% Addendum 222: Monad Closure Polynomial and the R1–R8 Constraint Chain
% P18-T2 Uniqueness Programme
% L. F. Vlegels, Independent Researcher

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{booktabs,longtable,array}
\usepackage{enumitem}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\usepackage{geometry}
\geometry{margin=1in}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{remark}[theorem]{Remark}
\theoremstyle{remark}
\newtheorem{openproblem}[theorem]{Open Problem}

\newcommand{\Ohat}{\hat{O}}
\newcommand{\ALPHA}{\alpha^{-1}}
\newcommand{\mcp}{4\pi^3 + \pi^2 + \pi}
\newcommand{\mudef}{\mu \;=\; 4\pi^3 + \pi^2 + \pi}
\newcommand{\Bezout}{\textsc{B\'{e}zout}}

\title{%
  Addendum 222: The Monad Closure Polynomial\\
  and the R1--R8 Constraint Chain\\[4pt]
  \large P18-T2 Uniqueness Programme}
\author{L.~F.~Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}
\date{18 May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum 219 showed that Axiom~ID (the short/long root energy
identification) is derivable from the monad closure polynomial
\[
  f(e) \;=\; 4e^3 + e^2 + e \;=\; \mu, \qquad e = \pi,\quad
  \mu = 4\pi^3+\pi^2+\pi = \ALPHA,
\]
and conjectured that this polynomial in turn arises from the R1--R8
constraints on the master operator $\Ohat$.  The present addendum
resolves the question by a systematic examination of Papers~00--31.

We find \textbf{Outcome~B (Partially Resolved)}: the geometric quantity
$\Omega = \mcp$ is derived from the self-intersection structure of
$S^3 \hookrightarrow B^4$ via the Bézout Frame Axiom (Paper~31,
Theorem~6.3), which in turn rests on the Hopf fibration structure
encoded in requirements R1--R3.  The R1--R8 requirements therefore
imply the monad closure formula, and Axiom~ID follows.

The single remaining gap is the \emph{physical identification} step:
the corpus derives $\Omega$ as a geometric quantity from topology and
shows it matches the experimental $\ALPHA$ to $2.2 \times 10^{-6}$
relative error.  Whether this match is exact (i.e., whether the
physical fine-structure constant $\alpha$ is equal to $\mu^{-1}$ to
all decimal places, as opposed to coinciding to measured precision) is
not deduced from R1--R8 alone; it remains an empirical identification.

Consequently, \textbf{Theorem~P218.1 (the P18-T2 master operator
closure) is conditional}: unconditional on the geometric derivation
of $\Omega$ from R1--R8, but conditional on the empirical
identification $\Omega \equiv \ALPHA$.  We formulate this precisely as
Conjecture~P222.1 and identify the single remaining step required to
make it unconditional.

Companion file: \texttt{verify/verify\_P222.py} (60 assertions,
all passing, at \texttt{mp.dps = 60}).
\end{abstract}

\tableofcontents
\newpage

% ─────────────────────────────────────────────────────────────────────────────
\section{Background and Motivation}
\label{sec:background}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{The P18-T2 programme}

Paper~18 assembles a canonical master operator
\begin{equation}\label{eq:master}
  \Ohat \;=\; D_{B^4}^2 + \Delta_{S^3} + \Delta_{S^1}
            + V_{\text{self}} + \alpha\rho + \tfrac{3}{4}T
            + \kappa\mathcal{R} + \tfrac{3\pi}{20}\mathcal{M}
\end{equation}
from eight coverage requirements R1--R8 (bulk dynamics, boundary
dynamics, fiber dynamics, self-consistency, density coupling, family
symmetry, scale invariance, moment hierarchy).  Theorem~P218.1
states that this assembly satisfies R1--R8 with all four coefficients
$(\alpha, \tfrac{3}{4}, \kappa, \tfrac{3\pi}{20})$ determined by
geometry.  Whether the assembly is the \emph{unique} admissible one is
recorded as Open Problem~P18-T2 (\S4.4 of Paper~18).

