% Addendum 144: Uniqueness of alpha^{-1} — A Choice-Point Analysis
% L. F. Vlegels, Independent Researcher
% Follows Papers 01, 03, 18, 29, 143 of the LumenOS TOE corpus.

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\title{\textbf{Addendum 144: Uniqueness of $\alpha^{-1}$}\\[4pt]
\large A Choice-Point Analysis of the $(B^4,S^3)$ Derivation}
\author{L. F. Vlegels \\ Independent Researcher}
\date{May 2026}

\begin{document}
\maketitle
\label{addendum:144}

\begin{abstract}
Paper~143 nominated the uniqueness of $\alpha^{-1}=4\pi^3+\pi^2+\pi$ as the sharpest
candidate G\"{o}del sentence (G1): no other admissible $(B^4,S^3)$ density choice
yields the same fine-structure constant.
We read Papers~01 and~03 directly and locate every point at which a choice is made.
Four independent choice points are identified; two cannot be justified from $S^3$
geometry alone.
We state the minimal additional axiom needed to close the gap, assess its
naturalness, and evaluate whether $J_3(\mathbb{O})$ representation theory provides
the missing constraint.
The outcome is a precise failure-mode report: G1 remains open, but the two
load-bearing gaps are now named.
\end{abstract}

\section{Setup}
\label{sec:setup}

The G1 candidate asserts: under any admissible density on $(B^4,S^3)$ consistent
with the Hopf fibration and the three-layer ontology (bulk, boundary, edge),
the integral defining $\alpha^{-1}$ is forced to equal $4\pi^3+\pi^2+\pi$.

The source derivation (P01 \S2, P03 \S2) proceeds as follows.
Introduce the normalized radial coordinate $x\in[0,1]$ on $B^4$.
Define the cubic phase density
\begin{equation}
  \rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x,
  \label{eq:rho}
\end{equation}
and integrate with flat (Lebesgue) measure to obtain
\begin{equation}
  \int_0^1 \rho(x)\,dx
    = \tfrac{16\pi^3}{4} + \tfrac{3\pi^2}{3} + \tfrac{2\pi}{2}
    = 4\pi^3 + \pi^2 + \pi
    = 137.036303\ldots
  \label{eq:alpha_integral}
\end{equation}
The experimental value is $\alpha^{-1}_{\mathrm{exp}}=137.035999084(21)$,
a discrepancy of $2.2\times10^{-6}$ ($0.00022\%$).
P01/P03 call~\eqref{eq:alpha_integral} an ``exact identity'' to
``$0.0002\%$ precision'': the precision language is correct; the exactness
language is not — this distinction is CP-1 below.

The integer coefficients $(16,3,2)$ are justified post hoc:
$16=2\cdot\dim\mathbb{O}$ from octonionic structure (four arguments, P01 \S3.1);
$3=\dim(\mathrm{adj}\,SU(2))$ from gauge-field representation theory (P01 \S3.2);
$2=\dim(\mathrm{fund}\,SU(2))$ or a $\mathbb{Z}_2$ count (P01 \S3.3, two
interpretations).

\section{The Derivation of $\alpha^{-1}$}
\label{sec:derivation}

The logical structure is a four-step construction:
\begin{enumerate}
  \item Restrict $\rho(x)$ to a degree-3 polynomial with terms $c_n x^n$,
        $n\in\{1,2,3\}$.
  \item Fix the integration measure to be flat $dx$ on $[0,1]$.
  \item Impose $\int_0^1\rho\,dx = 4\pi^3+\pi^2+\pi$ (exact).
  \item Identify $c_n = (n+1)\cdot(\text{summand for term }n)$ algebraically.
\end{enumerate}
Step~3 forces $c_3=16\pi^3$, $c_2=3\pi^2$, $c_1=2\pi$ by trivial algebra.
Step~4 then \emph{checks} that these integers match known algebraic structures.
The derivation verifies consistency; it does not derive $\alpha^{-1}$ from
the geometry independently.

\section{Choice Point Analysis}
\label{sec:choicepoints}

\begin{choicepoint}[Approximate identity treated as exact]\label{cp:exact}
The formula $4\pi^3+\pi^2+\pi=137.036303\ldots$ exceeds the experimental value
by $\Delta=3.05\times10^{-4}$ (0.22\,ppm).
Two resolutions exist: (a) the formula is the \emph{bare} theoretical value and
QED radiative corrections shift it to $\alpha^{-1}_{\mathrm{exp}}$ (analogous to
$g-2$ corrections to a tree-level formula); or (b) the formula is an
approximation to a different exact expression.
The TOE does not currently decide between (a) and (b).
G1 is not assessable until CP-1 is resolved.
\end{choicepoint}

