% Paper 33: Self-Intersection Weights and Monad Closure — TOE canon. Léon Fernando Vlegels, CC BY 4.0.
% Status: current canon with one confirmed false formula: Vol(S^{d-1}) = pi^d, as printed in this paper, is wrong at
% every d (correct: 2*pi^{d/2}/Gamma(d/2)); downstream uses are affected and recorded. The Hopf absorption claim is
% structural and unverified.
% Registry: P033_2 (confirmed-load-bearing), P033_3_c (confirmed-load-bearing) — ledger: Paper 40 and addenda/verify/tbs_registry.json.
% Verifier: verify_P033.py (historical series; includes expected-fail checks that document recorded gaps).

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% Shorthand
\newcommand{\alphinv}{\alpha^{-1}}
\newcommand{\dplus}[1]{(#1)_+}

\title{Self-Intersection Weights and Monad Closure: \\[0.5em]
\large A Derivation of $\alpha^{-1}$ via the Nicomachus Structure}

\author{L\'{e}on Fernando Vlegels \\
\small Independent Researcher}
\date{2026-04}

\begin{document}
\maketitle
\tableofcontents
\newpage

\begin{abstract}
We establish a new derivation of the fine-structure constant
$\alphinv = 4\pi^3 + \pi^2 + \pi$ by connecting the self-intersection
counts of Paper 08 to a general formula for the density coefficients.
The main result is:
\[
  \alphinv \;=\; \sum_{d=1}^{3} (d+1)^{\max(d-2,\,0)} \cdot \pi^d
    \;=\; \pi \;+\; \pi^2 \;+\; 4\pi^3,
\]
where the integer coefficients $(d+1)^{\max(d-2,0)} \in \{1,1,4\}$ arise
from the self-intersection counts $c_d$ of Paper 08 after dividing by
the integration factor $k = d+1$.  A Corollary identifies this as a
Hopf-fibration factorisation: the self-intersection weight $k^{\max(d-1,1)}$
decomposes into an algebraic factor $k$ (absorbed by integration) and a
geometric screening factor $k^{\max(d-2,0)}$ (the Hopf fibration acting
on the self-intersection structure).  The total exponent at each dimension
is always $3$, a cubic structure that connects to the Nicomachus
identity for sums of cubes.  Paper~32 of this volume is cited as a
companion result.
\end{abstract}

% -----------------------------------------------------------------------
\section{Introduction}
% -----------------------------------------------------------------------

Most physical formalisms treat observation as a primitive.  An observer
is posited, a measurement postulate is attached to it, and the geometry
of the theory is then arranged around the posited act.  The corpus takes
the opposite route.  Here observation is not a primitive but a geometric
event: an observer is a projection from the four-ball $B^4$ into its
boundary, and an act of observation is a coincidence of that projection
with itself.  Nothing is observed from outside the geometry, because
there is no outside; the geometry meets itself, and the meeting is the
act.  On this reading, every quantity that depends on how often the
geometry can meet itself is a quantity about observation, and the
fine-structure constant turns out to be exactly such a quantity.

The fine-structure constant $\alphinv = 4\pi^3 + \pi^2 + \pi \approx 137.036$
is the single most precisely measured dimensionless constant in physics.
Paper 01 of this corpus \cite{Paper01} derives it as the integral of the
cubic phase density
\[
  \rho(x) \;=\; 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x, \qquad x \in [0,1].
\]
The three integer coefficients $(16, 3, 2)$ were identified in Paper 08
\cite{Paper08} as self-intersection counts arising from the $(16,3,2)$
triple, the number of self-configurations of observer projections in
$B^4$.  The formula $c_d = (d+1)^{\max(d-1,1)}$ (Theorem 3.1 of Paper
08) encodes these counts for $d = 1, 2, 3$:
\begin{align*}
  d=3\colon&\quad k = 4,\quad c_3 = 4^2 = 16, \\
  d=2\colon&\quad k = 3,\quad c_2 = 3^1 = 3, \\
  d=1\colon&\quad k = 2,\quad c_1 = 2^1 = 2.
\end{align*}

The present paper asks: \emph{what formula governs the coefficients
that appear in $\alphinv$ after integration?}  That is, why are the
integrated contributions $4\pi^3$, $\pi^2$, and $\pi$, and not some
other multiples?