Addendum~219 showed that Axiom~ID is equivalent to the monad closure
polynomial identity (see \S\ref{sec:polynomial} below) and that, if
this polynomial follows from R1--R8, Axiom~ID becomes a theorem, which
upgrades Paper~6's mass-family result P6 from
\textsc{Conjectured} to \textsc{Proven} and makes the P18-T2 closure
unconditional.  The question of this addendum is therefore:

\begin{quote}
\emph{Does the polynomial $f(x) = 4x^3 + x^2 + x$ evaluated at
$x = \pi$ yield $\mu = \ALPHA$ as a consequence of the R1--R8
constraints, or is $\mu$ an independent input?}
\end{quote}

\subsection{Papers consulted}

The following papers were read in full for this analysis:

\begin{center}
\begin{tabular}{llp{8cm}}
\toprule
Paper & Title (short) & Key contribution to this question \\
\midrule
P00 & Monad Rosetta Map & GSP Axioms G1--G4; Axiom G4 introduces $\mu$ \\
P01 & Perfect Stable Sphere & $\mu$ identified with $\ALPHA$ to 0.0002\% \\
P03 & Mathematical Foundations & Spectral interpretation conjectured (open) \\
P08 & Time-Observation Bootstrap & Self-intersection counting values $c_d$ \\
P18 & Master Operator & R1--R8 coverage requirements; assembly \\
P30 & Geometric Observer Network & $\Omega_{\text{monad}} = \mu$ as fixed constant \\
P31 & Nicomachus Monad & Bézout Frame Axiom; $\Omega = \mu$ derived \\
P14 & Cosmic Breathing & $\pi\cdot\mu$ as breath period \\
\bottomrule
\end{tabular}
\end{center}


% ─────────────────────────────────────────────────────────────────────────────
\section{The Monad Closure Polynomial}
\label{sec:polynomial}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{Statement}

\begin{definition}[Monad closure polynomial]\label{def:mcp}
The monad closure polynomial is
\begin{equation}\label{eq:mcp}
  f(x) \;=\; 4x^3 + x^2 + x \;=\; x\,(4x^2 + x + 1).
\end{equation}
\end{definition}

\begin{lemma}[Polynomial identity, tautological]\label{lem:tautology}
Let $\mu = 4\pi^3 + \pi^2 + \pi$.  Then $f(\pi) = \mu$.
\end{lemma}

\begin{proof}
Direct substitution: $f(\pi) = 4\pi^3 + \pi^2 + \pi = \mu$.
\end{proof}

\begin{remark}
Lemma~\ref{lem:tautology} is a tautology.  The non-trivial content lies
entirely in the question of \emph{where $\mu$ itself comes from}: is it
a derived spectral quantity, or is it a postulated normalization constant?
That is the substance of this addendum.
\end{remark}

\subsection{Uniqueness of the root}

\begin{proposition}[Unique positive root]\label{prop:unique}
$\pi$ is the unique positive real number satisfying $f(x) = \mu$.
\end{proposition}

\begin{proof}
The quadratic factor $4x^2 + x + 1$ has discriminant
$\Delta = 1 - 16 = -15 < 0$ and positive leading coefficient, so it is
positive for all $x \in \mathbb{R}$.  Hence $f(x) = x\cdot(4x^2+x+1)$
is a product of $x$ and a positive function, giving $f'(x) = 12x^2 +
2x + 1$ with discriminant $4 - 48 < 0$, so $f$ is strictly increasing
on $(0,\infty)$.  By the intermediate value theorem and strict
monotonicity, $f(x) = \mu$ has exactly one solution on $(0,\infty)$,
which is $x = \pi$.
\end{proof}

\begin{remark}[Spectral interpretation]
Although $\pi$ is the unique positive root of $f(x) = \mu$ when $\mu$
is given, the standard spectral families on $S^3$ do \emph{not}
contain $\mu$ as an eigenvalue at an integer mode:
\begin{itemize}
  \item Scalar Laplacian: eigenvalues $l(l+2)$ for $l \in \mathbb{Z}_{\ge 0}$.
        The equation $l(l+2) = \mu$ gives $l \approx 10.75$, not an integer.
  \item Spinor Laplacian: eigenvalues $(n + \tfrac{3}{2})^2$.
        The equation $(n + \tfrac{3}{2})^2 = \mu$ gives $n \approx 10.21$,
        not an integer.
  \item Hopf base $\mathbb{CP}^1$: eigenvalues $k(k+1)$.
        The equation $k(k+1) = \mu$ gives $k \approx 11.22$, not an integer.
\end{itemize}
Hence $\mu$ is \emph{not} a standard eigenvalue of the spectral
geometries of $B^4$ or $S^3$.  This is consistent with Outcome~B: the
route from R1--R8 to $\mu$ passes through the Bézout self-intersection
count, not through a simple spectral gap.
\end{remark}


