% Addendum 218b — Amended Closure Statement for the P18-T2 Uniqueness Programme
% Upgrades Theorem P218.1 from conditional to unconditional within the TOE.
% Author: Léon Fernando Vlegels
% © Léon Fernando Vlegels. CC BY 4.0.

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\newtheorem{theorem}{Theorem}[section]
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%─── notation (identical to P218) ─────────────────────────────────────────────
\newcommand{\Ee}{e}
\newcommand{\Om}{\omega}
\newcommand{\aI}{\mu}
\newcommand{\hv}{h}
\newcommand{\Emu}{E_\mu}
\newcommand{\Sinf}{S_\infty}
\newcommand{\Rinf}{R_\infty}
\newcommand{\rr}{r}
\newcommand{\ppm}{\,\text{ppm}}
\newcommand{\Oop}{\hat{O}}
\newcommand{\codata}{206.768\,283\,0}
\newcommand{\Bezout}{\textsc{B\'{e}zout}}

\title{\textbf{Addendum P218b}\\[4pt]
Amended Closure Statement for the P18-T2 Uniqueness Programme:\\
\emph{Unconditional Establishment Within the TOE}\\[2pt]
{\normalsize Amending Addendum P218 in Light of P219--P222}}
\author{L\'eon Fernando Vlegels}
\date{18 May 2026}

\begin{document}
\maketitle
\tableofcontents
\bigskip

\noindent\textbf{Abstract.}
Addendum P218 established the P18-T2 uniqueness programme
\emph{conditionally}, subject to Axiom~ID (Root--Lepton Energy
Identification).  Addenda P219--P222 have since resolved that condition
entirely within the TOE's own axioms.  Specifically, P222 derives
$\mu = 4\pi^3+\pi^2+\pi$ from the Hopf fibration structure implicit in
R1--R3, via the \Bezout{} Frame Axiom (Paper~31, Theorem~6.3); this makes
Paper~00 Axiom~G4 a redundant theorem rather than an independent postulate,
and makes Axiom~ID itself a geometric consequence of R1--R8.
This addendum amends P218 accordingly.  Result P6 is upgraded from
\textsc{conjectured} to \textsc{derived} (conditional only on
$\alpha=\Omega^{-1}$, the TOE's own foundational identification).
\textbf{Theorem P218b.1} states that within the TOE's axiom system,
including the geometric identification $\alpha = \Omega^{-1}$, the
P18-T2 uniqueness programme is \textsc{established unconditionally}.
Three open problems from P218 are collapsed to final status; one new
foundational question is stated.

\bigskip\noindent\emph{This is a targeted amendment.  All notation,
definitions, and results of P218 remain in force; this document overrides
only the items listed here.}

%─────────────────────────────────────────────────────────────────────────────
\section{Post-P218 Developments}
\label{sec:post218}
%─────────────────────────────────────────────────────────────────────────────

The following addenda were issued after P218's formal closure date of
18 May 2026.

\smallskip
\begin{center}
\begin{tabular}{@{}p{1.3cm}p{7.8cm}p{2.8cm}@{}}
\toprule
Doc & Central result & Status \\
\midrule
P219 &
Axiom~ID derivable from monad closure polynomial
$4\Ee^3+\Ee^2+\Ee=\aI$; $\Ee=\pi$ is the unique positive real root;
reversed root assignment ruled out by $7\times10^6\,\sigma$. &
RESOLVED \\[4pt]
P220 &
OP-rSinf-100: $\rr\cdot\Sinf \approx c/100$ with
$\varepsilon \approx -2.20\times10^{-7}$; deviation is
$\approx 0.048\,\sigma_{\rm CODATA}$ and therefore experimentally
unfalsifiable at current precision. &
CONJECTURED / UNFALSIFIABLE \\[4pt]
P221 &
OP-G2A2-lattice: resolvent $G(\lambda^*) = \Sinf$ exactly at
$\lambda^*\approx 4255.7$; $\lambda^*$ is derivable from TOE constants
but is not itself a TOE primitive.  Problem closed (Theorem P221.1,
Outcome~B). &
CLOSED \\[4pt]
P222 &
$\mu = 4\pi^3+\pi^2+\pi$ is derived from R1--R3 via the \Bezout{}
Frame Axiom (Paper~31 Thm.~6.3) with derivation chain
$\text{R1--R3} \to \mathbb{Z}_k\text{-monodromy} \to T_k(t)=0
\to c_d = (d{+}1)^{\max(d-1,1)} \to \rho \to \Omega = \mu$.
Axiom~G4 of Paper~00 becomes a redundant theorem. &
DERIVED \\
\bottomrule
\end{tabular}
\end{center}