The question is sharper than it first appears.  The counts $c_d$ are
fixed by Paper 08; the integral over $[0,1]$ is fixed by Paper 01; so
the integrated coefficients are determined.  What is not given in
advance is whether the determined values have a closed form of their
own, one expressible in the same variables $k = d+1$ that govern the
counts.  If they do, the passage from the density to the constant is
not mere arithmetic but a second instance of the same dimensional
grading, and the structure of $\alphinv$ can be read at the level of
the coefficients rather than reconstructed from the integral each time.

The answer is a new formula, $c_d/k = k^{\max(d-2,0)}$, which we call
the \emph{screened self-intersection weight}.  The screening factor
$k^{\max(d-2,0)}$ arises from the Hopf fibration absorbing one power of
$k$ from the self-intersection count: the integration over $[0,1]$
divides out exactly the factor $k$.  The resulting structure has a
total exponent of $3$ at every dimension, which we recognise as the
Nicomachus cubic property.

The paper proceeds in five steps.
Section~\ref{sec:background} recalls the self-intersection counting of
Paper 08 and develops the reading of those counts as literal acts of
observation, including the sense in which each act installs a reference
frame on the boundary.  Section~\ref{sec:main} proves the screened
weight formula and recovers $\alphinv$ from it.
Section~\ref{sec:hopf} factorises the self-intersection weight into
three exponents that always sum to $3$, and records, honestly, where
the geometric reading of that factorisation outruns what the arithmetic
proves.  Section~\ref{sec:staticmotion} separates the result into a
static spectrum and a dynamic spectrum, with the status of the
geometric-measure claim recorded in the registry note attached there.
Section~\ref{sec:nicomachus} states the cubic correspondence that gives
the paper its title, and Section~\ref{sec:corpus} places the result
among its neighbours in the corpus.

% -----------------------------------------------------------------------
\section{Background: Self-Intersection Counting}
\label{sec:background}
% -----------------------------------------------------------------------

We recall the relevant result from Paper 08.  One point of discipline
first: the counts used below are \emph{imported}, not derived.  Paper 08
constructs the observer-projection counting; the present paper takes
the counts as given and works out their consequences.  Nothing in what
follows re-establishes the counting itself, and a reader who doubts the
counts should direct that doubt at Paper 08, where it belongs.

\begin{definition}[Self-intersection count]
For dimension $d \in \{1,2,3\}$, let $k = d+1$.  The
\emph{self-intersection count} $c_d$ is the number of self-configurations
of observer projections from $B^{d+1}$ onto its boundary $S^d$,
as defined in Paper 08, Theorem~3.1 \cite{Paper08}:
\[
  c_d \;=\; k^{\max(d-1,\,1)}.
\]
\end{definition}

Explicitly:
\[
\begin{array}{c|ccc}
  d & k = d+1 & \max(d-1,1) & c_d \\ \hline
  1 & 2 & 1 & 2^1 = 2 \\
  2 & 3 & 1 & 3^1 = 3 \\
  3 & 4 & 2 & 4^2 = 16
\end{array}
\]

These are precisely the coefficients of $\pi$, $\pi^2$, and $\pi^3$ in
$\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$, up to the dimensional
factor $\pi^d$:
\[
  \rho(x) \;=\; \sum_{d=1}^{3} c_d \cdot \pi^d \cdot x^d.
\]

\subsection{The construction, step by step}

The shape of the construction behind Definition~2.1 is worth setting
out in order, because the rest of the paper trades on it.

First, fix a dimension $d \in \{1,2,3\}$ and set $k = d+1$.  The
ambient object is the ball $B^{d+1}$, and its boundary is the sphere
$S^d$.  The ball is the bulk; the sphere is everything the bulk can
present of itself.

Second, an observer is a projection from the ball onto its boundary.
The projection carries interior points to boundary points.  It is the
act by which the bulk is rendered on the surface, and it is the only
kind of seeing the framework admits: there is no external vantage from
which $B^{d+1}$ could be inspected, so whatever is registered must be
registered on $S^d$, by the object itself.

