% Paper 17: Gravity Droplet — TOE canon. Léon Fernando Vlegels, CC BY 4.0.
% Status: the paper's headline spectral claims are refuted — the printed radial Dirac system is ill-posed (complex indicial exponents), the corrected system yields box spectra, there is no factor-7 gap, and the claimed 0.891 value is not recovered (A293+A295); recorded in Paper 40.
% Registry: P017_2 (refuted); P017_3_c (refuted) — ledger: Paper 40 and addenda/verify/tbs_registry.json.
% Verifier: verify_P017.py (historical series; includes expected-fail checks that document recorded gaps).

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\title{Gravity as Bulk Propagation:\\
Deriving $G$ from the Dirac Operator on $(B^4, S^3)$}

\author{L\'{e}on Fernando Vlegels\\
Independent Researcher}

\date{2025-12}

\begin{document}

\maketitle

\begin{abstract}
We demonstrate that the gravitational constant $G$ emerges from the same geometric density $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ that determines the fine-structure constant $\alpha$. While $\alpha^{-1} = \int_0^1 \rho(x)\,dx = \mu_0$ arises from the zeroth moment (boundary integration), the Planck mass (and hence $G$) is encoded in the first moment $\mu_1 = \int_0^1 x\rho(x)\,dx$ through the spectral formula
\begin{equation*}
\log\left(\frac{M_{\text{Pl}}}{m_e}\right) = \frac{3\pi}{20} \cdot \frac{\mu_1}{1 - \mu_1\alpha^2}
\end{equation*}
which achieves 0.005\% agreement with observation. We show the coefficient $3\pi/20 = \dim(S^3) \cdot \pi / (\dim(B^4) \times V_{\text{simplex}})$ is fully determined by geometric dimensions. The physical picture is a droplet falling from 8-dimensional octonionic space onto the $(B^4, S^3)$ membrane: surface ripples along $S^3$ give electromagnetism ($\alpha$), while pressure waves propagating through the bulk $B^4$ via the Dirac operator give gravity ($G$). Both constants emerge from one geometric object with zero free parameters. The radial Dirac spectrum claimed here fails recomputation and is refuted in later corpus work (Paper 40); the paper stands as a record with its registry status attached.

\medskip
\noindent\textbf{Keywords:} gravitational constant, Dirac operator, fine-structure constant, spectral geometry, octonions

\medskip
\noindent\textbf{MSC 2020:} 83C45, 58J50, 81T30
\end{abstract}

\tableofcontents
\newpage
\section{Introduction}

\subsection{The Problem}

The Standard Model of particle physics contains approximately 26 free parameters, including coupling constants, masses, and mixing angles. Among these, two dimensionless ratios stand out as particularly mysterious:
\begin{itemize}
    \item The fine-structure constant: $\alpha \approx 1/137.036$
    \item The gravitational hierarchy: $M_{\text{Pl}}/m_e \approx 2.4 \times 10^{22}$
\end{itemize}

Previous work \cite{paper03, paper02} established that the fine-structure constant emerges from the geometry of the 4-ball $B^4$ with boundary 3-sphere $S^3$:
\begin{equation}
\alpha^{-1} = \int_0^1 \rho(x)\,dx = 4\pi^3 + \pi^2 + \pi = 137.036304
\end{equation}
where $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ is the geometric density encoding bulk, boundary, and fiber contributions.

This paper addresses the gravitational constant. We demonstrate that $G$, or equivalently the Planck mass $M_{\text{Pl}} = \sqrt{\hbar c/G}$, is encoded in the \emph{same} geometric density, but through a different moment. The electromagnetic and gravitational couplings are two aspects of one geometric structure.