\subsection{The Bézout derivation chain (P222 summary)}

Paper~31 Theorem~6.3 establishes that the self-intersection configuration
counts of $S^3 \hookrightarrow B^4$ are:
\[
  c_d \;=\; (d+1)^{\max(d-1,1)},\quad d\in\{1,2,3\}
  \quad\Longrightarrow\quad c_1 = 2,\quad c_2 = 3,\quad c_3 = 16.
\]
These are \emph{Bézout intersection numbers}: exactly $c_d$ solutions,
all real, to a system of Chebyshev equations in the Hopf modulus
coordinates. The $\mathbb{Z}_k$ monodromy required for totality is
established corpus-internally from R1, R2, R3, and R6.  The
density polynomial $\rho(x) = c_3\pi^3 x^3 + c_2\pi^2 x^2 + c_1\pi x
= 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ integrates to:
\[
  \Omega \;=\; \int_0^1 \rho(x)\,dx
  \;=\; 4\pi^3 + \pi^2 + \pi \;=\; \mu.
\]
Hence $\mu$ is a geometric consequence of R1--R3 (via Paper~31), not an
independent normalisation axiom.

%─────────────────────────────────────────────────────────────────────────────
\section{Upgraded Programme Status Table}
\label{sec:table}
%─────────────────────────────────────────────────────────────────────────────

The following table supersedes the programme summary of P218 \S4.  Rows
P1--P5, P7 are unchanged; P6 is upgraded.

\bigskip
\begin{longtable}{@{}p{1.1cm}p{7.6cm}p{1.7cm}p{2.2cm}@{}}
\toprule
Label & Statement & Source & Status \\
\midrule
\endfirsthead
\toprule Label & Statement & Source & Status \\ \midrule
\endhead
\bottomrule \endfoot

P1 &
Prefactor $207 = \dim(G_2)\cdot(\dim(G_2){+}1)-h^\vee(A_2) = 14{\cdot}15-3$.
&P211 & \textsc{proven} \\[5pt]

P2 &
Correction numerator: first correction has numerator $\Om-h^\vee(A_2)$.
&P210 & \textsc{proven} \\[5pt]

P3 &
$h^\vee(A_2)=3$ is unique; next-best integer gives $208\times$ worse gap.
&P210 & \textsc{proven} \\[5pt]

P4 &
$k$-independence of $\rr$: $T_{k+1}/T_k = -\rr$ for all $k\ge1$.
&P214.1 & \textsc{proven} \\[5pt]

P5 &
Series factorisation $T_k = (-\rr)^k\cdot\sigma$, $\sigma=\aI/\Ee^5$.
&P215 Lem.~2.1 & \textsc{proven} \\[5pt]

P6 &
\textbf{[Upgraded.]}\ Exponent increment $(\Delta n_\Ee, \Delta n_\aI)=(+2,-2)$ per
$k$-step accounts for the $G_2$ root-length ratio $|\alpha_s|^2/|\alpha_\ell|^2
= 1/h^\vee$ and the Coxeter factor.  Axiom~ID, which assigns
$|\alpha_s|^2\leftrightarrow\Emu$ and $|\alpha_\ell|^2\leftrightarrow\aI^2$,
is itself derived from the monad closure polynomial (P219) whose
coefficients are fixed by R1--R3 via the \Bezout{} Frame Axiom (P222).
\emph{Status conditional only on $\alpha=\Omega^{-1}$}, which is the
TOE's own foundational identification. &
P217.1 + P219 + P222 &
\textsc{derived}
(cond.\ $\alpha{=}\Omega^{-1}$) \\[5pt]

P7 &
CODATA agreement: $\Rinf = 206.768\,285\,300\ldots$, gap $\delta_\infty/\sigma=0.500$.
&P216 & \textsc{verified} \\

