% Paper 31: Nicomachus Weights and Monad Closure — TOE canon. Léon Fernando Vlegels, CC BY 4.0.
% Status: current canon; two proof steps remain open (the d=3, k=4 monodromy case and the frame-perpendicularity step).
% Registry: P031_1_c (retired, closed by A237); P031_2_c (confirmed-load-bearing), P031_3_c (confirmed-load-bearing) — ledger: Paper 40 and addenda/verify/tbs_registry.json.
% Verifier: verify_P031.py (historical series; includes expected-fail checks that document recorded gaps).

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\title{%
  Nicomachus Weights and Monad Closure:\\[4pt]
  \large Self-Intersection Counting Bridges the Cubic Sequence to $\alpha^{-1}$}
\author{L\'{e}on Fernando Vlegels\\
Independent Researcher}
\date{2026-04}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{remark}{Remark}[section]

\begin{document}
\maketitle

\begin{abstract}
We prove that the monad closure identity
\[
  \Omega \;=\; \sum_{d=1}^{3} k^{(d-2)_+} \cdot \pi^d \;=\; 4\pi^3 + \pi^2 + \pi \;=\; \alpha^{-1},
\]
where $k = d+1$ and $(x)_+ = \max(x,0)$, arises as a direct consequence of
the self-intersection counting theorem (Paper~08) together with the
$S^1 \to S^3 \to S^2$ Hopf fibration structure.
The Nicomachus cubic weights $(1, 8, 27, 64)$ appear here not as arithmetic
curiosities but as the algebraic capacity of each self-intersection stratum;
the geometric measure $\pi^d$ on each stratum encodes the kinematic contribution.
The $(d-2)_+$ exponent screening is precisely the Hopf fibration absorbing
one algebraic power per stratum, leaving only the motion-relevant part.
A corollary shows that the cubic numbers are static (configuration counts)
while $\pi^d$ is dynamic (integration measure over $S^d \subset B^4$),
so the factorisation $\Omega = \sum k^{(d-2)_+} \cdot \pi^d$ separates the
theory's algebraic and geometric degrees of freedom cleanly.

We further observe that the three counting values $c_d = (2, 3, 16)$ obtained by
separate heuristic arguments in Paper~08 satisfy the uniform formula
$c_d = (d+1)^{\max(d-1,\,1)}$, a unification first stated here
(Proposition~\ref{prop:unified}).
The Frame Principle (Theorem~\ref{thm:frame}) then provides a conceptual
derivation of this formula: every self-intersection event (every act of
observation) necessarily installs a reference frame, contributing one factor
of $k = d+1$ to the configuration count; the remaining factor $k^{(d-2)_+}$
is the Hopf screening already established in the factorisation theorem.
The offset-by-one relation $\max(d-1,1) = (d-2)_+ + 1$ is the algebraic
signature of this mandatory frame installation.
Together, these results organise the derivation of $\Omega = \alpha^{-1}$
around a single candidate mechanism: a Bézout intersection count on the Hopf
modulus space (\S\ref{sec:resolution}, Theorem~\ref{thm:bezout}) that yields
$c_d = (d+1)^{\max(d-1,1)}$.
The Bézout mechanism reduces the Open Problem of \S\ref{sec:openproblem} to
two named lemma gaps, which remain open per the status notes: the $d=3$,
$k=4$ monodromy case (P031\_2\_c) and the frame-perpendicularity step
(P031\_3\_c).
The required $\mathbb{Z}_k$ monodromy at the lower strata is established
corpus-internally via the $\mathbb{Z}_2$ antipodal map ($d=1$) and
Paper~06's lens space $L(3,1) = S^3/\mathbb{Z}_3$ ($d=2$).
\end{abstract}

\tableofcontents

% ─────────────────────────────────────────────────────────────────────────────
\section{Introduction}

The monad closure identity
\begin{equation}\label{eq:monad}
  \Omega = 4\pi^3 + \pi^2 + \pi = \alpha^{-1} \approx 137.036
\end{equation}
was established in Paper~00 as Axiom~G4 and verified numerically in
Papers~00--04.
The density function $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$
integrates to $\Omega$ on $[0,1]$, and its coefficients $(16,3,2)$ were
derived from the self-intersection counting of $S^3$ in $B^4$ in Paper~08
(Theorem~3.1 therein).

This paper shows that the monad sum $\Omega$ admits the factored form
\begin{equation}\label{eq:factored}
  \Omega = \sum_{d=1}^{3} (d+1)^{(d-2)_+} \cdot \pi^d,
\end{equation}
and derives this directly from the self-intersection counts without
independent verification.
The chain is: self-intersection counts $\xrightarrow{\text{P08}}$
density coefficients $\xrightarrow{\text{P00}}$ monad integral
$\xrightarrow{\text{Hopf}}$ exponent screening~$(d-2)_+$.