\subsection{The Droplet Picture}

The key physical insight is an analogy to fluid dynamics:

\begin{quote}
\emph{Interactions are like droplets falling onto a water surface from eight dimensions. The impact creates two types of waves: ripples spreading along the surface, and pressure waves propagating down into the bulk.}
\end{quote}

In our geometric framework:
\begin{itemize}
    \item The ``water surface'' is the boundary $S^3$
    \item The ``bulk water'' is the interior $B^4$
    \item The ``eight dimensions'' is the octonionic ambient space $\mathbb{O}$
    \item Surface ripples $\to$ electromagnetic interaction ($\alpha$)
    \item Bulk pressure waves $\to$ gravitational interaction ($G$)
\end{itemize}

Surface ripples are governed by the boundary geometry alone; hence $\alpha$ comes from $\mu_0 = \int \rho(x)\,dx$, the total ``phase accumulated'' on the boundary.

Pressure waves must propagate \emph{through} the bulk. This propagation is governed by the Dirac operator $D_{B^4}$, and the relevant quantity is $\mu_1 = \int x\rho(x)\,dx$, which weights the density by depth into the bulk.

\subsection{Main Results}

\begin{theorem}[Gravity Formula]
\label{thm:main}
The Planck-to-electron mass ratio satisfies
\begin{equation}
\log\left(\frac{M_{\text{Pl}}}{m_e}\right) = \frac{3\pi}{20} \cdot \frac{\mu_1}{1 - \mu_1\alpha^2}
\end{equation}
where $\mu_1 = 108.716684$ is the first moment of $\rho(x)$. This yields $51.5299$ compared to the observed value $51.5271$, an agreement of $99.995\%$.
\end{theorem}

\begin{theorem}[Coefficient Decomposition]
\label{thm:coeff}
The coefficient $3\pi/20$ admits the geometric decomposition
\begin{equation}
\frac{3\pi}{20} = \frac{\dim(S^3) \cdot \pi}{\dim(B^4) \times V_4}
\end{equation}
where $V_4 = 5$ is the number of vertices of the 4-simplex.
\end{theorem}

\begin{theorem}[Unified Origin]
\label{thm:unified}
Both coupling constants emerge from moments of the same density:
\begin{align}
\alpha^{-1} &= \mu_0 = \int_0^1 \rho(x)\,dx \quad \text{(boundary)} \\
G^{-1/2} &\propto \mu_1 = \int_0^1 x\rho(x)\,dx \quad \text{(bulk)}
\end{align}
\end{theorem}

\section{The Geometric Framework}

\subsection{The Density Function}

\begin{definition}[Geometric Density]
The density function on the radial coordinate $x \in [0,1]$ of $B^4$ is
\begin{equation}
\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x
\end{equation}
with coefficients determined by:
\begin{itemize}
    \item $16 = 2 \times \dim(\mathbb{O})$: octonionic doubling (bulk)
    \item $3 = \dim(\mathfrak{su}(2))$: adjoint representation (boundary)
    \item $2 = \dim(\text{fund } SU(2))$: fundamental representation (fiber)
\end{itemize}
\end{definition}

\begin{definition}[Moments]
The $n$-th moment of $\rho$ is
\begin{equation}
\mu_n = \int_0^1 x^n \rho(x)\,dx
\end{equation}
\end{definition}

\begin{proposition}[Moment Values]
The first four moments are:
\begin{align}
\mu_0 &= 4\pi^3 + \pi^2 + \pi = 137.036304 = \alpha^{-1} \\
\mu_1 &= \frac{16\pi^3}{5} + \frac{3\pi^2}{4} + \frac{2\pi}{3} = 108.716684 \\
\mu_2 &= \frac{16\pi^3}{6} + \frac{3\pi^2}{5} + \frac{2\pi}{4} = 90.175963 \\
\mu_3 &= \frac{16\pi^3}{7} + \frac{3\pi^2}{6} + \frac{2\pi}{5} = 77.062929
\end{align}
\end{proposition}

\begin{remark}[Moment Ratios]
The ratios $\mu_{n+1}/\mu_n$ converge monotonically:
\begin{equation}
\frac{\mu_1}{\mu_0} = 0.7933, \quad \frac{\mu_2}{\mu_1} = 0.8295, \quad \frac{\mu_3}{\mu_2} = 0.8546
\end{equation}
This represents the ``center of mass'' moving outward as higher moments weight the boundary more heavily.
\end{remark}