\end{longtable}

%─────────────────────────────────────────────────────────────────────────────
\section{Theorem P218b.1}
\label{sec:main}
%─────────────────────────────────────────────────────────────────────────────

\begin{theorem}[P218b.1 — Unconditional P18-T2 Closure Within the TOE]
\label{thm:218b}
Within the axiom system of the TOE, including the identification
$\alpha = \Omega^{-1}$ where
$\Omega = 4\pi^3+\pi^2+\pi$ is derived geometrically from
R1--R3 via \textnormal{Paper~31 Theorem~6.3}, the P18-T2 uniqueness
programme is \textsc{established unconditionally}.
More precisely:
\begin{enumerate}
\item[\textnormal{(i)}]
The coverage requirements R1--R8 of Paper~18, together with
Paper~31's \Bezout{} Frame Axiom, imply $\mu = 4\pi^3+\pi^2+\pi$
as a geometric theorem rather than a postulate \textnormal{(P222)}.
\item[\textnormal{(ii)}]
The monad closure polynomial $4x^3+x^2+x=\mu$ has $x=\pi$ as its
unique positive real root; the electron-sector energy $\Ee=\pi$ is
therefore derived from the monad, and Axiom~ID is a geometric
consequence of R1--R8 within the TOE \textnormal{(P219)}.
\item[\textnormal{(iii)}]
All seven programme results P1--P7 hold with P6 upgraded from
\textsc{conjectured} to \textsc{derived} \textnormal{(\S\ref{sec:table})}.
\item[\textnormal{(iv)}]
The unique zero-free-parameter formula
\[
  \Rinf \;=\; 207\!\left(1 - \frac{\Om}{\Ee^3\aI}
  + \frac{\hv\aI}{\Ee^3(\aI^2+\hv\Emu)}\right)
\]
satisfies $|\Rinf - T_{\rm CODATA}|/\sigma = 0.500 < 1$.
\item[\textnormal{(v)}]
Paper~00 Axiom~G4 ($M[\rho]=\mu$ as normalization postulate) is a
redundant theorem within the TOE: it follows from P222 without
independent input.
\end{enumerate}
The single residual foundational question --- whether $\alpha$ equals
$\Omega^{-1}$ exactly or only to measured precision --- is an open problem of
the TOE itself, not of the P18-T2 programme
\textnormal{(see \S\ref{sec:residual})}.
\end{theorem}

\begin{proof}[Proof sketch]
Part~(i) is the main result of P222.  The derivation chain is:
$\text{R1--R3}\xrightarrow{\mathbb{Z}_k}T_k(t){=}0
\xrightarrow{\text{B\'ezout}}c_d
\xrightarrow{\rho}\Omega=\mu$.
The $\mathbb{Z}_k$ monodromy values ($k=2,3,4$) are established
corpus-internally; the Chebyshev bound is achieved exactly.

Part~(ii) is P219 Proposition~2.1.  The quadratic factor $4x^2+x+1$
has discriminant $\Delta = 1-16 = -15 < 0$ (no real roots), so
$f(x) = x(4x^2+x+1)$ is strictly increasing on $(0,\infty)$ and
$f(x)=\mu$ has a unique positive solution $x=\pi$.

Parts~(iii)--(iv) follow from P218 with P6 upgraded as above.
Part~(v) is immediate from (i): if $\Omega=\mu$ is proved geometrically,
the normalisation $M[\rho]=\mu$ is no longer an axiom.
\end{proof}

\begin{corollary}
R1--R8 \emph{together with} the TOE identification $\alpha=\Omega^{-1}$
admit a unique admissible assembly of $\Oop$, resolving Paper~18 \S4.4.
\end{corollary}

%─────────────────────────────────────────────────────────────────────────────
\section{Open Problems: Collapsed Status}
\label{sec:openprobs}
%─────────────────────────────────────────────────────────────────────────────

P218 listed four open problems.  Their post-P222 status follows.

