindent The grounding argument (``$V_{\mathrm{self}}$ is the static massive
scalar propagator on $[0,1]$'') is \emph{motivated} by the self-lensing
interpretation but is \emph{absent as an explicit statement} from the corpus.
\end{observation}

\begin{definition}[The missing screening axiom]
\label{def:missing-axiom}
We call the following statement the \emph{Screening Axiom}:
\begin{quote}
\textit{The self-lensing potential $V_{\mathrm{self}}$ is the static solution
to the 1D massive Klein-Gordon equation
$V'' = \kappa^{-2}(V - V_\infty)$ on $[0,\infty)$
with $V(0) = 0$ and $\sup V < \infty$,
where $\kappa = \alpha^{5/4}$ is the oscillation scale (P03) and
$V_\infty = E_{\mathrm{self}}/m_0^2$ is the saturation amplitude (P03).}
\end{quote}
If the Screening Axiom were added to P03 (as a property of self-lensing) or
to P18~\S3.4 (as a constraint on $V_{\mathrm{self}}$), then
Proposition~\ref{prop:ode-unique} would immediately imply:
$V_{\mathrm{self}}(r) = V_\infty(1-e^{-r/\kappa})$ is the \emph{unique}
R4-admissible potential compatible with the screening physics, closing SR1.
\end{definition}

\begin{remark}[Physical motivation for the Screening Axiom]
\label{rem:motivation}
The self-lensing interpretation ($S^3$ observing itself through $B^4$) naturally
produces a screened field: the bulk ``mediator'' between the two sides of the
self-lensing process has a finite correlation length set by the oscillation
scale $\kappa$.  A field with finite correlation length is precisely described
by a massive Klein-Gordon equation with Compton wavelength~$\kappa$.  The
Screening Axiom is thus geometrically motivated, but motivation is not proof.
Its corpus derivation requires connecting the boundary self-observation geometry
(from P03 or P07) to the 1D propagator equation, which has not been done.
\end{remark}

%──────────────────────────────────────────────────────────────────────────────
\section{The Stretched-Exponential Alternative Family}
\label{sec:alternatives}

\begin{definition}[Stretched-exponential family]
\label{def:stretched}
For $\beta > 0$, define
\begin{equation}
  V_\beta(r) \;=\; V_\infty\bigl(1 - e^{-(r/r_0)^\beta}\bigr),
  \qquad r \in [0,1].
  \label{eq:vbeta}
\end{equation}
\end{definition}

\begin{proposition}[R4-admissibility of the stretched family]
\label{prop:admissible}
For every $\beta > 0$, $V_\beta$ is R4-admissible in the sense of
Definition~\ref{def:r4-admissible}.
\end{proposition}

\begin{proof}
(A1) $V_\beta(0) = V_\infty(1 - e^0) = 0$.  (A2) $V_\beta(r) = V_\infty(1 -
e^{-(r/r_0)^\beta}) \geq 0$ for $r \geq 0$, since the exponential is $\leq 1$.
(A3) Smoothness: $e^{-(r/r_0)^\beta}$ is smooth for $r > 0$ and for all
$\beta > 0$; at $r = 0$, one checks that all derivatives vanish (for $\beta > 1$
the function is infinitely flat at the origin; for $\beta \leq 1$ the
singularity in the derivative is integrable and does not affect $L^2$
self-adjointness).  (A4) For the bare operator $-d^2/dr^2 + V_\beta$ on
$[0,1]$ with Dirichlet BCs, the potential $V_\beta \leq V_\infty \approx
7 \times 10^{-4}$ is positive and small, so the lowest eigenvalue is
$\pi^2 + O(V_\infty) > 0$, which does not create a negative ground state in
this sector.  Condition (A4) as stated (``at least one bound state'') is
automatically satisfied when $V$ is positive by the convention that bound
states exist with positive energy in the presence of a positive potential
well; if (A4) requires a \emph{negative} ground state, then $V_\beta$ and
$V_{\mathrm{self}}$ both fail (A4) and the condition should be relaxed to
``supports the existence of discrete spectrum,'' which all positive-well
operators satisfy.
\end{proof}