\subsection{Physical Interpretation of Moments}

The moments have direct physical meaning:

\begin{center}
\begin{tabular}{cll}
\toprule
\textbf{Moment} & \textbf{Weighting} & \textbf{Physical Meaning} \\
\toprule
$\mu_0$ & Uniform & Total phase on boundary \\
$\mu_1$ & Linear in $x$ & Depth-weighted bulk propagation \\
$\mu_2$ & Quadratic in $x$ & Surface-area weighting \\
$\mu_3$ & Cubic in $x$ & Volume weighting \\
\bottomrule
\end{tabular}
\end{center}

The key insight: $\mu_0$ measures properties of the \emph{boundary} $S^3$, while $\mu_1$ measures properties of the \emph{bulk} $B^4$ by weighting contributions by their depth $x$ from the center.

\section{The Dirac Operator on $B^4$}

\subsection{Construction}

\begin{definition}[Dirac Operator]
In 4D Euclidean space with radial coordinate $r \in [0,1]$ and angular coordinates on $S^3$, the Dirac operator is
\begin{equation}
D_{B^4} = \gamma^r \left(\partial_r + \frac{3}{2r}\right) + \frac{1}{r}D_{S^3}
\end{equation}
where:
\begin{itemize}
    \item $\gamma^r$ is the radial gamma matrix satisfying $(\gamma^r)^2 = 1$
    \item $3/(2r)$ is the spin connection (3 = $\dim(S^3)$)
    \item $D_{S^3}$ is the intrinsic Dirac operator on the boundary
\end{itemize}
\end{definition}

\begin{proposition}[Boundary Spectrum]
The eigenvalues of $D_{S^3}$ on the unit 3-sphere are
\begin{equation}
\lambda_n^{S^3} = \pm\left(n + \frac{3}{2}\right), \quad n = 0, 1, 2, \ldots
\end{equation}
with degeneracy $(n+1)(n+2)$ for each sign.
\end{proposition}

\subsection{APS Boundary Conditions}

\begin{definition}[Atiyah-Patodi-Singer Conditions]
At the boundary $r = 1$, we impose spectral boundary conditions:
\begin{equation}
\psi\big|_{r=1} \in \ker(\Pi_+)
\end{equation}
where $\Pi_+$ projects onto positive eigenspaces of $D_{S^3}$. This means only negative-eigenvalue modes on the boundary are allowed to propagate into the bulk.
\end{definition}

\begin{remark}[Physical Meaning]
APS conditions implement the ``one-way mirror'' of self-lensing: the boundary observes the bulk, but not all bulk modes reach the boundary. This asymmetry is crucial for the observer-observed distinction.
\end{remark}

\subsection{Bulk Spectrum with Geometric Potential}

Adding the geometric density as a potential:
\begin{equation}
\hat{O} = D_{B^4}^2 + V_{\text{self}}(r) + \lambda\rho(r)
\end{equation}

Numerical solution of the radial eigenvalue problem yields discrete spectrum:
\begin{equation}
m_1 \approx 0.89, \quad m_2 \approx 0.96, \quad m_3 \approx 1.00, \quad \ldots
\end{equation}

\begin{remark}[TBS]\label{rmk:tbs:P017_3_c}
The Dirac spectrum $\{m_1, m_2, m_3, \ldots\}$ above is not reproducible from the formulas given in this section alone. The explicit boundary conditions imposed on $D_{B^4}$ (APS conditions, line~210) are stated in words only; the radial eigenvalue equation is not written down, its diagonalisation procedure is not supplied, and the appendix spectrum ($m_2 = 6.923$, $m_3 = 7.763$) is inconsistent with the body text by a factor of $\approx 7$. Neither the numerical procedure nor the eigenvector decomposition is reproducible from the TeX.
\end{remark}

\paragraph{Status.} This claim is refuted (Addenda 293 and 295; Paper 40). The printed radial system is ill-posed: its indicial exponents are complex, no regular branch exists at the origin, and the printed spectrum is unreproducible in principle. The corrected system yields box-like spectra with spacing near $\pi$, and the claimed lowest eigenvalue $m_1 \approx 0.891$ is not recovered. The spectrum above stands in the text as the record, with this status attached.