% ─────────────────────────────────────────────────────────────────────────────
\section{Source Analysis: Where Does $\mu$ Enter the Corpus?}
\label{sec:source}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{Paper 00: Axiom G4 as the normalization condition}

Paper~00 introduces $\mu$ through the Generative Symmetry Principle
(GSP), built from four axioms G1--G4.  Axioms G1--G3 fix the
form of the cubic density
\[
  \rho(x) = a_3 x^3 + a_2 x^2 + a_1 x
\]
with representation-proportional coefficients $a_3 = \lambda \cdot 16\pi^3$,
$a_2 = \lambda \cdot 3\pi^2$, $a_1 = \lambda \cdot 2\pi$.

\begin{axiom}[G4, Paper~00]\label{ax:G4}
The total integral of the phase density reproduces the geometric
decomposition:
\[
  M[\rho] \;=\; \int_0^1 \rho(x)\,dx \;=\; 4\pi^3 + \pi^2 + \pi \;=:\; \mu.
\]
\end{axiom}

The GSP Theorem of Paper~00 then shows that Axioms G1--G4 uniquely
determine $\lambda = 1$, hence $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$.
\emph{Within Paper~00, Axiom~G4 is a normalization postulate that
introduces $\mu$ by name.}  No derivation of this value from a more
primitive principle appears in Paper~00 itself.

\subsection{Paper 01: Empirical identification of $\mu$ with $\ALPHA$}

Paper~01 (``The Perfect Stable Sphere'') identifies $\mu = 4\pi^3 +
\pi^2 + \pi$ as a formula that \emph{matches} the experimentally measured
fine-structure constant:
\[
  \mu = 137.036303776\ldots
  \qquad \text{vs.} \qquad
  \ALPHA_{\text{exp}} = 137.035999084(21).
\]
Relative agreement: $|\mu - \ALPHA_{\text{exp}}| / \ALPHA_{\text{exp}}
\approx 2.22 \times 10^{-6}$.  Paper~01 is explicit about the
epistemological status: ``Rather than attempting to derive $\alpha$ from
first principles within an assumed framework, we take its experimental
value as input and ask what geometric structure naturally accommodates
it.''  The identification $\mu = \ALPHA$ is therefore \emph{empirical
in Paper~01}, not derived from a spectral condition.

\subsection{Paper 03: Spectral interpretation as a conjecture}

Paper~03 contains the following:
\begin{quote}
  \textbf{Conjecture~[Spectral Interpretation].}
  The coefficients $16\pi^3, 3\pi^2, 2\pi$ may correspond to spectral
  invariants of a 4-dimensional Laplace-type operator, analogous to heat
  kernel expansion coefficients.
\end{quote}
and explicitly flags four steps needed to realize this program
(constructing the Dirac operator, computing its heat kernel, deriving
the integer coefficients $16, 3, 2$ from topological invariants, and
connecting the radial coordinate to spectral flow) as \textbf{future
work}.  As of the publication of Paper~03, the spectral route to $\mu$
is unproven.

\subsection{Paper 08: Heuristic self-intersection counts}

Paper~08 derives the density coefficients from self-intersection
counting of $S^3$ in $B^4$.  The three counting values
\begin{center}
\begin{tabular}{clc}
\toprule
$d$ & Heuristic source & $c_d$ \\
\midrule
1 & two tangency points           & $2$ \\
2 & $\dim(\mathrm{SO}(3)) = 3$   & $3$ \\
3 & $2^4$ ambient orientations    & $16$ \\
\bottomrule
\end{tabular}
\end{center}
are obtained by \emph{three independent heuristics} with no shared
derivation.  Paper~08 itself records in \S10.2 that ``a rigorous proof
of the counting formula is outstanding.''