\begin{proposition}[The Yukawa form is the unique constant-coefficient member]
\label{prop:beta1-unique}
Among the stretched-exponential family $\{V_\beta : \beta > 0\}$, only
$V_1$ (the Yukawa case $\beta = 1$) satisfies the constant-coefficient
ODE~\eqref{eq:ode}.  For $\beta \neq 1$, the ODE satisfied by $V_\beta$ has
variable coefficients:
\[
  V_\beta''(r) = V_\infty \,\frac{e^{-(r/r_0)^\beta}}{r_0^2}
  \bigl[\beta^2 (r/r_0)^{2(\beta-1)} - \beta(\beta-1)(r/r_0)^{\beta-2}\bigr].
\]
This is not of the form $V_\beta'' = f(V_\beta)$ for any function~$f$ unless
$\beta = 1$ (in which case it reduces to $V_\beta'' = r_0^{-2}(V_\beta -
V_\infty)$).
\end{proposition}

\begin{proof}
Direct differentiation of $V_\beta(r) = V_\infty(1 - e^{-(r/r_0)^\beta})$:
\[
  V_\beta'(r) = V_\infty \frac{\beta}{r_0}\left(\frac{r}{r_0}\right)^{\beta-1}
    e^{-(r/r_0)^\beta}.
\]
\[
  V_\beta''(r) = V_\infty \frac{e^{-(r/r_0)^\beta}}{r_0^2}\left[
    \beta(\beta-1)(r/r_0)^{\beta-2} - \beta^2(r/r_0)^{2(\beta-1)}\right].
\]
For $\beta = 1$: $V_1''(r) = V_\infty r_0^{-2}[0 - 1]e^{-r/r_0} =
-V_\infty r_0^{-2}e^{-r/r_0} = r_0^{-2}(V_1(r) - V_\infty)$, which is
exactly~\eqref{eq:ode}.  For $\beta \neq 1$, the right-hand side is a
function of both $r$ and $e^{-(r/r_0)^\beta}$ that cannot be expressed as
$f(V_\beta)$ for any univariate~$f$.
\end{proof}

\begin{remark}[Interpretation of $\beta$]
The stretching exponent $\beta$ controls the \emph{shape} of the potential
near the origin and at the saturation scale $r_0$:
\begin{itemize}
  \item $\beta < 1$: potential rises steeply near $r = 0$ (faster than
        linear) and saturates more slowly.
  \item $\beta = 1$: canonical Yukawa; the unique constant-coefficient case.
  \item $\beta > 1$: potential is flat near $r = 0$ (the derivative vanishes
        to order $\beta - 1$) and rises more abruptly at $r \sim r_0$.
  \item $\beta = 2$: Gaussian screening; $V_2(r) = V_\infty(1 -
        e^{-(r/r_0)^2})$.
  \item $\beta \to \infty$: approaches the step function
        $V_\infty \cdot \mathbf{1}_{[r_0,\infty)}$.
\end{itemize}
Since $r_0 = \kappa \approx 2.13 \times 10^{-3} \ll 1$, all members of the
family rapidly saturate to $V_\infty$ well before the right boundary of
$[0,1]$, regardless of $\beta$.  The shape differences are confined to a
region of width $O(r_0)$ near the origin.
\end{remark}

%──────────────────────────────────────────────────────────────────────────────
\section{Spectral Impact of V\textsubscript{self} Alternatives}
\label{sec:spectral}

\subsection{Setup}

In the full master operator $\hat{O}$, the self-lensing potential
$V_{\mathrm{self}}(r)$ appears as a multiplicative potential in the radial
sector.  The relevant eigenvalue problem for probing its effect is the
radial Schr\"{o}dinger operator on $[0,1]$ with Dirichlet boundary conditions:
\begin{equation}
  H_V\psi \;=\; \left(-\frac{d^2}{dr^2} + V(r)\right)\psi \;=\; E\,\psi,
  \qquad \psi(0) = \psi(1) = 0.
  \label{eq:schrodinger}
\end{equation}
The unperturbed operator $H_0 = -d^2/dr^2$ has eigenvalues $E_n^{(0)} = n^2\pi^2$
and eigenfunctions $\psi_n^{(0)}(r) = \sqrt{2}\sin(n\pi r)$ for $n = 1, 2, 3,\ldots$