Third, a self-configuration is a coincidence of the projection with
itself: a way in which the observer's image meets the observer.  This
is the step at which observation becomes literal self-intersection.
The projection does not report on something other than itself; it
crosses itself, and the crossing is the event.  There is no observed
system distinct from the observing one.  The geometry is both.

Fourth, the self-configurations are counted.  Paper 08 establishes
that the count is $c_d = k^{\max(d-1,1)}$, giving the values $2$, $3$,
and $16$ tabulated above.  These integers are the entire combinatorial
input to the present paper.

\subsection{Frame installation}

Each coincidence in the third step is localised.  It singles out a
place on the boundary together with the directions along which the
images cross there.  A marked point equipped with crossing directions
is the minimal datum of a reference frame: a place from which, and an
orientation in which, subsequent structure can be registered.  On the
reading the corpus attaches to the counts, then, the statement that
every act of observation installs a reference frame is not a metaphor
draped over the construction; the intersection datum \emph{is} the
frame.  Before the first coincidence the sphere carries no marked
point and no preferred direction.  After it, the sphere carries both.

This reading also fixes what motion means here.  Once frames exist,
relating one frame to another is transport, and transport across the
sphere is motion: the continuous sweep of phase between one installed
frame and the next.  The counts $c_d$ are indifferent to that sweep;
they record how many coincidences the geometry admits, not how the
geometry moves between them.  The division of labour between the
discrete count and the continuous sweep is made precise in
Section~\ref{sec:staticmotion}, where it appears as the split between
the static factors and the $\pi^d$ factors.  We flag now, and the
registry note in that section records, that the geometric measure
assigned to the $\pi^d$ factors there is a recorded error; the split
itself does not depend on it.

% -----------------------------------------------------------------------
\section{The Main Theorem}
\label{sec:main}
% -----------------------------------------------------------------------

The passage from the counts to the constant is the integral of Paper 01.
What the following theorem adds is the closed form of what survives the
integral: each count $c_d$ enters the constant divided by exactly
$k = d+1$, and the quotient is again a pure power of $k$.

\begin{theorem}[Screened self-intersection formula]\label{thm:main}
For each $d \in \{1,2,3\}$, let $k = d+1$.  Define the
\emph{screened weight}
\[
  w_d \;=\; \frac{c_d}{k} \;=\; k^{\max(d-2,\,0)}.
\]
Then
\[
  \alphinv \;=\; \int_0^1 \rho(x)\,dx
    \;=\; \sum_{d=1}^{3} w_d \cdot \pi^d
    \;=\; \pi \;+\; \pi^2 \;+\; 4\pi^3.
\]
\end{theorem}

\begin{proof}
Integrating term by term:
\[
  \int_0^1 c_d \cdot \pi^d \cdot x^d\,dx
    \;=\; c_d \cdot \pi^d \cdot \frac{1}{d+1}
    \;=\; c_d \cdot \pi^d \cdot \frac{1}{k}
    \;=\; w_d \cdot \pi^d.
\]
Computing $w_d$ for each $d$:
\[
\begin{array}{c|ccccc}
  d & k & c_d & 1/k & w_d = c_d/k & k^{\max(d-2,0)} \\ \hline
  1 & 2 & 2   & 1/2 & 1           & 2^0 = 1 \\
  2 & 3 & 3   & 1/3 & 1           & 3^0 = 1 \\
  3 & 4 & 16  & 1/4 & 4           & 4^1 = 4
\end{array}
\]
Summing:
\[
  \alphinv \;=\; 1 \cdot \pi + 1 \cdot \pi^2 + 4 \cdot \pi^3
    \;=\; \pi + \pi^2 + 4\pi^3
    \;=\; 4\pi^3 + \pi^2 + \pi.
\]
\end{proof}

\begin{remark}
The screened weights $w_d \in \{1, 1, 4\}$ are determined entirely by
the self-intersection counts and the dimension.  No free parameters appear.
The formula $w_d = k^{\max(d-2,0)}$ is a dimensionally graded power law
in $k = d+1$, with exponent zero (weight 1) at dimensions $d \le 2$
and exponent one (weight $k$) at $d = 3$.
\end{remark}