\subsection{Paper 31: The Bézout Frame Axiom (the key resolution)}

Paper~31 (``Nicomachus Monad'', 2026-05-12 update) provides the critical
advancement.  It shows:

\begin{enumerate}[label=(\roman*)]
\item The three counting values satisfy the \emph{uniform formula}
      $c_d = (d+1)^{\max(d-1,1)}$ (Proposition~3.3 of Paper~31).
\item This formula is derived from a Bézout intersection count on the
      Hopf modulus space (Theorem~6.3 of Paper~31):
\end{enumerate}

\begin{theorem}[Bézout Frame Axiom, Paper~31~Thm.~6.3]\label{thm:bezout}
The self-intersection configuration count satisfies
\[
  c_d \;=\; (d+1)^{\max(d-1,\,1)}, \qquad d \in \{1,2,3\},
\]
as the Bézout intersection number of a system of $n = \max(d-1,1)$
equations of degree $k = d+1$ in the Hopf modulus coordinates, each
being the Chebyshev polynomial $T_k(t_j) = 0$.  All $k^n$ solutions
are real; the Bézout bound is achieved exactly.
\end{theorem}

The $\mathbb{Z}_k$ monodromy at each stratum required for
Theorem~\ref{thm:bezout} is established corpus-internally:
$d=1$ from the antipodal $\mathbb{Z}_2$ map; $d=2$ from
Paper~06's lens space $L(3,1) = S^3/\mathbb{Z}_3$; $d=3$ from
Paper~28's diagonal $\mathbb{Z}_4$ action on $S^3 \subset \mathbb{C}^2$.

\begin{corollary}[Monad closure from Bézout, Paper~31~Cor.~6.4]\label{cor:nico}
The monad closure sum is
\begin{equation}\label{eq:nico}
  \Omega \;=\; \sum_{d=1}^{3} (d+1)^{(d-2)_+} \cdot \pi^d
         \;=\; \pi + \pi^2 + 4\pi^3 \;=\; \mu.
\end{equation}
\end{corollary}

The derivation chain is:
\[
  \underbrace{S^1 \hookrightarrow S^3 \to S^2}_{\text{Hopf (R1--R3)}}
  \;\xrightarrow{\mathbb{Z}_k\text{ monodromy}}\;
  T_k(t) = 0
  \;\xrightarrow{\text{Bézout}}\;
  c_d = (d+1)^{\max(d-1,1)}
  \;\xrightarrow{\text{density}}\;
  \rho(x)
  \;\xrightarrow{\text{integral}}\;
  \Omega = \mu.
\]

\subsection{Relation of the Bézout derivation to R1--R8}

The R1--R8 requirements of Paper~18 cover:
R1 (bulk $B^4$), R2 (boundary $S^3$), R3 (fiber $S^1$), R4
(self-lensing), R5 (density coupling), R6 ($\mathbb{Z}_3$ family
symmetry), R7 (renormalization), R8 (moment hierarchy).

The Bézout argument uses:
\begin{itemize}
  \item \textbf{R1}: the $B^4$ bulk defines the ambient space in which
        $S^3$ self-intersects.
  \item \textbf{R2}: the boundary $S^3$ is the manifold whose
        self-intersections are being counted.
  \item \textbf{R3}: the Hopf fibration $S^1 \hookrightarrow S^3 \to
        S^2$ is the structural input for the modulus space and the
        Chebyshev frame condition.
  \item \textbf{R6}: the monodromy groups $\mathbb{Z}_2, \mathbb{Z}_3,
        \mathbb{Z}_4$ are all subgroups of the $\mathbb{Z}_3$ family
        symmetry structure or implied by the lens space geometry.
\end{itemize}
Requirements R4, R5, R7, R8 concern the coupling coefficients of $\Ohat$
and do not bear on the derivation of $\mu$ from the Bézout count.

\textbf{Conclusion of this section}: The geometric quantity $\Omega$ is
derivable from R1--R3 (and implicitly R6) via the Bézout Frame Axiom.
This makes $\Omega = \mcp$ a consequence of the coverage requirements, not
an independent postulate.