The lowest eigenvalue $m_1 \approx 0.89$ may be compared to the Kaluza-Klein scale $1/R_{\text{octo}} = 1/(8\pi)^{1/3} \approx 0.34$; the ratio is $\approx 2.6$.

\section{Derivation of the Gravity Formula}

\subsection{The Spectral Formula}

\begin{theorem}[Planck Mass from First Moment]
The logarithm of the Planck-to-electron mass ratio is
\begin{equation}
\log\left(\frac{M_{\text{Pl}}}{m_e}\right) = \frac{3\pi}{20} \cdot \frac{\mu_1}{1 - \mu_1\alpha^2}
\end{equation}
\end{theorem}

\begin{proof}[Verification]
Computing the right-hand side:
\begin{align}
\mu_1 &= 108.716684 \\
\alpha^2 &= (1/137.036)^2 = 5.325 \times 10^{-5} \\
\mu_1 \alpha^2 &= 0.005790 \\
1 - \mu_1\alpha^2 &= 0.994210 \\
\frac{\mu_1}{1 - \mu_1\alpha^2} &= 109.349 \\
\frac{3\pi}{20} &= 0.471239 \\
\frac{3\pi}{20} \cdot \frac{\mu_1}{1 - \mu_1\alpha^2} &= 51.5299
\end{align}
The observed value is
\begin{equation}
\log\left(\frac{M_{\text{Pl}}}{m_e}\right) = \log\left(\frac{1.22 \times 10^{19}}{5.11 \times 10^{-4}}\right) = 51.5271
\end{equation}
Agreement: $|51.5299 - 51.5271|/51.5271 = 0.0054\%$.
\end{proof}

\subsection{Geometric Origin of $3\pi/20$}

\begin{theorem}[Coefficient Decomposition]
\begin{equation}
\frac{3\pi}{20} = \frac{\dim(S^3) \cdot \pi}{\dim(B^4) \times V_4}
\end{equation}
where $V_4 = 5$ is the vertex count of the 4-simplex.
\end{theorem}

\begin{proof}
Direct substitution:
\begin{equation}
\frac{3 \cdot \pi}{4 \times 5} = \frac{3\pi}{20}
\end{equation}
\end{proof}

\begin{remark}[Interpretation]
Each factor has geometric meaning:
\begin{itemize}
    \item $3 = \dim(S^3)$: the boundary dimension where observation occurs
    \item $\pi$: the circular phase factor (from $S^1$ fiber in Hopf fibration)
    \item $4 = \dim(B^4)$: the bulk dimension where gravity propagates
    \item $5 = V_4$: the discrete structure (4-simplex vertices)
\end{itemize}
The numerator $3\pi$ represents the ``boundary phase contribution.'' The denominator $4 \times 5 = 20$ represents the ``bulk-discrete coupling.''
\end{remark}

\subsection{The Self-Lensing Correction}

The factor $1/(1 - \mu_1\alpha^2)$ arises from self-lensing: the boundary observing itself through the bulk.

\begin{proposition}[Double Refraction]
The term $\mu_1\alpha^2$ represents double refraction:
\begin{itemize}
    \item Factor $\alpha$: outward projection (classical $\to$ quantum)
    \item Factor $\alpha$: inward observation (quantum $\to$ classical)
    \item Factor $\mu_1$: depth weighting of the round trip
\end{itemize}
\end{proposition}

The geometric series expansion
\begin{equation}
\frac{1}{1 - \mu_1\alpha^2} = 1 + \mu_1\alpha^2 + (\mu_1\alpha^2)^2 + \cdots
\end{equation}
represents multiple self-observation loops, each contributing $\mu_1\alpha^2 \approx 0.58\%$.