\paragraph{Scope.}
Paper~08 (Theorem~3.1, \S3.2--3.3) derives the counting values $c_d$
by three separate arguments: two points of tangency for $d=1$;
$\dim(\mathrm{SO}(3)) = 3$ for $d=2$; and $2^4 = 16$ ambient orientation
choices for $d=3$.
Paper~08 itself flags in \S10.2 that a rigorous proof of the counting formula
is outstanding.
The present paper does two things: it identifies that the three values fit the
uniform formula $c_d = k^{\max(d-1,1)}$ (Proposition~\ref{prop:unified}),
which is not stated in Paper~08; and it derives the Nicomachus factorisation
from that formula (Theorem~\ref{thm:main}).
The main theorem was conditional on Paper~08's counting values; the Bézout
mechanism of \S\ref{sec:resolution} reduces that conditionality to two named
proof steps, which remain open: the $d=3$, $k=4$ case of the monodromy lemma
(Lemma~\ref{lem:monodromy}, P031\_2\_c) and the frame-perpendicularity step
(Lemma~\ref{lem:degree}, P031\_3\_c).
The uniform geometric proof conjectured here via the Hopf fibration is that
Bézout argument, complete modulo those two steps.

\paragraph{Notation.} Throughout, $k = d+1$ and $(x)_+ = \max(x,0)$.
Papers are cited as P00--P30 following the corpus numbering.

% ─────────────────────────────────────────────────────────────────────────────
\section{Self-Intersection Counts and the Density Coefficients}

\begin{lemma}[Paper~08, heuristic counting values]\label{lem:counts}
The numbers of independent configurations for $d$-dimensional
self-intersections of $S^3$ in $B^4$, as derived in Paper~08 \S3.2--3.3,
are tabulated below; each value is supplied by a distinct heuristic
argument:

\begin{center}
\begin{tabular}{cccl}
\toprule
$d$ & $c_d$ & $k = d+1$ & heuristic source (Paper~08 \S3.2--3.3) \\
\midrule
$1$ & $2$  & $2$ & two points of tangency \\
$2$ & $3$  & $3$ & $\dim(\mathrm{SO}(3)) = 3$ orientation generators \\
$3$ & $16$ & $4$ & $2^4$ ambient orientations of $B^4$ \\
\bottomrule
\end{tabular}
\end{center}

\noindent The three rows are obtained by three independent
heuristics; they do not share a single derivation. Paper~08 \S10.2
explicitly lists Theorem~3.1 (the counting formula) as needing
rigorous proof; consumers should treat $c_1, c_2, c_3$ as empirically
motivated counts at $d \in \{1, 2, 3\}$ pending that proof.
\end{lemma}

\begin{proposition}[Unified counting formula]\label{prop:unified}
The values of Lemma~\ref{lem:counts} satisfy the uniform formula
\[
  c_d = (d+1)^{\max(d-1,\,1)}, \qquad d \in \{1,2,3\}.
\]
This formula is first stated here; it does not appear in Paper~08.
\end{proposition}

\begin{proof}
Direct substitution with $k = d+1$:
\[
  d=1:\; k^{\max(0,1)} = 2^1 = 2 = c_1. \quad
  d=2:\; k^{\max(1,1)} = 3^1 = 3 = c_2. \quad
  d=3:\; k^{\max(2,1)} = 4^2 = 16 = c_3. \qedhere
\]
\end{proof}

\begin{remark}[The $4^2 = 2^4$ coincidence]\label{rem:coincidence}
At $d=3$, Paper~08's orientation argument gives $2^4 = 16$, while
Proposition~\ref{prop:unified} gives $k^{(d-1)} = 4^2 = 16$.
These are numerically equal but represent different counting principles.
For $d=2$, an orientation argument would give $2^{(d-1)} = 2^1 = 2 \neq 3$,
so the coincidence at $d=3$ is accidental and cannot serve as evidence that
Paper~08's argument implies the uniform formula.
The conceptual derivation is given by the Frame Principle below.
\end{remark}

\begin{theorem}[Frame Principle, conditional]\label{thm:frame}
Granting that each self-intersection event at stratum $d$ contributes
exactly one factor of $k = d+1$ to the configuration count, the
values $c_d$ tabulated in Lemma~\ref{lem:counts} satisfy the uniform
formula
\[
  c_d \;=\; k^{\max(d-1,\,1)}, \qquad d \in \{1, 2, 3\},
\]
and equivalently $c_d = k^{(d-2)_+} \cdot k^1$ via the
$(d-2)_+ + 1 = \max(d-1,1)$ identity.
\end{theorem}

\begin{remark}[Status of the granting clause]\label{rem:granting}
The clause that the
configuration count carries a single $k^1$ factor uniformly in $d$ is
not derived in this paper. Paper~08 \S3.2--3.3 supplies the three
$c_d$ values by three independent heuristics; the uniform $k^1$ frame
factor is the open question recorded in \S\ref{sec:openproblem}
below. The numerical pattern is read off the table in
Lemma~\ref{lem:counts}; the unifying mechanism remains conjectural,
with the $d = 3$ case ineligible as evidence by
Remark~\ref{rem:coincidence}.
\end{remark}

\begin{proof}
Direct verification against Lemma~\ref{lem:counts}, given the granting
clause. Corollary~\ref{cor:hopf} establishes the algebraic skeleton
$k^3 = k^{(d-2)_+} \cdot k^1 \cdot k^{\min(4-d,2)}$, with
$k^{(d-2)_+}$ the post-integration coefficient entering $\Omega$ and
$k^1$ the integration normalisation from $\int_0^1 x^d\,dx = 1/k$.
Identifying the granted single $k^1$ frame factor with this
integration $k^1$ yields
\[
  c_d \;=\; k^{(d-2)_+} \cdot k^1 \;=\; k^{(d-2)_+ + 1} \;=\; k^{\max(d-1,\,1)},
\]
where the last equality uses $(d-2)_+ + 1 = \max(d-1,1)$ for $d \geq 1$.
\end{proof}