% ─────────────────────────────────────────────────────────────────────────────
\section{Formal Assessment: Outcome B}
\label{sec:formal}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{Two-layer structure of the problem}

There are two distinct questions that must be carefully separated:

\begin{enumerate}[label=(\Alph*)]
\item \label{itm:geo}
      \textbf{Geometric question.}  Does the Hopf self-intersection
      structure of $(B^4, S^3)$ imply that the monad closure sum is
      $\Omega = \mcp$?
\item \label{itm:phys}
      \textbf{Physical identification.}  Is the physical fine-structure
      constant $\alpha$ equal to $\Omega^{-1}$ exactly (not merely to
      measured precision)?
\end{enumerate}

The answer to~\ref{itm:geo} is \textbf{yes} (conditional on the
Bézout Frame Axiom, which is established in Paper~31).  The answer
to~\ref{itm:phys} is \textbf{open}: as of this addendum, the
identification rests on the $2.22 \times 10^{-6}$ relative match
between $\Omega$ and $\ALPHA_{\text{exp}}$.

\subsection{Main result}

\begin{conjecture}[P222.1: Monad closure polynomial from R1--R8]\label{conj:main}
Let $\Ohat$ be a canonical assembly satisfying R1--R8.  Then:
\begin{enumerate}[label=(\roman*)]
\item The monad closure sum $\Omega = \mcp$ is a consequence of the
      Hopf fibration structure implicit in R1--R3, via the Bézout
      Frame Axiom (Paper~31, Theorem~6.3).
\item The monad closure polynomial $f(\pi) = \mu$ holds, and $\pi$ is
      the unique positive root of $f(x) = \mu$ (Proposition~\ref{prop:unique}).
\item \textbf{Gap}: Whether the physical fine-structure constant $\alpha$
      equals $\Omega^{-1}$ exactly (rather than to $2.22 \times 10^{-6}$
      precision) is not established by R1--R8 alone, and remains
      an open problem.
\end{enumerate}
Conditional on resolving~(iii), Axiom~ID is a theorem, Paper~6 result
P6 upgrades to \textsc{Proven}, and Theorem~P218.1 is unconditional.
\end{conjecture}

\begin{remark}[Why this is Outcome~B, not Outcome~A]
Outcome~A (proved) would require that R1--R8 force $\alpha = \Omega^{-1}$
exactly.  This would require computing the spectrum of $\Ohat$ exactly
and showing that the ground-state normalization equals $\Omega$ at
infinite precision.  Paper~18 Remark~5.4.1 records a factor-of-66
discrepancy in fermion mass ratios, indicating that the spectrum of
$\Ohat$ has not been computed exactly.  Until the spectral computation
is completed, the physical identification step remains empirical.
\end{remark}

\begin{remark}[Why this is Outcome~B, not Outcome~C]
Outcome~C (defined, not derived) would mean $\mu$ is a raw postulate
with no geometric explanation.  Paper~31's Bézout derivation refutes
this: $\Omega$ is derived from the topology of the Hopf fibration and
the self-intersection geometry of $(B^4, S^3)$.  The derivation is
conditional on Theorem~6.3 of Paper~31 (the Bézout Frame Axiom), which
is corpus-internal and relies on the $\mathbb{Z}_k$ monodromy structure
established in Papers~06 and~28.
\end{remark}

\subsection{The single remaining gap}

\begin{openproblem}[OP-AxID, refined]\label{op:axid}
Show that the physical fine-structure constant $\alpha$ satisfies
$\alpha = \Omega^{-1}$ exactly, where $\Omega = \mcp$ is the monad
closure sum derived from the Bézout Frame Axiom.  Equivalently: derive
the numerical value of $\alpha$ from the spectrum of the master operator
$\Ohat$ acting on $(B^4, S^3)$-sections, and confirm that the
ground-state normalization yields $\Omega$ without remainder.
\end{openproblem}

The gap has a precise nature: it is the gap between a \emph{topological}
derivation (which gives $\Omega$ from counting and measure) and a
\emph{spectral} derivation (which would give $\alpha$ from eigenvalues
of $\Ohat$).  The topological derivation is complete; the spectral
derivation is not.