\smallskip
\begin{center}
\begin{tabular}{@{}p{3.4cm}p{5.2cm}p{3.0cm}@{}}
\toprule
Problem & Outcome & Final status \\
\midrule
\textbf{OP-AxID} &
Derivation of Axiom~ID from R1--R8. &
\textsc{resolved} (P219+P222; conditional on $\alpha{=}\Omega^{-1}$) \\[5pt]
\textbf{OP-rSinf-100} &
Near-miss $\rr\cdot\Sinf\approx c/100$, $\varepsilon\approx-2.20\times10^{-7}$;
no PSLQ form in natural basis.
Deviation is $0.048\,\sigma_{\rm CODATA}$, below detectability at
current experimental precision. &
\textsc{conjectured}: coincidence not identifiable with current measurements \\[5pt]
\textbf{OP-G2A2-lattice} &
Resolvent $G(\lambda^*){=}\Sinf$ at
$\lambda^*\approx 4255.7$ (P221 Thm.~P221.1); $\lambda^*$ not a
TOE primitive. &
\textsc{closed} (P221) \\[5pt]
\textbf{OP-Exactness} &
$\Rinf$ overshoots CODATA-2018 by $0.011\ppm$ ($0.5\sigma$); future
measurement with $\sigma < 10^{-6}$ required to decide whether this is
the exact value or a structural residual. &
\textsc{open}: awaits precision measurement \\
\bottomrule
\end{tabular}
\end{center}

%─────────────────────────────────────────────────────────────────────────────
\section{Residual Foundational Question}
\label{sec:residual}
%─────────────────────────────────────────────────────────────────────────────

Theorem~P218b.1 is unconditional \emph{within the TOE}.  The TOE includes
the identification $\alpha = \Omega^{-1}$ as a physical axiom: the
measured fine-structure constant $\alpha$ is declared equal to
$\Omega^{-1} = (4\pi^3+\pi^2+\pi)^{-1}$.  P222 shows that $\Omega$
is geometrically derived; whether the \emph{physical} $\alpha$ equals
$\Omega^{-1}$ exactly --- to all decimal places, not merely to the
$2.22\times10^{-6}$ relative agreement measured at CODATA-2018 precision
--- is \emph{not} a question about the P18-T2 programme.  It is the
central empirical question of the TOE as a whole.

\begin{openprob}[TOE-AxID: Exact identification $\alpha = \Omega^{-1}$]
\label{op:toe}
Is the physical fine-structure constant $\alpha$ equal to
$(4\pi^3+\pi^2+\pi)^{-1}$ exactly, or only to the currently-measured
precision of $2.22\times10^{-6}$?  This is distinct from the P18-T2
programme, which is established within the TOE's axioms regardless of
the answer.  Candidate routes: (a) spectral action computation for
$\Oop$ on $(B^4, S^3)$-sections; (b) direct derivation of $\alpha$
from the eigenspectrum of $\Oop$, closing the factor-66 discrepancy of
Paper~18 Remark~4.3; (c) future measurement of $\alpha$ to $<10^{-8}$
relative precision.
\end{openprob}

\noindent This question does not affect the status of Theorem~P218b.1.

%─────────────────────────────────────────────────────────────────────────────
\begin{thebibliography}{9}

\bibitem{paper31}
L.~F.~Vlegels.
\textit{Paper 31: Nicomachus Monad.}
LumenOS TOE Corpus, 2026.  \textbf{Theorem~6.3}: Bézout Frame Axiom.

\bibitem{P218}
L.~F.~Vlegels.
\textit{Addendum P218: Closure of the P18-T2 Uniqueness Programme.}
LumenOS addenda corpus, 18 May 2026.

\bibitem{P219}
L.~F.~Vlegels.
\textit{Addendum P219: OP-AxID Spectral Derivation.}
LumenOS addenda corpus, 18 May 2026.

\bibitem{P220}
L.~F.~Vlegels.
\textit{Addendum P220: Closure Attempt on OP-rSinf-100.}
LumenOS addenda corpus, 18 May 2026.

\bibitem{P221}
L.~F.~Vlegels.
\textit{Addendum P221: OP-G2A2-lattice Resolvent.}
LumenOS addenda corpus, 18 May 2026.

\bibitem{P222}
L.~F.~Vlegels.
\textit{Addendum P222: Monad Closure Polynomial and the R1--R8 Constraint Chain.}
LumenOS addenda corpus, 18 May 2026.

\bibitem{codata2018}
CODATA 2018.
$m_\mu/m_e = 206.768\,283\,0\,(46)$.
\textit{NIST Reference on Constants, Units, and Uncertainty}, 2019.

\end{thebibliography}

\end{document}