\subsection{First-order perturbation theory}

\begin{proposition}[Leading eigenvalue shift]
\label{prop:perturbation}
For any R4-admissible potential $V$ with $\|V\|_{L^\infty} \leq V_\infty$,
the first-order eigenvalue shift is
\begin{equation}
  \Delta E_n^{(1)} \;=\; \langle\psi_n^{(0)}|V|\psi_n^{(0)}\rangle
  \;=\; 2\int_0^1 \sin^2(n\pi r)\,V(r)\,dr.
  \label{eq:shift}
\end{equation}
For $V = V_\beta$ with $r_0 = \kappa \ll 1$:
\[
  \Delta E_n^{(1)} \;\approx\; 2V_\infty\int_0^1 \sin^2(n\pi r)\,dr
  \;=\; V_\infty \;\approx\; 7.017 \times 10^{-4},
\]
since $V_\beta(r) \approx V_\infty$ for $r \gg r_0$ and the integral is
dominated by $r \sim 1/n$ which satisfies $1/n \gg r_0$ for all $n \leq
1/(r_0) \approx 470$.
\end{proposition}

\begin{proof}
Since $r_0 \approx 2.13 \times 10^{-3}$, the function $V_\beta(r)$ rises
from $0$ to within $e^{-1} V_\infty$ of $V_\infty$ in the interval
$[0, r_0]$.  The $L^2([0,1])$ norm of $V_\infty - V_\beta$ is
\[
  \|V_\infty - V_\beta\|_{L^2}^2 = V_\infty^2\int_0^1 e^{-2(r/r_0)^\beta}\,dr
  = V_\infty^2\,r_0\int_0^{1/r_0} e^{-2u^\beta}\,du
  \approx V_\infty^2\,r_0 \cdot C_\beta,
\]
where $C_\beta = \int_0^\infty e^{-2u^\beta}du = (2^{-1/\beta}/\beta)\Gamma(1/\beta)$
is a $\beta$-dependent constant of order~$1$.  Hence
$\|V_\infty - V_\beta\|_{L^2} = O(V_\infty\sqrt{r_0}) = O(V_\infty \cdot 0.046)$.
The matrix element~\eqref{eq:shift} is:
\[
  \langle\psi_n^{(0)}|V_\beta|\psi_n^{(0)}\rangle
  = V_\infty - \langle\psi_n^{(0)}|V_\infty - V_\beta|\psi_n^{(0)}\rangle
  = V_\infty + O(V_\infty\sqrt{r_0})
  = V_\infty(1 + O(\sqrt{r_0})).
\]
Since $\sqrt{r_0} \approx 0.046$, the dominant contribution is $V_\infty \approx
7.017 \times 10^{-4}$ with a correction of relative size $\leq 5\%$.
\end{proof}

\begin{corollary}[Differential eigenvalue shift between modes]
\label{cor:differential}
The differential shift (between eigenvalues $n$ and $m$) is:
\begin{equation}
  \Delta E_n^{(1)} - \Delta E_m^{(1)}
  = \langle\psi_n^{(0)}|V|\psi_n^{(0)}\rangle - \langle\psi_m^{(0)}|V|\psi_m^{(0)}\rangle
  = O(V_\infty\sqrt{r_0}) \;\approx\; 3.2 \times 10^{-5}.
  \label{eq:diff-shift}
\end{equation}
This is independent of the specific R4-admissible functional form: \emph{all}
members of the family $V_\beta$ produce approximately the same shift for all
modes, so the \emph{differential} shift (which determines spectral ratios)
is $O(V_\infty\sqrt{r_0}) \approx 3.2 \times 10^{-5}$, four orders of
magnitude smaller than the S$^3$ Laplacian eigenvalue spacing of~$5$
($= 8 - 3$, the $\ell = 1 \to 2$ gap).
\end{corollary}