Candidate routes to closing the gap include:
\begin{enumerate}
\item Construct the explicit Dirac operator on $(B^4, S^3)$ (flagged as
      future work in Paper~03), compute its heat kernel coefficients, and
      show they equal $16\pi^3, 3\pi^2, 2\pi$.
\item Show that the ground state normalization of $\Ohat$ equals
      $\int_0^1 |\psi_0|^2 \rho(x)\,dx = \Omega$ via the spectral
      action principle (Paper~18, \S5.2).
\item Demonstrate that no physically admissible deformation of $\Ohat$
      moves the ground-state normalization away from $\Omega$ (spectral
      rigidity from the moment hierarchy~R8).
\end{enumerate}


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

All numerical claims are verified at \texttt{mp.dps = 60} in
\texttt{verify/verify\_P222.py} (60 assertions, all passing).  Key
numerical facts:

\begin{center}
\begin{tabular}{llr}
\toprule
Quantity & Expression & Value \\
\midrule
$\mu$ & $4\pi^3 + \pi^2 + \pi$ & $137.03630\ldots$ \\
$\ALPHA_{\text{exp}}$ & CODATA 2018 & $137.03599\ldots$ \\
Relative gap & $|\mu - \ALPHA_{\text{exp}}|/\ALPHA_{\text{exp}}$ &
  $2.22 \times 10^{-6}$ \\
$f(\pi) - \mu$ & polynomial identity residual & $<10^{-55}$ \\
$\pi$ uniqueness & only positive root of $f(x)=\mu$ &
  proved (disc $= -15 < 0$) \\
Nicomachus sum & $\sum_{d=1}^3 (d+1)^{(d-2)_+}\pi^d - \mu$ &
  $< 10^{-58}$ \\
$c_1, c_2, c_3$ & Bézout counts & $2, 3, 16$ \\
$c_3 = 4^2 = 2^4$ & coincidence confirmed & $16 = 16$ \\
$T_k$ real roots & exactly $k$ in $(-1,1)$ for $k=2,3,4$ &
  verified \\
$S^3$ spectrum & $l(l+2) \neq \mu$ for $l \in \{0,\ldots,20\}$ &
  verified \\
BREATH\_PERIOD & $\pi \cdot \mu$ & $430.51\ldots$ \\
Layer fractions sum & FRAC\_EDGE + FRAC\_BULK + FRAC\_BNDRY & $1$ \\
\bottomrule
\end{tabular}
\end{center}


% ─────────────────────────────────────────────────────────────────────────────
\section{Derivation Chain in Full}
\label{sec:chain}
% ─────────────────────────────────────────────────────────────────────────────

For reference, we record the complete derivation chain from
R1--R8 to the monad closure polynomial, with the status of each step:

\begin{longtable}{cllp{4cm}l}
\caption{Derivation chain: R1--R8 $\to$ $\mu = f(\pi)$}\label{tab:chain} \\
\toprule
Step & Input & Output & Mechanism & Status \\
\midrule
\endfirsthead
\toprule
Step & Input & Output & Mechanism & Status \\
\midrule
\endhead
1 & R1--R3 & $S^1 \hookrightarrow S^3 \to S^2$ &
  Hopf fibration from $B^4$ geometry &
  \textsc{Established} \\[3pt]
2 & Hopf fibration & $\mathbb{Z}_k$ monodromy &
  Papers 06, 28 (lens space, diagonal action) &
  \textsc{Established} \\[3pt]
3 & $\mathbb{Z}_k$ monodromy & $T_k(t) = 0$ frame condition &
  Hopf modulus perpendicularity &
  \textsc{Paper~31 Lem.~6.2} \\[3pt]
4 & $T_k(t) = 0$ & $c_d = k^{\max(d-1,1)}$ &
  Bézout + totally-real property &
  \textsc{Paper~31 Thm.~6.3} \\[3pt]
5 & $c_d$ counts & $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ &
  Density = $c_d \cdot \pi^d$ coefficients &
  \textsc{Paper~31 Thm.~3.1} \\[3pt]
6 & $\rho(x)$ & $\Omega = \int_0^1 \rho = \mu$ &
  GSP G4 reconstruction &
  \textsc{Analytic} \\[3pt]
7 & $\Omega = \mu$ & $f(\pi) = \mu$ &
  Substitution (tautological) &
  \textsc{Tautological} \\[3pt]
8 & $\mu$ & $\alpha = \mu^{-1}$ &
  Physical identification &
  \textsc{Empirical ($2.22 \times 10^{-6}$)} \\
\bottomrule
\end{longtable}

Steps 1--7 are geometric/topological and collectively constitute the
Outcome~B derivation.  Step~8 is the empirical identification gap that
prevents Outcome~A.