Two boundaries of the theorem should be stated as plainly as the
theorem itself.  First, the proof is arithmetic: term-by-term
integration of a polynomial, followed by the observation that the
quotients $c_d/k$ close into the single expression $k^{\max(d-2,0)}$.
Nothing geometric is invoked in the proof, and nothing geometric is
established by it.  Second, the theorem does not explain \emph{why}
the divisor produced by integration coincides with one power of the
base $k$ that generates the counts.  That coincidence is the hinge of
the paper.  The next section gives it a name and a proposed mechanism,
and records exactly how much of the mechanism is proven.

% -----------------------------------------------------------------------
\section{Corollary: Hopf Factorisation}
\label{sec:hopf}
% -----------------------------------------------------------------------

The transition from $c_d$ to $w_d$ has a natural geometric interpretation.

There are two ways to read the division by $k$.  The algebraic reading
is the one the proof of Theorem~\ref{thm:main} already contains:
$\int_0^1 x^d\,dx = 1/(d+1) = 1/k$, so integration removes one power of
$k$ as a matter of calculus.  The geometric reading proposes that this
removal is the arithmetic shadow of the Hopf fibration
$S^1 \to S^3 \to S^2$ acting on the self-intersection structure, the
fibre absorbing one unit of the algebraic capacity at each level.  The
corollary below is a statement of the first kind: it is exponent
bookkeeping, and it is exact.  The second reading is an organising
assertion layered on top of the bookkeeping; its status is recorded at
the end of this section and should be read before the assertion is
relied on.

\begin{corollary}[Hopf factorisation of self-intersection weights]
\label{cor:hopf}
For each $d \in \{1,2,3\}$, the self-intersection weight $c_d$ satisfies
\[
  c_d \cdot k^{\min(4-d,\,2)} \;=\; k^3,
\]
where:
\begin{itemize}[nosep]
  \item $k^{\max(d-2,0)}$ is the \emph{screened weight} $w_d$ (the
        contribution to $\alphinv$);
  \item $k^1$ is the \emph{integration factor} (absorbed by
        $\int_0^1 x^d\,dx = 1/k$);
  \item $k^{\min(4-d,2)}$ is the \emph{Hopf complement factor}
        (the portion of the algebraic capacity absorbed by the fibration).
\end{itemize}
In all cases the three exponents sum to exactly $3$:
\[
  \max(d-2,0) + 1 + \min(4-d,2) \;=\; 3.
\]
\end{corollary}

\begin{proof}
For each $d$ we verify the exponent sum and the product identity.

\textit{Case $d=1$, $k=2$:}
Exponents: $\max(-1,0) + 1 + \min(3,2) = 0 + 1 + 2 = 3$.
Product: $2^0 \cdot 2^1 \cdot 2^2 = 1 \cdot 2 \cdot 4 = 8$. But $c_1 = 2$\ldots

We correct the statement: the three-way factorisation holds at the level
of the \emph{exponent sum}, not as a literal product identity for $c_d$.
Explicitly:

\[
\begin{array}{c|ccc|c|c}
  d & \max(d-2,0) & 1 & \min(4-d,2) & \text{sum} & c_d \\ \hline
  1 & 0 & 1 & 2 & 3 & k^{\max(d-1,1)} = 2^1 = 2 \\
  2 & 0 & 1 & 2 & 3 & k^{\max(d-1,1)} = 3^1 = 3 \\
  3 & 1 & 1 & 1 & 3 & k^{\max(d-1,1)} = 4^2 = 16
\end{array}
\]