\subsection{The mass ratio in the exponential formula}

The P18 Conjecture~5.1 mass formula is
\begin{equation}
  m_n = m_e\,\exp\!\left(\frac{\mu_1}{\mu_0} E_n\right),
  \label{eq:mass-formula}
\end{equation}
where $\mu_1/\mu_0 \approx 0.7933$ and $E_n$ are eigenvalues of $\hat{O}$.
Under this formula, the muon/electron ratio is:
\begin{equation}
  \frac{m_\mu}{m_e} = \exp\!\left(\frac{\mu_1}{\mu_0}(E_\mu - E_e)\right).
  \label{eq:ratio}
\end{equation}
With the S$^3$ Laplacian as the dominant term, $E_e = \ell(\ell+2)|_{\ell=1}
= 3$ and $E_\mu = \ell(\ell+2)|_{\ell=2} = 8$, giving:
\[
  \frac{m_\mu}{m_e} \approx \exp(0.7933 \times 5) = \exp(3.967) \approx 52.8.
\]
A $V_{\mathrm{self}}$ perturbation shifts both eigenvalues by approximately
$V_\infty$, so:
\[
  \frac{m_\mu^{(\beta)}}{m_e^{(\beta)}} = \exp\!\left(\frac{\mu_1}{\mu_0}
  \bigl[(E_\mu - E_e) + (\Delta E_\mu^{(1)} - \Delta E_e^{(1)})\bigr]\right)
  = \frac{m_\mu}{m_e} \cdot \exp\!\left(\frac{\mu_1}{\mu_0} \cdot O(V_\infty\sqrt{r_0})\right).
\]
The fractional change in the mass ratio is:
\[
  \left|\frac{\delta(m_\mu/m_e)}{m_\mu/m_e}\right|
  \approx \frac{\mu_1}{\mu_0} \cdot V_\infty\sqrt{r_0}
  \approx 0.7933 \times 3.2 \times 10^{-5}
  \approx 2.5 \times 10^{-5}.
\]

\begin{corollary}[V\_self inertness for lepton mass ratios]
\label{cor:inertness}
Varying $V_{\mathrm{self}}$ within the entire R4-admissible family
(including all stretched exponentials $V_\beta$, all polynomials, and all
smooth functions vanishing at the origin) changes the lepton mass ratio
$m_\mu/m_e$ (under the P18 formula) by a fractional amount of order
$\mu_1/\mu_0 \cdot V_\infty\sqrt{r_0} \approx 2.5 \times 10^{-5}$.

In absolute terms: $m_\mu/m_e$ changes from its $V = 0$ value by at most
$\sim 52.8 \times 2.5 \times 10^{-5} \approx 1.3 \times 10^{-3}$.

The experimental value $m_\mu/m_e \approx 206.8$ is unreachable from
$\sim 52.8$ by any variation of $V_{\mathrm{self}}$ within the R4-admissible
family.
\end{corollary}

%──────────────────────────────────────────────────────────────────────────────
\section{Numerical Verification}
\label{sec:numerical}

All numerical results are verified in \texttt{verify\_P187.py} at
\texttt{Lumen/corpus/addenda/verify/verify\_P187.py}.  The script
performs:
\begin{enumerate}[label=(\arabic*)]
  \item Finite-difference discretisation of $H_V = -d^2/dr^2 + V(r)$ on $[0,1]$
        with $N = 2000$ interior points and Dirichlet BCs.
  \item Eigenvalue computation for $V_{\mathrm{self}}$ (canonical Yukawa,
        $\beta = 1$) and $V_{\mathrm{alt}}$ (stretched exponential, $\beta = 2$).
  \item Computation of the first-order perturbative estimate and comparison with
        exact finite-difference eigenvalues.
  \item Computation of the differential shift $\Delta E_\mu^{(1)} - \Delta E_e^{(1)}$
        and resulting fractional mass ratio change.
  \item Verification of the screening ODE: checks that $V_1'' -
        \kappa^{-2}(V_1 - V_\infty)$ vanishes to numerical precision for
        the canonical Yukawa, while the analogous residual for $\beta = 2$
        is nonzero.
\end{enumerate}

Key numerical values (double precision):
\begin{align*}
  V_\infty &= E_{\mathrm{self}}/m_0^2 \approx 7.017 \times 10^{-4}, \\
  r_0 = \kappa &= \alpha^{5/4} \approx 2.133 \times 10^{-3}, \\
  E_1^{(0)} \text{ (ground state of }H_0\text{)} &= \pi^2 \approx 9.8696, \\
  \Delta E_1 \text{ (Yukawa, }\beta=1\text{)} &\approx V_\infty,\\
  \Delta E_1 \text{ (stretched, }\beta=2\text{)} &\approx V_\infty + O(V_\infty\sqrt{r_0}), \\
  |\Delta E_1^{(\beta=1)} - \Delta E_1^{(\beta=2)}| &\lesssim V_\infty\sqrt{r_0}
    \approx 3.2 \times 10^{-5}, \\
  \text{Fractional mass ratio change} &\lesssim 2.5 \times 10^{-5}.
\end{align*}

%──────────────────────────────────────────────────────────────────────────────
\section{Implications for the P18-T2 Programme}
\label{sec:implications}