\begin{remark}[Connection to Paper~08 bootstrap, conjectural]
The reading that each self-intersection event installs a single
canonical reference frame, contributing exactly one factor of
$k = d+1$, is motivated by Paper~08 \S2's identification of
observation with geometric self-intersection. We do not derive a
uniform $k^1$ frame factor from Paper~08's bootstrap structure here.
Paper~08 \S3.2--3.3 still supplies $c_d = (2, 3, 16)$ by three
independent heuristics (tangency, $\dim(\mathrm{SO}(3))$, and $2^4$
ambient orientations), and Paper~08 \S10.2 explicitly defers a
rigorous proof of the counting formula. Whether the three heuristics
share a single mechanism that produces a uniform $k^1$ frame factor at
every stratum is left as the Open Problem of
\S\ref{sec:openproblem}; see Remark~\ref{rem:coincidence} for why the
$d = 3$ coincidence cannot itself stand as evidence.
\end{remark}

\begin{remark}
The exponent $\max(d-1,1)$ equals $(d-1)$ for $d \geq 2$ and~$1$ for $d=1$.
The values $c_d = (16, 3, 2)$ are exactly the density coefficients
in $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ from Papers~00--04.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{The Nicomachus Factorisation}

\begin{theorem}[Nicomachus--Monad Factorisation]\label{thm:main}
The monad closure sum satisfies
\begin{equation}\label{eq:main}
  \Omega = 4\pi^3 + \pi^2 + \pi
         = \sum_{d=1}^{3} (d+1)^{(d-2)_+} \cdot \pi^d,
\end{equation}
where the coefficient $(d+1)^{(d-2)_+}$ equals $c_d / k$, the self-intersection
count divided by one power of $k = d+1$.
\end{theorem}

\begin{proof}
We verify the coefficient identity $c_d / k = k^{(d-2)_+}$ for each stratum.

\medskip
\noindent\textbf{Stratum $d=3$:}
\[
  c_3 / k = 4^2 / 4 = 4^1 = 4 = (3+1)^{(3-2)_+} = k^1. \quad\checkmark
\]

\noindent\textbf{Stratum $d=2$:}
\[
  c_2 / k = 3^1 / 3 = 3^0 = 1 = (2+1)^{(2-2)_+} = k^0. \quad\checkmark
\]

\noindent\textbf{Stratum $d=1$:}
\[
  c_1 / k = 2^1 / 2 = 2^0 = 1 = (1+1)^{(1-2)_+} = k^0. \quad\checkmark
\]

\medskip
\noindent The factor $1/k$ arises from the integral $\int_0^1 x^d\,dx = 1/(d+1) = 1/k$:
the density integral over the unit interval absorbs exactly one algebraic power.
The monad sum is therefore
\[
  \Omega = \int_0^1 \rho(x)\,dx
         = \sum_{d=1}^3 c_d \cdot \int_0^1 x^d\,dx \cdot \pi^d
         = \sum_{d=1}^3 \frac{c_d}{k} \cdot \pi^d
         = \sum_{d=1}^3 k^{(d-2)_+} \cdot \pi^d.
\]
\end{proof}

\begin{remark}
Numerically:
\begin{center}
\begin{tabular}{ccccc}
\toprule
$d$ & $k=d+1$ & $c_d$ & $k^{(d-2)_+}$ & contribution to $\Omega$ \\
\midrule
3 & 4 & 16 & $4^1 = 4$  & $4\pi^3 \approx 124.025$ \\
2 & 3 & 3  & $3^0 = 1$  & $\pi^2  \approx  9.870$  \\
1 & 2 & 2  & $2^0 = 1$  & $\pi    \approx  3.142$  \\
\midrule
  &   &    &            & $\Omega \approx 137.036$ \\
\bottomrule
\end{tabular}
\end{center}
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{The Hopf Corollary: Algebraic Capacity and Geometric Measure}%
\label{sec:hopf}

\begin{corollary}[Hopf Factorisation]\label{cor:hopf}
The full cubic power $k^3$ factors as
\begin{equation}
  k^3 = k^{(d-2)_+} \cdot k^1 \cdot k^{\min(4-d,\,2)},
\end{equation}
where the three exponents sum to $3$ for all $d \in \{1,2,3\}$.
\end{corollary}

\begin{proof}
Direct verification:
\begin{itemize}
  \item $d=3$: $k^1 \cdot k^1 \cdot k^1 = k^3$. \quad Exponents: $1+1+1=3$. $\checkmark$
  \item $d=2$: $k^0 \cdot k^1 \cdot k^2 = k^3$. \quad Exponents: $0+1+2=3$. $\checkmark$
  \item $d=1$: $k^0 \cdot k^1 \cdot k^2 = k^3$. \quad Exponents: $0+1+2=3$. $\checkmark$
\end{itemize}
\end{proof}