The exponent of $k$ in $c_d = k^{\max(d-1,1)}$ equals
$\max(d-1,1) = \max(d-2,0) + 1 + 0$ for $d \ge 2$ and $= 1$ for $d=1$.
The ``Hopf complement'' exponent $\min(4-d,2)$ counts how much of the
full cubic capacity $k^3$ is \emph{not} expressed in $c_d$:
\[
  k^3 \;=\; c_d \cdot k^{3 - \max(d-1,1)} \;=\; c_d \cdot k^{\min(4-d,2)}.
\]
Thus $c_d \cdot k^{\min(4-d,2)} = k^3$ in all cases, and the exponent sum
$\max(d-1,1) + \min(4-d,2) = 3$ holds.  Including the integration
factor $1/k$:
\[
  w_d \cdot k^{\min(4-d,2)} \;=\; \frac{c_d}{k} \cdot k^{\min(4-d,2)}
    \;=\; k^{\max(d-2,0)} \cdot k^{\min(4-d,2)}
    \;=\; k^{3-1},
\]
so the screened weight, integration factor, and complement factor
together reconstruct $k^3$ (the full cubic capacity) at every dimension.
\end{proof}

The proof's own correction deserves emphasis rather than concealment.
The corollary as first stated reads as a product identity for $c_d$;
the case $d=1$ refutes that reading immediately, and the proof says so
in the open before restating the claim at the level where it is true,
the level of exponent sums.  What survives is exact and is verified
case by case in the table: the exponent carried by $c_d$, the exponent
removed by integration, and the complement exponent always total $3$,
and the product identity $c_d \cdot k^{\min(4-d,2)} = k^3$ holds in the
corrected form.  A reader who wants only what is proven should take
the exponent sum and the corrected product identity and nothing more.

\begin{remark}[Geometric interpretation]
The Hopf fibration $S^1 \to S^3 \to S^2$ acts on the observer projections
in $B^4$.  The fibre $S^1$ absorbs one ``unit'' of the algebraic
self-intersection structure at each dimensional level.  The integration
over $[0,1]$ (i.e.\ division by $k$) reflects this absorption: the
Hopf fibre dimension is $1$, and the integration factor is $1/k = 1/(d+1)$,
exactly the reciprocal of the base-sphere dimension plus one.
What remains after absorption is the screened weight $w_d$, which
carries the dimensional information into $\alphinv$.
\end{remark}

\begin{remark}[TBS]\label{rmk:tbs:P033_3_c}
The claim that the Hopf fibration \emph{absorbs} one power of $k$ from the self-intersection count is a structural assertion, not a construction. The self-intersection counts $c_d$ are imported from Paper~08 (itself flagged), and the identification of the integration factor $1/(d+1)$ with the Hopf fibre dimension is only an analogy. A rigorous derivation of the Hopf absorption mechanism, one establishing that a specific geometric map reduces $c_d$ to $w_d = c_d/k$, does not appear here or in any later paper in the corpus.
\end{remark}

\paragraph{Status.} Recorded in the registry as P033\_3\_c (confirmed-load-bearing), confirmed by A295. The Hopf absorption claim is structural and unverified: no construction reducing $c_d$ to $w_d = c_d/k$ has been supplied. The claim should be read as an organising assertion, not a derived result. Ledger: Paper 40.

% -----------------------------------------------------------------------
\section{The Static-Motion Interpretation}
\label{sec:staticmotion}
% -----------------------------------------------------------------------

The factorisation of Corollary~\ref{cor:hopf} admits a natural
geometric decomposition into static (algebraic) and dynamic (geometric)
factors.

The decomposition returns to the reading developed in
Section~\ref{sec:background}.  The integer factors are the installed
frames: discrete, combinatorial, indifferent to how anything moves.
The $\pi^d$ factors are the transport between frames: the continuous
phase swept in moving across the sphere from one coincidence to the
next.  Observation supplies the frames; motion is the transport that
relates them; and the constant is what the two spectra produce
together.  One element of the proposition's justification, the
identification of $\pi^d$ as the volume of $S^{d-1}$, is the subject of
the registry note that follows it, and the note governs how that
element is to be read.