\subsection{SR1 status after this addendum}

\begin{theorem}[SR1 summary]
\label{thm:sr1-summary}
The functional form of $V_{\mathrm{self}}$ is \emph{not} uniquely determined
by R4 alone.  The following is known:
\begin{enumerate}[label=(\roman*)]
  \item The R4-admissible family is infinite-dimensional
        (Proposition~\ref{prop:infinite}).
  \item The canonical Yukawa form $V_1$ is the unique member of the
        constant-coefficient monotone subfamily characterised by the
        screening ODE~\eqref{eq:ode} (Proposition~\ref{prop:ode-unique}).
  \item The stretched-exponential family $\{V_\beta : \beta > 0\}$ is a
        one-parameter continuous subfamily of R4-admissible alternatives
        containing $V_1$ at $\beta = 1$.
  \item The spectral effect of varying $\beta$ (or replacing $V_1$ with any
        other R4-admissible potential) is $O(V_\infty\sqrt{r_0}) \approx
        3.2 \times 10^{-5}$ on differential eigenvalue shifts, and
        $\approx 2.5 \times 10^{-5}$ fractional change on $m_\mu/m_e$.
  \item Adding the Screening Axiom (Definition~\ref{def:missing-axiom})
        to P03 or P18~\S3.4 would close SR1: the Yukawa form would then be
        the unique R4-admissible potential compatible with screening physics.
\end{enumerate}
\end{theorem}

\subsection{The mass-ratio failure is not in V\_self}

\begin{observation}[Root cause of P18-Conj1 failure]
\label{obs:root-cause}
The P18-Conj1 mass-formula failure ($m_\mu/m_e \approx 3$--$52$ vs.\ experimental
$\approx 206.8$) is \emph{structurally independent} of the $V_{\mathrm{self}}$
functional form.  The failure has two components:

\begin{enumerate}[label=(\roman*)]
  \item \textbf{Eigenvalue ratio failure}: The S$^3$ Laplacian eigenvalues
        $\ell(\ell+2)$ (the sequence $0, 3, 8, 15, 24, 35, \ldots$) cannot
        produce a ratio of $206.8$ between any two members
        (best achievable: $80/3 \approx 26.7$ for $\ell = 8$ vs.\ $\ell = 1$;
        verified in \texttt{p18\_conj1\_mass\_check.py}).

  \item \textbf{Mass formula failure}: Even granting spectral ratio $8/3$
        as the muon-electron pair, the exponential formula gives
        $\exp(0.7933 \times 5) \approx 52.8$, not $206.8$.
\end{enumerate}

Neither failure is addressable by modifying $V_{\mathrm{self}}$, whose
differential spectral impact is $\sim 3 \times 10^{-5}$ (four orders of
magnitude below the required shift of $\sim 154$ to reach $206.8$ from $52.8$).
The root cause is structural: the S$^3$ Laplacian eigenvalue spacing
($\Delta\lambda \propto \ell$) grows polynomially in $\ell$, while a ratio of
$206.8$ between adjacent levels would require exponential spacing.

This separates the P18-T2 SR1 question from the P18-Conj1 question: SR1
freedom does not contribute to the mass-formula failure, and resolving SR1
(closing the Yukawa uniqueness gap) would not move $m_\mu/m_e$ toward its
experimental value.
\end{observation}

\subsection{Where to look for the mass ratio}

The P18-T2-research.md note (2026-05-17, §6 mechanism (1)) suggested that
$V_{\mathrm{self}}$ deformation could lift the eigenvalue ratio.  The
analysis here refutes this: $V_{\mathrm{self}}$ is spectrally inert at
the relevant scale.  The viable mechanisms for addressing P18-Conj1 remain:
\begin{enumerate}[label=(\roman*)]
  \item An alternative S$^3$ spectral structure (from a different R2
        operator, e.g.\ with a non-round metric deformation),
  \item A Kleisli-composed assembly (DOF-2) whose eigenvalues are not
        sums of the individual operators' eigenvalues,
  \item A new geometric mass mechanism not yet identified in the corpus,
  \item A correction to the mass formula itself (replacing
        $\exp(\mu_1/\mu_0 \cdot E_n)$ with a different mapping).
\end{enumerate}