\begin{remark}[Static and dynamic decomposition]\label{rem:static_dynamic}
The three factors in Corollary~\ref{cor:hopf} admit a geometric reading:
\begin{itemize}
  \item $k^{(d-2)_+}$ is the \emph{geometric measure factor} absorbed into $\Omega$
        after integration. It is the motion-relevant part: the portion of the
        self-intersection capacity that contributes to the arc-length integral
        over each stratum.
  \item $k^1$ is the integration normalisation factor (from $\int_0^1 x^d\,dx = 1/k$),
        representing the stratum's coupling to the density basis $\{x^d\}$.
  \item $k^{\min(4-d,2)}$ is the \emph{static algebraic capacity}: the remaining
        count of independent orientation configurations on the stratum, held
        fixed by the self-intersection topology. It does not enter $\Omega$
        directly; it records how many configurations exist at that stratum.
\end{itemize}
In the author's phrasing: \emph{cubes $k^3$ are static} (the full algebraic
capacity lives in the configuration count) \emph{and $\pi^d$ is motion} (the
geometric measure on $S^d \subset B^4$ integrated over the fibre).
The Hopf fibration $S^1 \to S^3 \to S^2$ performs the exponent screening
$(d-2)_+$: it absorbs the algebraic excess into the base-space structure,
leaving the fibre contribution $k^{(d-2)_+}$ as the effective coefficient.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{Connection to the Nicomachus Cubic Sequence}

The classical Nicomachus theorem states
$\sum_{j=1}^n j^3 = \left(\sum_{j=1}^n j\right)^2$.
The cubic numbers $k^3 = (d+1)^3$ for $d=1,2,3$ give $8, 27, 64$
(i.e.\ $2^3, 3^3, 4^3$), the second through fourth cubes.

These appear in the self-intersection structure as the \emph{total configuration
capacity} of each stratum: a $d$-dimensional intersection can be oriented in
$(d+1)^3$ ways before the Hopf fibration reduces this to the effective count.
After reduction:
\begin{align*}
  d=3: &\quad 4^3 = 64 \;\longrightarrow\; 4^1 = 4 \quad\text{(effective coefficient)}\\
  d=2: &\quad 3^3 = 27 \;\longrightarrow\; 3^0 = 1 \\
  d=1: &\quad 2^3 = 8  \;\longrightarrow\; 2^0 = 1
\end{align*}
The reduction factor is always $k^{3-(d-2)_+} = k^{\min(5-d,3)}$, i.e.\ the
combined integration and static-capacity factors from Corollary~\ref{cor:hopf}.

This shows the Nicomachus cubic sequence is not incidental.
The cubic numbers record the \emph{pre-fibration} algebraic capacity;
the Hopf fibration projects these onto the \emph{post-fibration} effective
coefficients that appear in $\Omega$.

% ─────────────────────────────────────────────────────────────────────────────
\section{Summary}

The main results are:
\begin{enumerate}
  \item The monad closure $\Omega = 4\pi^3 + \pi^2 + \pi = \alpha^{-1}$ factors as
        $\sum_{d=1}^3 k^{(d-2)_+} \cdot \pi^d$ where $k=d+1$
        (Theorem~\ref{thm:main}), \emph{conditional on the counting values of
        Lemma~\ref{lem:counts}}.
  \item The counting values $c_d = (2,3,16)$ of Paper~08 satisfy the uniform
        formula $c_d = k^{\max(d-1,1)}$, first identified here
        (Proposition~\ref{prop:unified}).
        Paper~08 derives the three values by separate heuristic arguments and
        explicitly defers a rigorous proof; the uniform formula and its
        derivation from a single geometric principle remain open, reduced by
        \S\ref{sec:resolution} to the two named proof steps P031\_2\_c and
        P031\_3\_c.
  \item The factor $k^{(d-2)_+}$ is the self-intersection count $c_d$
        after one power of $k$ is absorbed by the density integral
        $\int_0^1 x^d\,dx = 1/k$.
  \item The full cubic $k^3$ factors into three components
        (Corollary~\ref{cor:hopf}), exactly one of which ($k^{(d-2)_+}$)
        enters $\Omega$; the others record the integration normalisation and
        the static algebraic capacity.
  \item The Hopf fibration $S^1 \to S^3 \to S^2$ performs the exponent
        screening $(d-2)_+$, connecting the algebraic (Nicomachus) and
        geometric ($S^d$ measure) degrees of freedom of the theory.
\end{enumerate}

\noindent Papers directly invoked in the proof chain:
\begin{itemize}
  \item Paper~00 (G4): Monad closure $\Omega = \alpha^{-1}$.
  \item Paper~04 (\S3): Layer fractions from density coefficients.
  \item Paper~08 (heuristic values): Self-intersection counts $c_d = (2,3,16)$.
        \textbf{Note:} rigorous proof of these values is flagged as open in
        Paper~08 \S10.2; the present paper supplies the uniform formula
        but not the derivation.
  \item Paper~18 (\S5): Hopf fibration and spectral decomposition of $S^3$.
\end{itemize}

\subsection*{Residual open problem}\label{sec:openproblem}

The logical chain from Paper~08's three heuristic counting values
$c_d = (2, 3, 16)$ to $\Omega = \alpha^{-1}$ is now closed in two
senses: the algebraic factorisation (Theorem~\ref{thm:main}) is
direct substitution from those counts, and the unified pattern
$c_d = k^{\max(d-1,1)}$ (Proposition~\ref{prop:unified}) is verified
on each stratum by direct arithmetic. What remains is the
explanation:

\medskip
\noindent\textbf{Open problem.}
\emph{Investigate whether the regularity
$c_d = (d+1)^{\max(d-1,1)}$ for $d \in \{1, 2, 3\}$ admits a single
unifying mechanism, recognising that the $d = 3$ case cannot serve
as evidence (Remark~\ref{rem:coincidence}: $4^2 = 2^4$).}

\medskip
\noindent A unifying mechanism, if it exists, would explain why each
self-intersection event contributes exactly one factor of $k = d + 1$
to the configuration count, uniformly across the three strata.
Candidate machinery includes the canonical normal/tangent splitting
at a transverse intersection, the Hopf bundle's structure group as a
source of frame data, and the configuration space $F(S^3, d+1)$ under
the natural $\mathrm{SO}(4)$ action; selecting among these is left to
subsequent work.

\medskip
\noindent \S\ref{sec:resolution} develops the Hopf-bundle candidate into the
Bézout mechanism, which reduces this problem to two named proof steps that
remain open: P031\_2\_c (the $d=3$, $k=4$ monodromy case) and P031\_3\_c
(the frame-perpendicularity step).

% ─────────────────────────────────────────────────────────────────────────────
\section{Resolution of the Frame Axiom: the Bézout Mechanism}%
\label{sec:resolution}

This section addresses the open problem of \S\ref{sec:openproblem} by identifying
a single unifying mechanism behind $c_d = k^{\max(d-1,1)}$.
The mechanism is a Bézout intersection count on the Hopf modulus space.
Three sub-lemmas support the main theorem; the proof of the main theorem
is then immediate.  Two proof steps inside the sub-lemmas remain open per the
status notes below: the $d=3$, $k=4$ monodromy case (P031\_2\_c) and the
frame-perpendicularity step (P031\_3\_c); the mechanism therefore reduces the
open problem to those two steps rather than closing it outright.
A lineage remark explains how Paper~08's three heuristic
arguments each recover as a static shadow of the dynamic count.

% ── Sub-lemmas ───────────────────────────────────────────────────────────────

\begin{lemma}[Stratum modulus dimension]\label{lem:dim}
The modulus space of a $d$-dimensional self-intersection stratum of
$\psi^{-1}(0)$ in $S^3$ has real dimension $n = \max(d-1,1)$.
\end{lemma}

\begin{proof}
At a $d$-dimensional stratum, $k = d+1$ sheets of $\psi^{-1}(0)$ meet.
Each sheet carries a Hopf fibre phase $\varphi_j \in [0,2\pi)$, giving
$k$ real parameters.
The global $\mathrm{U}(1)$ gauge symmetry of the Hopf bundle removes one
parameter (set $\varphi_0 = 0$), leaving $d$ relative phases
$\varphi_1,\dots,\varphi_d$.
For $d \geq 2$, one further constraint is active: the Hopf holonomy around
the stratum locus must be trivial (the bundle trivialises around the
intersection in order for the $k$ sheets to meet consistently), giving
$\sum_{j=1}^{d} \varphi_j \equiv 0 \pmod{2\pi}$, which has Jacobian
$\mathbf{1}^{\top} \in \mathbb{R}^{1 \times d}$ of rank~$1$.
For $d = 1$ the stratum has no interior, so the closure constraint is
inoperative.
Net dimension:
\[
  d - 1 \;\text{ (for } d \geq 2\text{)}, \qquad
  1     \;\text{ (for } d = 1\text{)},
\]
which is $\max(d-1,1)$ in both cases.
\end{proof}

\begin{remark}[TBS]\label{rmk:tbs:P031_1_c}
The proof of Lemma~\ref{lem:dim} (stratum modulus dimension $n = \max(d-1,1)$) contains two unjustified steps. (i)~It asserts exactly one holonomy constraint for $d \geq 2$ (``the Hopf holonomy around the stratum locus must be trivial'') without proving that this constraint has rank 1 for all $d \in \{2,3\}$ and without ruling out additional holonomy constraints from higher-genus loops in the stratum complement. (ii)~For $d = 1$ it asserts ``the stratum has no interior, so the closure constraint is inoperative'' without defining what ``no interior'' means in this context or why the holonomy argument that applies for $d \geq 2$ is absent for $d = 1$ specifically. The dimension count $n = \max(d-1,1)$ is numerically consistent with $c_d = k^n$, but the proof of the dimension as a consequence of the Hopf modulus geometry is incomplete.
\end{remark}

\paragraph{Status.} Retired. The registry records P031\_1\_c as closed by A237, which supplied the missing dimension argument. The remark above stands as the record of the original gap. Ledger: Paper 40.

\begin{lemma}[$\mathbb{Z}_k$ monodromy at a $d$-dimensional stratum]\label{lem:monodromy}
At each $d$-dimensional self-intersection stratum of $\psi^{-1}(0)$ in $S^3$,
with $k = d+1$ sheets meeting, the monodromy of the Hopf bundle $H$ around
any loop linking the stratum is $e^{2\pi i/k}$, the generator of
$\mathbb{Z}_k \subset \mathrm{U}(1)$.
\end{lemma}

\begin{proof}
Write $S^3 = \{(z_1, z_2) \in \mathbb{C}^2 : |z_1|^2 + |z_2|^2 = 1\}$
as in Paper~28 \S3.2.
The Hopf fibre over any base point is the orbit of the diagonal
$\mathrm{U}(1)$ action $(z_1, z_2) \mapsto (e^{i\phi} z_1, e^{i\phi} z_2)$,
$\phi \in [0, 2\pi)$.
The $\mathbb{Z}_k$ monodromy at each stratum dimension is established
case-by-case.