\begin{proposition}[Static-motion decomposition]\label{prop:staticmotion}
In the formula $\alphinv = \sum_{d=1}^3 w_d \cdot \pi^d$:
\begin{enumerate}[label=(\roman*),nosep]
  \item The factors $k^{\max(d-2,0)} \in \{1,1,4\}$ are \emph{static}:
        they count the algebraic capacity of the self-intersection
        structure: discrete, combinatorial, dimensionally graded.
  \item The factors $\pi^d$ are \emph{dynamic}: they measure the
        geometric volume of $S^{d-1}$ (the $(d-1)$-sphere of unit
        radius), which governs the phase dynamics of observers on $S^3$.
\end{enumerate}
The fine-structure constant $\alphinv$ is the inner product of the
static capacity spectrum with the dynamic phase-volume spectrum.
\end{proposition}

\begin{proof}[Justification]
The factor $\pi^d$ appears naturally as the $d$-dimensional
solid-angle contribution: the volume of the $d$-dimensional unit ball
is $V_d = \pi^{d/2}/\Gamma(d/2+1)$, giving surface areas proportional
to $\pi^{\lfloor d/2 \rfloor}$ (up to rational factors depending on $d$).
For the restricted set $d \in \{1,2,3\}$ relevant here, $\pi^d$ is the
exact measure.  The static factors $k^{\max(d-2,0)}$ are integers determined
by self-intersection counting (Paper 08) and carry no geometric phase
information.  Their product $w_d \cdot \pi^d$ is therefore a
``static capacity $\times$ geometric measure'' factorisation at each
layer $d$.
\end{proof}

\begin{remark}[TBS]\label{rmk:tbs:P033_2}
The formula $\mathrm{Vol}(S^{d-1})=\pi^d$ is false: for $d=1$, $\mathrm{Vol}(S^0)=2\neq\pi$; for $d=2$, $\mathrm{Vol}(S^1)=2\pi\neq\pi^2$; for $d=3$, $\mathrm{Vol}(S^2)=4\pi\neq\pi^3$. The correct formula $\mathrm{Vol}(S^{d-1})=2\pi^{d/2}/\Gamma(d/2)$ must be substituted throughout.
\end{remark}

\paragraph{Status.} The formula $\mathrm{Vol}(S^{d-1})=\pi^d$ as printed and used in this paper is false, and this is confirmed at every $d$, not merely suspected; the correct identity is $\mathrm{Vol}(S^{d-1})=2\pi^{d/2}/\Gamma(d/2)$. The registry records this as P033\_2 (confirmed-load-bearing, A295), and the downstream uses of $\pi^d$ as a sphere volume are affected and recorded with it. The printed formula stays in the text as the record of what was asserted; it must not be cited as a volume identity. Ledger: Paper 40.

What the proposition retains after the registry note is its
load-bearing content: the split itself.  The factors
$k^{\max(d-2,0)}$ are integers fixed by counting and carry no phase;
the factors $\pi^d$ are the transcendental part and carry all of it.
That separation, static count against continuous phase, does not rest
on reading $\pi^d$ as a sphere volume and survives the note intact.
What does not survive is any use of the proposition as a source for
sphere volumes; per the note, the printed identification stands in the
text as a record of what was asserted and is not to be cited.

\begin{remark}
This interpretation echoes the Hopf fibration: the Hopf map sends
$S^3$ (a geometric, phase-carrying object) to $S^2$ (the base), with
fibre $S^1$ carrying the remaining phase.  The static factors track
the $S^2$ structure; the dynamic $\pi^d$ factors track the full $S^3$
phase.  The Hopf fibration, as the geometric mechanism behind the
TOE layer structure \cite{Paper01}, naturally enforces this
static-dynamic split.
\end{remark}

% -----------------------------------------------------------------------
\section{The Nicomachus Connection}
\label{sec:nicomachus}
% -----------------------------------------------------------------------

We now explain the title's reference to Nicomachus of Gerasa.  The
ancient \emph{Nicomachus identity} states
\[
  \sum_{j=1}^{n} j^3 \;=\; \left(\sum_{j=1}^{n} j\right)^2.
\]
In particular, the partial sums of cubes $1^3 + 2^3 + \cdots$ equal
perfect squares.  The sequence of cubes $1, 8, 27, 64, \ldots$ is
central to Nicomachus's arithmetic treatise.

The connection to our formula arises from Corollary~\ref{cor:hopf}:
at every dimension $d \in \{1,2,3\}$, the three exponents of $k$ sum
to exactly $3$.  This means the full algebraic capacity at each level
is \emph{cubic in $k$}: the product $c_d \cdot k^{\min(4-d,2)} = k^3$.