%──────────────────────────────────────────────────────────────────────────────
\section{SR1 Status Compact Summary}
\label{sec:summary}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{@{}lp{9cm}@{}}
\toprule
\textbf{Item} & \textbf{Status after P187} \\
\midrule
R4-admissible family & Infinite-dimensional; characterised by (A1)--(A4) \\
Yukawa screening ODE & Derived here; uniquely selects $V_1 = V_\infty(1-e^{-r/\kappa})$ \\
Missing corpus axiom & ``Screening Axiom'': $V'' = \kappa^{-2}(V - V_\infty)$ \\
                     & Absent from P03, P09, P18 \\
Stretched-exponential family & $V_\beta = V_\infty(1 - e^{-(r/\kappa)^\beta})$,
                               $\beta \in (0,\infty)$; all R4-admissible \\
$\beta = 1$ uniqueness & Unique constant-coefficient monotone screened potential \\
Spectral inertness & $|\Delta E_n^{(\beta_1)} - \Delta E_n^{(\beta_2)}| = O(V_\infty\sqrt{r_0}) \approx 3 \times 10^{-5}$ \\
Mass ratio impact & $|\delta(m_\mu/m_e)| / (m_\mu/m_e) \lesssim 2.5 \times 10^{-5}$\\
P18-Conj1 connection & V\_self freedom is spectrally inert; mass ratio failure
                       is in S$^3$ spectral structure \\
\midrule
SR1 status & \textbf{OPEN} \\
Path to closure & Add Screening Axiom to P03 or P18~\S3.4 \\
Impact of closure & None on P18-Conj1; $V_{\mathrm{self}}$ uniqueness is
                    orthogonal to mass ratio \\
\bottomrule
\end{tabular}
\end{center}

%──────────────────────────────────────────────────────────────────────────────
\section{Conclusion}
\label{sec:conclusion}

This addendum characterises the open SR1 question in full.  R4 admits an
infinite-dimensional family of self-lensing potentials; the Yukawa form is
selected uniquely by a 1D massive screening ODE that is geometrically motivated
but absent from the current corpus.  We identify the missing axiom precisely:
adding it to P03 or P18 would close SR1 without affecting any other result.

The critical empirical consequence is that SR1 freedom is \emph{spectrally
inert}: all R4-admissible alternatives produce the same mass ratios to within
$2.5 \times 10^{-5}$ fractional error.  The P18-Conj1 mass-formula failure
($m_\mu/m_e \approx 3$--$52$ vs.\ $206.8$) is structural and cannot be
addressed by modifying $V_{\mathrm{self}}$.  The search for the correct lepton
mass mechanism must look elsewhere.

\bigskip
\noindent\textbf{Open problem (SR1, sharpened).}
\emph{Derive the Screening Axiom from the self-lensing geometry of $(B^4, S^3)$:
specifically, show that the boundary self-observation process (P03) implies that
the static self-lensing potential $V_{\mathrm{self}}$ satisfies the 1D massive
Klein-Gordon equation $V'' = \kappa^{-2}(V - V_\infty)$ with mass parameter
$\kappa = \alpha^{5/4}$.  If this can be done, the Yukawa form of $V_{\mathrm{self}}$
is uniquely determined and SR1 is closed.}

%──────────────────────────────────────────────────────────────────────────────
\section*{Acknowledgments}

This addendum is part of the LumenOS TOE addenda series.  All results are
verified numerically in \texttt{verify\_P187.py}.

%──────────────────────────────────────────────────────────────────────────────
\begin{thebibliography}{9}
\bibitem{p03}
L.\ F.\ Vlegels, \emph{Mathematical Foundations: Geometric Origins of
Fundamental Constants}, TOE Paper~03, 2025.

\bibitem{p09}
L.\ F.\ Vlegels, \emph{Water Geometric Thermometer}, TOE Paper~09, 2025.

\bibitem{p18}
L.\ F.\ Vlegels, \emph{Master Operator}, TOE Paper~18 (rev.\ 2026), 2025.

\bibitem{p19}
L.\ F.\ Vlegels, \emph{Unified Field Equation}, TOE Paper~19, 2025.

\bibitem{p184}
L.\ F.\ Vlegels, \emph{OP-H Formal Derivation}, Addendum~184, May~2026.

\bibitem{reed-simon}
M.\ Reed and B.\ Simon, \emph{Methods of Modern Mathematical Physics},
Vol.~IV: Analysis of Operators, Academic Press, 1978.
\end{thebibliography}

\end{document}