\subsection{Status of Axiom G4}

Within Paper~00, Axiom~G4 (\S4.3 above) introduces $\mu$ as a
normalization postulate.  In the light of Paper~31, however,
Axiom~G4 is now \emph{post-hoc explained}: Step~6 in
Table~\ref{tab:chain} shows that $M[\rho] = \mu$ follows from the
Bézout counts (Steps~1--5), so G4 need not be taken as a primitive
axiom.  It is instead a theorem conditional on Paper~31.  This is a
genuine upgrade in the logical structure of the corpus: $\mu$ migrates
from postulate (G4) to derived quantity (Steps 1--6).


% ─────────────────────────────────────────────────────────────────────────────
\section{Implications for P218.1 and Theorem Status}
\label{sec:implications}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{Status of P6 and Theorem P218.1}

Addendum~219 showed:
\begin{itemize}
  \item Axiom~ID $\Leftrightarrow$ monad closure polynomial $f(\pi) = \mu$.
  \item Axiom~ID $\Rightarrow$ P6 \textsc{Proven}.
  \item P6 \textsc{Proven} $\Rightarrow$ Theorem~P218.1 unconditional.
\end{itemize}

Conjecture~P222.1 of this addendum now gives:
\begin{itemize}
  \item R1--R3 (via Bézout) $\Rightarrow$ $\Omega = \mu$ (geometric).
  \item $\Omega = \mu$ + physical identification $\Rightarrow$ Axiom~ID.
\end{itemize}

The combined result is:

\begin{theorem}[P222.2, conditional]\label{thm:main}
If the physical fine-structure constant $\alpha$ equals $\Omega^{-1} =
(4\pi^3 + \pi^2 + \pi)^{-1}$ exactly (Open Problem~\ref{op:axid}),
then Axiom~ID is a consequence of R1--R3, Paper~6 result P6 is
\textsc{Proven}, and Theorem~P218.1 is unconditional.
\end{theorem}

\begin{proof}
By the Bézout Frame Axiom (Theorem~\ref{thm:bezout}), $\Omega = \mu$
follows from R1--R3.  By Lemma~\ref{lem:tautology}, $f(\pi) = \mu$.
By Proposition~\ref{prop:unique}, $\pi$ is the unique positive root,
so the short/long root identification $e = \pi$ with $\mu = f(e)$ is
forced.  This is Axiom~ID.  The rest follows from Addendum~219.
\end{proof}

\subsection{What upgrades vs.\ what does not}

\begin{center}
\begin{tabular}{lll}
\toprule
Item & Previous status & Current status \\
\midrule
$\mu = \mcp$ derived & Postulate (G4, P00) & Theorem (P31~Cor.~6.4) \\
Axiom~ID & Conjectured (P219) & Conditional theorem (P222.2) \\
P6 & Conjectured & Conditional theorem \\
Theorem~P218.1 & Conditional & Conditional (weaker condition) \\
$\alpha = \Omega^{-1}$ exactly & Open & Open (OP-AxID) \\
\bottomrule
\end{tabular}
\end{center}

The condition on Theorem~P218.1 has been weakened: previously, it
required either an independent derivation of $\alpha$ or a proof of
Axiom~ID; now it requires only OP-AxID (the exact identification).
This is progress, because OP-AxID is a single well-defined spectral
computation, whereas the previous condition was open-ended.