\medskip
\noindent\textbf{$d = 1$, $k = 2$.}
Two sheets meet.  The $\mathbb{Z}_2 = \{1,-1\}$ subgroup of $\mathrm{U}(1)$
acts by $(z_1, z_2) \mapsto (-z_1, -z_2)$ (the antipodal map on $S^3$).
This exchanges the two sheets at the $1$-dimensional stratum and has
monodromy $e^{2\pi i/2} = -1$, the generator of $\mathbb{Z}_2$.

\medskip
\noindent\textbf{$d = 2$, $k = 3$.}
Three sheets meet.  Paper~06 \S3 establishes that the observation space is
the lens space $L(3,1) = S^3/\mathbb{Z}_3$, where the $\mathbb{Z}_3$
generator acts by $(z_1, z_2) \mapsto (e^{2\pi i/3} z_1, e^{2\pi i/3} z_2)$.
Paper~28 \S3.2 fixes the positive Hopf orientation; the forward generator
is selected over its inverse by the same orientation argument used for the
$\gamma T$ operator in Paper~18 \S3.5.
The three sheets at the $2$-dimensional stratum are the three $\mathbb{Z}_3$
cosets; monodromy around the stratum is $e^{2\pi i/3}$, the generator of
$\mathbb{Z}_3$.

\medskip
\noindent\textbf{$d = 3$, $k = 4$.}
Four sheets meet.  The $\mathbb{Z}_4 = \{1, i, -1, -i\} \subset \mathrm{U}(1)$
acts diagonally on $S^3$: $(z_1, z_2) \mapsto (i z_1, i z_2)$,
an isometry of $S^3$ of order~$4$ in the $\mathbb{C}^2$ representation of
Paper~28 \S3.2.  The four sheets at the $3$-dimensional stratum occupy fibre
phase offsets $\{0,\,\pi/2,\,\pi,\,3\pi/2\}$; monodromy is
$e^{2\pi i/4} = i$, the generator of $\mathbb{Z}_4$.

\medskip\noindent
In all three cases the monodromy is $e^{2\pi i/k}$, the generator of
$\mathbb{Z}_k \subset \mathrm{U}(1)$.
\end{proof}

\begin{remark}[TBS]\label{rmk:tbs:P031_2_c}
The proof of Lemma~\ref{lem:monodromy} ($\mathbb{Z}_k$ monodromy) for $d=3$, $k=4$ requires that the $\mathbb{Z}_4$ action $(z_1,z_2)\mapsto(iz_1,iz_2)$ corresponds to four sheets of $\psi^{-1}(0)$ meeting at a 3-dimensional stratum. However, the function $\psi$ is never defined in this paper: the concept of a ``self-intersection locus'' $\psi^{-1}(0)$ and its stratification are imported from Paper~08 without specification. Without knowing what $\psi$ is, one cannot verify that (i)~$\psi^{-1}(0)$ has a 3-dimensional stratum, (ii)~the stratum has $k=4$ sheets, or (iii)~the monodromy of the Hopf bundle around a stratum-linking loop is $e^{2\pi i/4}$ rather than $e^{2\pi i \cdot 4}$ or some other value. The case-by-case argument is therefore incomplete at $d=3$.
\end{remark}

\paragraph{Status.} Recorded in the registry as P031\_2\_c (confirmed-load-bearing), per A237 and A295. The $d=3$, $k=4$ monodromy case remains open: until $\psi$ and its stratification are specified, the lemma is incomplete at that case. Ledger: Paper 40.

\begin{lemma}[Hopf frame polynomial degree]\label{lem:degree}
The Hopf-compatible frame condition at a $d$-dimensional stratum yields a
polynomial equation of degree $k = d+1$ in each modulus coordinate.
\end{lemma}

\begin{proof}
The Hopf line bundle $H \to S^2$ has first Chern number $c_1(H) = 1$.
At a $d$-dimensional stratum, the frame condition for $k = d+1$ intersecting
sheets requires the existence of a consistent global section of the
tensor-power bundle $H^{\otimes k}$, which has
$c_1(H^{\otimes k}) = k = d+1$.
By degree theory (Riemann--Roch on $S^2$), a generic section of $H^{\otimes k}$
has exactly $k$ zeros.
On the Hopf fibre $S^1 \cong \mathrm{U}(1)$, these zeros are the roots of a
polynomial of degree $k$ in the real coordinate $t = \mathrm{Re}(z_0)$,
$z_0 \in S^1$.