In other words, the generating object at each dimensional level is
$k^3 = (d+1)^3$, the cube of the next integer.  The three values
$k^3 \in \{8, 27, 64\}$ for $d \in \{1, 2, 3\}$ are exactly the second,
third, and fourth cubes in the Nicomachus sequence $\{1, 8, 27, 64, \ldots\}$.

The scope of the correspondence should be stated with the same care as
its content.  What the paper uses is the cube sequence: the invariance
of the total exponent at $3$, hence the appearance of $8$, $27$, and
$64$ as the full capacities of the three levels.  The sum-of-cubes
identity displayed above is stated as the context in which that
sequence lives, not as a step in any derivation here.  The
correspondence is an organising one: it names the pattern the
factorisation exhibits and ties it to the oldest catalogue of that
pattern, and it claims no more than that.

\begin{proposition}[Nicomachus correspondence]\label{prop:nic}
The self-intersection structure of $B^4$ at dimensions $d = 1, 2, 3$
is governed by Nicomachus cubes: the full algebraic capacity at each
layer $d$ is $k^3 = (d+1)^3$, decomposing as
\[
  (d+1)^3 \;=\; \underbrace{(d+1)^{\max(d-2,0)}}_{\text{screened weight}}
    \;\times\;
    \underbrace{(d+1)^1}_{\text{integration}}
    \;\times\;
    \underbrace{(d+1)^{\min(4-d,2)}}_{\text{Hopf complement}}.
\]
\end{proposition}

\begin{remark}[Why three layers and three cubes]
The three-layer structure of the TOE (bulk $d=3$, boundary $d=2$, edge $d=1$)
corresponds precisely to the three Nicomachus cubes $(d+1)^3$ for
$d \in \{1,2,3\}$.  The Hopf fibration distributes the cubic capacity
unequally across the screened weight, the integration, and the complement;
but the total, the cube, is invariant.  This is a consequence of
the S$^3$ phase structure: any path around the fibre must restore the
full cubic period.
\end{remark}

% -----------------------------------------------------------------------
\section{Relation to the Corpus}
\label{sec:corpus}
% -----------------------------------------------------------------------

The result sits at a junction of several papers, and the junction is
worth mapping explicitly.

\emph{Paper 08} supplies the input.  The counts $c_d$ and the formula
$c_d = k^{\max(d-1,1)}$ are its Theorem 3.1; everything in the present
paper is downstream of that counting, and the present paper adds no
independent derivation of it.

\emph{Paper 01} supplies the route this paper reorganises.  There,
$\alphinv$ is the integral of the density $\rho(x)$; here, the same
constant is read off the screened weights without performing the
integral, the integration having been absorbed into the single division
by $k$.

\emph{Paper 31} is the companion statement of the same closure.  It
writes the identity with the positive-part notation, as
$\sum_{d=1}^{3} k^{\dplus{d-2}} \cdot \pi^d$, and frames the cubes
$8$, $27$, $64$ as the algebraic capacity of each self-intersection
stratum, the same role they play in
Proposition~\ref{prop:nic}.  The two papers state one structure from
two directions: Paper 31 from the arithmetic of the cubic sequence,
the present paper from the self-intersection construction.

\emph{Paper 34} develops the monad named in this paper's title in its
categorical form: the Kleisli monad over $S^3$, with its unit, its
bind, and the monad laws.  Where Paper 34 gives the monad its
compositional face, the present paper gives it its
intersection-geometric face: the carrier on which the monad acts is
the same sphere whose self-intersections are counted here.

\emph{Paper i}, the introductory paper of this volume, names the
present construction directly.  Its First Act section states that
noticing is not a report on a creation that already happened but the
act by which something definite is there at all, and it identifies
observation as literal self-intersection as this paper's construction
of that fact.  Read through that section, the content of
Section~\ref{sec:background} is the formal counterpart of the first
act: the coincidence of the projection with itself is the noticing,
the noticing installs the frame, and the constant assembled from the
counts of such coincidences is the arithmetic residue of the acts.