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

We set out to determine whether the monad closure polynomial $f(\pi) =
\mu$ arises from the R1--R8 constraints, or is an independent input.
The answer is \textbf{Outcome~B (Partially Resolved)}:

\begin{enumerate}
\item \textbf{The geometric quantity $\Omega = \mcp$ is derived from
      R1--R8.}  The Bézout Frame Axiom (Paper~31, Theorem~6.3) derives
      the self-intersection counts $c_d$ from the Hopf fibration
      structure required by R1--R3, with $\mathbb{Z}_k$ monodromy from
      Papers~06 and~28.  The density $\rho(x)$ and its integral $\Omega$
      then follow analytically.

\item \textbf{The single remaining gap is the physical identification.}
      Whether $\alpha = \Omega^{-1}$ exactly --- rather than to the
      measured $2.22 \times 10^{-6}$ relative precision --- is not
      established by the corpus.  Resolving OP-AxID (Open
      Problem~\ref{op:axid}) would make the entire chain unconditional.

\item \textbf{Axiom~G4 is now redundant.}  The Bézout derivation
      (Steps~1--6 of Table~\ref{tab:chain}) reconstructs G4 as a
      theorem.  The GSP axiom system of Paper~00 can be streamlined by
      replacing G4 with a reference to Paper~31.
\end{enumerate}

The polynomial identity $f(\pi) = \mu$ is thus not a free assumption: it
is a tautological consequence of the definition of $\mu$, and $\mu$
itself is derived from the geometry of the Hopf fibration.  The
remaining gap is purely about whether the physical constant $\alpha$
and the geometric constant $\Omega^{-1}$ agree beyond current
measurement precision.  That gap cannot be closed by pure geometry
alone; it requires either a derivation of $\alpha$ from the spectrum
of $\Ohat$ or a proof that the spectral action principle forces
$\alpha = \Omega^{-1}$ exactly.

\medskip
\noindent
\textbf{Session note.}  The \texttt{verify\_P222.py} script confirms 60
assertions at \texttt{mp.dps = 60}, including the polynomial identity,
Nicomachus factorization, Bézout counts, Chebyshev root structures,
spectral gap non-integrality, layer fraction normalization, and the
empirical identification gap.

\begin{thebibliography}{99}

\bibitem{P00}
L.~F.~Vlegels,
\textit{Rosetta Map of the Monad Identity and the Geometric Theory of Everything},
This volume, Paper~00 (2025).

\bibitem{P01}
L.~F.~Vlegels,
\textit{The Perfect Stable Sphere: A Geometric Foundation for Fundamental Physics},
This volume, Paper~01 (2025).

\bibitem{P03}
L.~F.~Vlegels,
\textit{Mathematical Foundations of Geometric Fundamental Physics},
This volume, Paper~03 (2025).

\bibitem{P06}
L.~F.~Vlegels,
\textit{The Fermion Sector: Topological Origin of Three Families},
This volume, Paper~06 (2025).

\bibitem{P08}
L.~F.~Vlegels,
\textit{Time, Observation, Bootstrap: Self-Intersection Counting},
This volume, Paper~08 (2025).

\bibitem{P14}
L.~F.~Vlegels,
\textit{The Cosmic Breathing Cycle: $\pi \times \alpha^{-1}$ and 432},
This volume, Paper~14 (2025).

\bibitem{P18}
L.~F.~Vlegels,
\textit{The Master Operator: Assembly of the Unified Field Operator},
This volume, Paper~18 (2025).

\bibitem{P28}
L.~F.~Vlegels,
\textit{Universal Wave Geometry: Emergent Lorentz Structure from $S^3$},
This volume, Paper~28 (2025).

\bibitem{P30}
L.~F.~Vlegels,
\textit{The Geometric Observer Network},
This volume, Paper~30 (2025).

\bibitem{P31}
L.~F.~Vlegels,
\textit{Nicomachus Weights and Monad Closure},
This volume, Paper~31 (2026).

\bibitem{A219}
L.~F.~Vlegels,
\textit{Addendum 219: Axiom ID and the Monad Closure Polynomial},
This volume, Addendum~219 (2026).

\bibitem{CODATA2018}
E.~R.~Tiesinga, P.~J.~Mohr, D.~B.~Newell, and B.~N.~Taylor,
\textit{CODATA Recommended Values of the Fundamental Physical Constants: 2018},
Rev.~Mod.~Phys.\ \textbf{93}, 025010 (2021).

\end{thebibliography}

\end{document}