\begin{choicepoint}[Monomial polynomial basis]\label{cp:basis}
The three-layer ontology motivates three graded contributions, but does not
uniquely select the functional form $\{x^3, x^2, x\}$.
Exponential, Bessel, or trigonometric bases all admit normalizations integrating
to $\alpha^{-1}$ with qualitatively similar layer interpretations.
The polynomial form is the minimal-degree choice consistent with the grading;
minimality is a natural additional assumption but is not forced by $S^3$ geometry.
\end{choicepoint}

\begin{choicepoint}[Flat integration measure]\label{cp:measure}
The natural volume measure on $B^4$ in radial coordinates is
$dV_4\propto r^3\,dr\propto x^3\,dx$.
Under this measure the same $\rho$ integrates to $\mu_3=77.06\ldots\neq\alpha^{-1}$.
P01 uses the volume element in one place (Approach~4 for $c_3$: $dV_4/dr=2\pi^2 r^3$
motivates the coefficient $16\pi^3$) but then integrates $\rho$ with flat $dx$.
This is an internal inconsistency: the coefficient is motivated by the volume
measure; the integral uses the flat measure.
CP-3 is the deepest structural gap in the derivation.
\end{choicepoint}

\begin{choicepoint}[Edge coefficient: two distinct justifications]\label{cp:c1}
P01 \S3.3 offers two independent motivations for $c_1=2\pi$:
$\dim(\mathrm{fund}\,SU(2))=2$ and $|\mathbb{Z}_2|=2$.
These are not equivalent (one is a complex representation dimension, the other
a group order), and no argument selects between them.
The existence of two justifications for the same number signals that neither
is a true derivation.
\end{choicepoint}

\begin{remark}[Coefficient 16: Approach~4 is circular]
Of the four arguments for $c_3=16\pi^3$, Approach~4 defines $R$ by
$2\pi^2 R^3=16\pi^3$, i.e.\ $R^3=8\pi$, then identifies $8=\dim\mathbb{O}$.
$R$ is chosen to make the coefficient work; the ``derivation'' reverses
the inference.
Approaches~1--3 (doubling, complexification, $SO(8)$ triality) give
$16=2\times8$ without circularity and should be taken as the canonical argument.
\end{remark}

\section{Uniqueness Theorem or Failure Mode}
\label{sec:uniqueness}

\begin{theorem}[Conditional uniqueness]\label{thm:conditional}
Assume:
\textup{(A1)} $\rho(x)=c_3x^3+c_2x^2+c_1x$ with $c_n=k_n\pi^n$, $k_n\in\mathbb{Z}_{>0}$;
\textup{(A2)} flat measure $dx$;
\textup{(A3)} $\alpha^{-1}=4\pi^3+\pi^2+\pi$ (exact);
\textup{(A4)} bulk$\to$degree~3, boundary$\to$degree~2, edge$\to$degree~1.
Then $\rho$ in~\eqref{eq:rho} is the unique density of this form, and
$\alpha^{-1}=4\pi^3+\pi^2+\pi$ is uniquely fixed.
\end{theorem}

\begin{proof}
Given (A3) and (A4), the constraint $c_n/(n+1)=\{4\pi^3,\pi^2,\pi\}$ for
$n=\{3,2,1\}$ uniquely determines $c_3=16\pi^3$, $c_2=3\pi^2$, $c_1=2\pi$.
The integer factors $k_n=(16,3,2)$ follow immediately, and (A1)
ensures no other polynomial of this form exists.
\end{proof}

This theorem is correct but its hypotheses include CP-2 (A1), CP-3 (A2),
and CP-1 (A3).
It is a consistency theorem, not a uniqueness proof from $S^3$ geometry alone.

\smallskip
\noindent\textbf{Failure mode (G1 statement).}
The derivation of $\alpha^{-1}$ requires the additional free choices of:
\begin{enumerate}
  \item[(i)] the flat measure $dx$ over the natural $S^3$ volume measure $x^3\,dx$;
  \item[(ii)] treating $4\pi^3+\pi^2+\pi$ as exact rather than as a 0.22\,ppm
        approximation.
\end{enumerate}
If (i) is dropped, the set of admissible $\alpha^{-1}$ values includes $\mu_3=77.06$
and infinitely many others depending on the chosen measure.
If (ii) is dropped, the exact theoretical value of $\alpha^{-1}$ is unknown.
G1 is false as a theorem of the current TOE axioms, and true only if (i) and (ii)
are elevated to axioms.

\begin{definition}[Minimal closure axiom (G1-fix)]
\textbf{Axiom G1-fix:} The admissible density on $(B^4,S^3)$ is the unique
degree-3 polynomial in $x\in[0,1]$ under flat measure whose graded integer
coefficients are $k_n\in\{2\cdot\dim\mathbb{O},\,\dim(\mathrm{adj}\,SU(2)),\,
|\mathbb{Z}_2|\}$ for $n=\{3,2,1\}$ respectively.
\end{definition}

Under G1-fix, $\alpha^{-1}=4\pi^3+\pi^2+\pi$ follows unconditionally.
The axiom is \emph{natural} (each clause references a canonical algebraic object)
but \emph{not forced} by the GON dynamics or the operator algebra $\hat{O}$ of
Paper~18.