\medskip
\noindent\emph{Identification of the polynomial.}
By Lemma~\ref{lem:monodromy}, the monodromy of $H$ around the
stratum-linking loop is $e^{2\pi i/k}$ (generator of $\mathbb{Z}_k$):
the $k$ sheets contribute equally spaced phase shifts.
The frame perpendicularity condition then requires $z_0^k = \pm i$, i.e.\
$k\varphi_0 = \pi/2 + m\pi$ for integer $m$, which in the coordinate
$t = \cos\varphi_0$ reads
\[
  T_k(t) \;=\; \cos(k \arccos t) \;=\; 0,
\]
the $k$-th Chebyshev polynomial of the first kind.
This has degree~$k$ in $t$.
\end{proof}

\begin{remark}[TBS]\label{rmk:tbs:P031_3_c}
The proof of Lemma~\ref{lem:degree} (Hopf frame polynomial degree $k$) identifies the ``frame perpendicularity condition'' with the equation $T_k(t) = 0$ (the $k$-th Chebyshev polynomial) via the intermediate step $z_0^k = \pm i$. Two gaps: (i)~The term ``frame perpendicularity condition'' is not defined in this paper. The claimed equation $k\varphi_0 = \pi/2 + m\pi$ appears to encode that the $k$ sheets are perpendicular to each other in the Hopf fibre, but why ``perpendicularity'' of $k$ sheets in a $\mathrm{U}(1)$ fibre translates to equally spaced phases (rather than, say, maximally separated phases under some other metric) is not argued. (ii)~The appeal to ``Riemann--Roch on $S^2$'' gives $k$ zeros for a section of $H^{\otimes k}$ over $S^2$, but these zeros are on the base $S^2$, not on the fibre $S^1$: conflating base-space zeros (a complex surface count) with fibre-space positions requires an identification that is not established.
\end{remark}

\paragraph{Status.} Recorded in the registry as P031\_3\_c (confirmed-load-bearing), per A237 and A295. The frame-perpendicularity step remains open; both gaps named in the remark stand. Ledger: Paper 40.

\begin{lemma}[Totally-real property]\label{lem:real}
The Chebyshev polynomial $T_k(t)$ is totally real: all $k$ roots
$t_m = \cos\!\bigl(\pi(2m+1)/(2k)\bigr)$, $m = 0,\dots,k-1$,
are real, distinct, and lie in $(-1,1)$.  They are uniformly spaced
in angle by $\pi/k$.
\end{lemma}

\begin{proof}
$T_k(\cos\theta) = \cos(k\theta) = 0 \iff k\theta = \pi/2 + m\pi$
$\iff \theta_m = \pi(2m+1)/(2k)$.
Since $\theta_m \in (0,\pi)$ for $m = 0,\dots,k-1$, the roots
$t_m = \cos\theta_m$ lie in $(-1,1)$ and are distinct (cosine is
injective on $(0,\pi)$).  The angular gap $\theta_{m+1}-\theta_m = \pi/k$ is
uniform.
\end{proof}

% ── Main theorem ─────────────────────────────────────────────────────────────

\begin{theorem}[Bézout Frame Axiom]\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 $n = \max(d-1,\,1)$
polynomial equations of degree $k = d+1$ in $n$ real variables,
each equation being the Chebyshev polynomial $T_k(t_j) = 0$.
All $k^n$ solutions are real (Lemma~\ref{lem:real}), so the Bézout
bound is achieved exactly.
\end{theorem}

\begin{proof}
By Lemma~\ref{lem:dim}, the modulus space of the $d$-dimensional stratum
has dimension $n = \max(d-1,1)$.
By Lemma~\ref{lem:degree}, the frame condition in each modulus coordinate
$t_j$ is $T_k(t_j) = 0$, a polynomial of degree $k = d+1$.
The $n$ equations $T_k(t_1) = \cdots = T_k(t_n) = 0$ are decoupled
(each involves a single variable), so by Bézout's theorem the number of
common solutions over $\mathbb{C}$ is $k^n$.
By Lemma~\ref{lem:real} every solution is real, so the real count equals
the complex count:
\[
  c_d \;=\; k^n \;=\; (d+1)^{\max(d-1,1)}.
\]
Numerical verification for $d \in \{1,2,3\}$:
\[
  c_1 = 2^1 = 2,\quad c_2 = 3^1 = 3,\quad c_3 = 4^2 = 16. \qedhere
\]
\end{proof}

\begin{corollary}[Open problem reduced]\label{cor:closed}
Theorem~\ref{thm:bezout} provides the single unifying mechanism requested
in the Open Problem of \S\ref{sec:openproblem}, modulo the two open proof
steps P031\_2\_c and P031\_3\_c.
The mechanism is: the Hopf modulus space for a $d$-dimensional stratum
carries $n = \max(d-1,1)$ independent frame coordinates, each governed
by the Chebyshev equation $T_{d+1}(t) = 0$ of degree $d+1$, giving
$c_d = (d+1)^{\max(d-1,1)}$ distinct real frame orientations by Bézout.
The $\mathbb{Z}_k$ monodromy at each stratum (Lemma~\ref{lem:monodromy})
is corpus-internal for $d=1$ (antipodal $\mathbb{Z}_2$) and $d=2$
(Paper~06's $L(3,1) = S^3/\mathbb{Z}_3$); the $d=3$ case, via the diagonal
$\mathbb{Z}_4$ action in Paper~28 \S3.2, remains open (P031\_2\_c), as does
the frame-perpendicularity step of Lemma~\ref{lem:degree} (P031\_3\_c).
\end{corollary}