\emph{Paper 32} shares the geometric ancestor; its role is taken up in
the conclusion.

% -----------------------------------------------------------------------
\section{Conclusion}
% -----------------------------------------------------------------------

We have derived the fine-structure constant $\alphinv = 4\pi^3 + \pi^2 + \pi$
from the self-intersection counts of Paper 08 via the screened weight
formula $w_d = k^{\max(d-2,0)}$, where $k = d+1$.  The derivation is
a new route to $\alphinv$ that is complementary to the direct integral of
Paper 01: rather than computing $\int_0^1 \rho(x)\,dx$, it expresses the
result as the inner product of a static capacity spectrum and a dynamic
phase-volume spectrum.

The Hopf factorisation (Corollary~\ref{cor:hopf}) shows that the
self-intersection weight $c_d$ decomposes into three factors whose
exponents always sum to $3$.  This cubic invariance is the Nicomachus
connection: the generating object at each layer is $(d+1)^3$, the cube
of the next integer, and the Hopf fibration distributes this cube
across the three factors of the factorisation.

Combined with Paper 32 \cite{Paper32}, which derives $\Lambda_0 = 1-\pi^2/32$
from the corner residual of $B^4$, the present result completes a
circuit: Paper 08 provides the self-intersection counts; the present
paper recovers $\alphinv$ from those counts; and Paper 32 uses the same
$B^4$ geometry to predict the cosmological constant.  The three results
share a common geometric ancestor, the unit $4$-ball and its embedding
in $\mathbb{R}^4$, and are mutually consistent.

The common ancestor deserves a closing word, because it is the claim
that binds the circuit.  The ball $B^4$ is one object, and each of the
three papers reads a different aspect of it.  Paper 08 reads its
capacity for self-coincidence: how many ways the projections of the
ball onto its boundary can meet themselves.  The present paper reads
what that capacity becomes under integration: the screened weights and,
through them, the constant.  Paper 32 reads the residual left where the
ball meets the ambient $\mathbb{R}^4$ in which it is embedded: the
corner, and the constant that the corner predicts.  None of the three
readings borrows its result from the others; what they share is the
object read.  That is the sense, and the only sense claimed here, in
which the circuit closes: one geometry, three measurements taken on it,
each landing where the others require.

Finally, the interpretive frame of the paper can be restated in one
sentence.  Observation is self-intersection; each act of observation
installs a reference frame on the boundary; transport between frames is
motion; and the fine-structure constant is the inner product of the
count of acts with the phase swept between them.  Where that sentence
exceeds what is proven, the registry notes in
Sections~\ref{sec:hopf} and~\ref{sec:staticmotion} say exactly where
and by how much, and the arithmetic that remains when the
interpretation is set aside, the screened weights, the exponent sum,
and the recovery of $\alphinv$, stands on its own.

% -----------------------------------------------------------------------
\begin{thebibliography}{99}

\bibitem{Paper00}
L.~F.\ Vlegels, \textit{The Monad Rosetta Map: $\emptyset \equiv 0 \equiv 1 \equiv \infty$},
This volume (2025).

\bibitem{Paper01}
L.~F.\ Vlegels, \textit{The Perfect Stable Sphere: $\rho(x) = 16\pi^3x^3 + 3\pi^2x^2 + 2\pi x$, $\alpha^{-1} = 4\pi^3 + \pi^2 + \pi$},
This volume (2025).

\bibitem{Paper04}
L.~F.\ Vlegels, \textit{Three-Layer Ontology of the $(B^4, S^3)$ Framework},
This volume (2025).

\bibitem{Paper08}
L.~F.\ Vlegels, \textit{Time-Observation Bootstrap: the $(16,3,2)$ Triple from Primes},
This volume (2025).

\bibitem{Paper18}
L.~F.\ Vlegels, \textit{The Master Operator $\hat{O}$ and Its Spectral Decomposition},
This volume (2025).

\bibitem{Paper32}
L.~F.\ Vlegels, \textit{Dark Energy as Geometric Residual: $\Lambda_0 = 1 - \pi^2/32$ from the $B^4$ Corner Structure},
This volume (2026).

\end{thebibliography}

\end{document}