\section{Connection to $J_3(\mathbb{O})$}
\label{sec:j3o}

The exceptional Jordan algebra $J_3(\mathbb{O})$ has $\dim_{\mathbb{R}}=27$ and
automorphism group $F_4$ ($\dim=52$).
Its Peirce decomposition with respect to the three primitive idempotents is
\begin{equation}
  J_3(\mathbb{O}) = \textstyle\bigoplus_{i=1}^3\mathbb{R}\,e_i
    \;\oplus\; \bigoplus_{i<j} V_{ij},\qquad V_{ij}\cong\mathbb{O},
  \label{eq:peirce}
\end{equation}
giving $3+3\!\cdot\!8=27$ dimensions.

\textbf{Integer pattern.}
Each diagonal idempotent $e_i$ is adjacent to exactly two off-diagonal sectors
($V_{ij}$ and $V_{ik}$), of combined dimension $2\!\cdot\!8=16$.
The diagonal subspace $\bigoplus\mathbb{R}\,e_i$ has dimension $3$.
Each $V_{ij}$ carries a canonical $\mathbb{Z}_2$ grading from the Cayley--Dickson
construction $\mathbb{H}\to\mathbb{O}$, contributing the factor $2$.
The integer pattern $(16,3,2)$ is therefore read off from the Peirce
decomposition without circularity or auxiliary choice.

\textbf{Does $J_3(\mathbb{O})$ fix the measure?}
The unique $F_4$-invariant inner product on $J_3(\mathbb{O})$ is the trace form
$\langle A,B\rangle=\mathrm{Tr}(A\circ B)$.
If the integration measure $d\mu$ on $[0,1]$ is required to be
$F_4$-equivariant with respect to the Peirce grading (i.e.\ to weight each
sector by its $F_4$-invariant volume), one expects the measure on the ``bulk''
radial direction to be flat, since the trace form on the off-diagonal sectors
$V_{ij}\cong\mathbb{O}$ is already the standard Euclidean measure on $\mathbb{R}^8$
(no radial warping).
This argument is suggestive but not yet a proof.

\begin{proposition}[$J_3(\mathbb{O})$ as partial constraint]\label{prop:J3}
The Peirce decomposition of $J_3(\mathbb{O})$ determines the integer factors
$(16,3,2)$ without circularity.
It does not independently fix the integration measure or the polynomial
functional form.
The hypothesis that the $F_4$-invariant measure on the radial interval $[0,1]$
is flat would close CP-3; this hypothesis is plausible from the trace-form
structure but requires explicit proof.
\end{proposition}

\section{Open Items}
\label{sec:open}

\begin{enumerate}
  \item \textbf{Resolve CP-1 (exactness).}
        Either derive the 0.22\,ppm shift from first principles (QED corrections
        to the bare geometric formula) or find the exact expression that equals
        $\alpha^{-1}_{\mathrm{exp}}$.
        Until this is done G1 is not well-posed.

  \item \textbf{Close CP-3 (measure) via $J_3(\mathbb{O})$.}
        Prove or disprove that the $F_4$-invariant volume form on the Peirce-graded
        radial direction is flat.
        This is the most tractable near-term target and the most structurally
        significant gap.

  \item \textbf{Close CP-2 (functional form) via minimality.}
        Prove that degree~3 is the minimum polynomial degree compatible with the
        three-layer Peirce grading and flat measure.
        (Degree~2 integrates to $\approx 13.01$, far from $\alpha^{-1}$, so
        minimality is plausible by inspection.)

  \item \textbf{Unify CP-4 (edge coefficient).}
        Adopt the $J_3(\mathbb{O})$ Peirce reading ($2$ from $\mathbb{Z}_2$
        grading of $V_{ij}$) as canonical and rule out the representation-dimension
        reading, or prove they are equivalent in this context.

  \item \textbf{Cross-reference Paper~18 ($\hat{O}$) assemblies.}
        If the master operator $\hat{O}$ uniquely selects an assembly on
        $(B^4,S^3)$, check whether that assembly determines the integration
        measure.
        This is the most direct internal route to closing G1 from within the TOE.
\end{enumerate}

\bigskip
\noindent\textbf{Summary.}
The derivation of $\alpha^{-1}=4\pi^3+\pi^2+\pi$ is internally consistent
and geometrically motivated but is not a uniqueness proof from $S^3$ geometry
alone.
Four choice points are identified; the two load-bearing ones are the exactness
of the numerical identity (CP-1) and the choice of flat integration measure (CP-3).
$J_3(\mathbb{O})$ eliminates the need for any auxiliary choice in deriving the
integer factors $(16,3,2)$ and provides a plausible route to closing CP-3 via
the $F_4$-invariant trace form.
Axiom G1-fix closes the proof conditionally but is not itself derivable from
the current TOE axioms.
G1 remains open.

\end{document}