% ── Lineage ──────────────────────────────────────────────────────────────────

\begin{remark}[Lineage: Paper~08 heuristics as static shadows]%
\label{rem:lineage}
Each of Paper~08's three heuristic arguments can be read as a static
projection of the dynamic Bézout count.

\begin{center}
\begin{tabular}{cllp{5cm}}
\toprule
$d$ & Paper~08 heuristic & Bézout mechanism & Connection \\
\midrule
$1$ & two tangency points
    & two roots of $T_2(t)=0$
    & A tangency point is a position in the $\mathbb{Z}_2$ monodromy
      orbit where the sheet is aligned with its monodromy image.
      Counted from outside (statically): two tangent directions.
      Counted dynamically: two equispaced fibre positions. Same object. \\[4pt]
$2$ & $\dim(\mathrm{SO}(3)) = 3$
    & three roots of $T_3(t)=0$
    & Three sheets meeting at the 2D stratum label the three generators
      of the $\mathrm{SO}(3)$ symmetry of the $S^2$ intersection.
      Paper~08 counts the static symmetry dimension; the Bézout mechanism
      counts the sheets directly.  The numbers agree because the sheet
      count $k = d+1 = 3$ equals the Lie algebra dimension of
      $\mathrm{SO}(k) = \mathrm{SO}(3)$ at this stratum. \\[4pt]
$3$ & $2^4$ ambient orientations
    & $4^2 = k^n$ monodromy orbits
    & These are numerically equal but mechanistically distinct
      (Remark~\ref{rem:coincidence}, Remark~\ref{rem:static_dyn_dist}).
      The static count labels four frozen ambient dimensions with $\pm$ signs.
      The dynamic count records the $4^2$ positions the Hopf frame
      occupies as the $\mathbb{Z}_4$ monodromy acts on two modulus slots. \\
\bottomrule
\end{tabular}
\end{center}
\end{remark}

% ── Static vs. dynamic ───────────────────────────────────────────────────────

\begin{remark}[Static vs.\ dynamic: why the dynamic count is primary]%
\label{rem:static_dyn_dist}
Paper~08's $2^4$ argument for $c_3 = 16$ is static: it assigns $\pm$ labels
to the four fixed axes of the ambient $B^4$ and counts the resulting
$2^4 = 16$ labellings.  No process is involved; the geometry is frozen.

The Bézout argument is dynamic: it asks how many positions the Hopf fibre
element occupies when parallel-transported around the stratum-linking loop
under the $\mathbb{Z}_4$ monodromy.  That transport is motion: it
requires the geometry to pass through itself.

Paper~08 \S4 establishes that self-observation is impossible in a static
geometry: ``a static geometry cannot have information propagation, cannot
have causality, cannot observe itself.''  Applied to the counting problem,
this principle selects the dynamic mechanism as primary.  The static $2^4$
argument is compatible with the correct answer ($c_3 = 16$) because
$4^2 = 2^4$ happens to hold numerically, but it arrives there without
the motion that Paper~08 \S4 identifies as necessary for observation.

The Nicomachus factorisation reinforces this.  The Nicomachus weights are
$k^{n-1} = c_d / k$, i.e.\ the dynamic count $c_d$ averaged over one frame
slot by the factor $1/k$.  Under the static $2^4$ factorisation, the
corresponding weight for $d=3$ would be $2^3 = 8$, not $4^1 = 4$; only
the dynamic $4^2$ factorisation is consistent with $\Omega = \alpha^{-1}$.
The monad identity thus independently confirms that the Bézout (dynamic)
mechanism is the correct one, not the ambient-orientation (static) one.
\end{remark}

% ── Summary of resolution ────────────────────────────────────────────────────

\begin{remark}[Resolution complete modulo P031\_2\_c and P031\_3\_c]
Lemma~\ref{lem:monodromy} supplies the $\mathbb{Z}_k$ monodromy that was the
remaining step: the $k = d+1$ sheets at a $d$-dimensional stratum produce
monodromy $e^{2\pi i/k}$, derived corpus-internally for $d=1$ (the
$\mathbb{Z}_2$ antipodal map) and $d=2$ (Paper~06's $L(3,1)$ lens space);
the $d=3$ case, argued from the diagonal $\mathbb{Z}_4$ action on
$S^3 \subset \mathbb{C}^2$ from Paper~28 \S3.2, remains open (P031\_2\_c),
as does the frame-perpendicularity step of Lemma~\ref{lem:degree}
(P031\_3\_c).
Together, Lemmas~\ref{lem:dim}--\ref{lem:real} and Theorem~\ref{thm:bezout}
give a proof of Proposition~\ref{prop:unified} that is complete modulo
P031\_2\_c and P031\_3\_c.
Once both steps close, the Open Problem of \S\ref{sec:openproblem} is fully resolved;
until then it stands reduced to those two steps.
FRAME-AXIOM is no longer an asserted conjecture but a conditional result:
the regularity $c_d = (d+1)^{\max(d-1,1)}$ is a theorem modulo the two
named open steps.
\end{remark}

\end{document}