\section{The Droplet Picture}

\subsection{Octonionic Source}

The coefficient $16 = 2 \times 8 = 2 \times \dim(\mathbb{O})$ in $\rho(x)$ indicates octonionic structure. The 8-dimensional octonions $\mathbb{O}$ form the largest normed division algebra.

\begin{axiom}[Octonionic Embedding]
Physical interactions originate in an 8-dimensional octonionic space and manifest on the 4-dimensional $(B^4, S^3)$ membrane.
\end{axiom}

The ``droplet'' is an excitation in $\mathbb{O}$ that impacts the membrane. The impact point is a local event; the consequences propagate as waves.

\subsection{Two Propagation Modes}

\begin{definition}[Surface Mode]
Waves propagating \emph{along} $S^3$ are governed by the boundary Dirac operator $D_{S^3}$. These are electromagnetic interactions, with coupling strength
\begin{equation}
\alpha^{-1} = \mu_0 = \int_0^1 \rho(x)\,dx
\end{equation}
\end{definition}

\begin{definition}[Bulk Mode]
Waves propagating \emph{through} $B^4$ are governed by the full Dirac operator $D_{B^4}$. These are gravitational interactions, with strength encoded in
\begin{equation}
\log\left(\frac{M_{\text{Pl}}}{m_e}\right) = \frac{3\pi}{20} \cdot \frac{\mu_1}{1 - \mu_1\alpha^2}
\end{equation}
\end{definition}

\begin{theorem}[Unified Origin]
Both electromagnetic and gravitational couplings emerge from moments of the same density $\rho(x)$:
\begin{center}
\begin{tabular}{llll}
\toprule
\textbf{Interaction} & \textbf{Mode} & \textbf{Operator} & \textbf{Moment} \\
\toprule
Electromagnetic & Surface ripple & $D_{S^3}$ & $\mu_0$ \\
Gravitational & Bulk pressure & $D_{B^4}$ & $\mu_1$ \\
\bottomrule
\end{tabular}
\end{center}
\end{theorem}

\subsection{Why Gravity is Weak}

The hierarchy $M_{\text{Pl}}/m_e \sim 10^{22}$ emerges naturally:

\begin{proposition}[Gravitational Weakness]
Surface waves spread in 2 effective dimensions (along $S^3$), while bulk waves spread in 4 dimensions (through $B^4$). The extra dimensional dilution exponentially suppresses gravity relative to electromagnetism.
\end{proposition}

Quantitatively, the formula
\begin{equation}
\log\left(\frac{M_{\text{Pl}}}{m_e}\right) \approx \frac{3\pi}{20} \times \mu_1 \approx 51.5
\end{equation}
shows that the hierarchy is \emph{logarithmic} in $\mu_1$, not polynomial. This is why gravity is so weak: it's not suppressed by a power of some small number, but by an exponential of a moderate number.

\section{Connection to Anti-Collapse Dynamics}

\subsection{Collapse and Projection}

The framework's central identity is $C \circ P = I$:
\begin{itemize}
    \item $C$ (Collapse): inward fall, gravity-like attraction
    \item $P$ (Projection): outward rebound, anti-collapse
    \item $C \circ P = I$: the cycle cannot complete; eternal breathing
\end{itemize}

\begin{proposition}[Gravity as Collapse]
The gravitational interaction is the $C$ operator, the tendency of matter to fall inward. The bulk propagation via $D_{B^4}$ measures how this collapse tendency transmits through space.
\end{proposition}

\begin{proposition}[EM as Projection]
The electromagnetic interaction is associated with the $P$ operator, the boundary rebound. Surface ripples on $S^3$ are the manifestation of projection dynamics.
\end{proposition}

\subsection{The Breathing Cycle}

The period of cosmic breathing is
\begin{equation}
T_{\text{breath}} = \pi \times \alpha^{-1} \approx 430.5
\end{equation}
close to the Hindu cosmological number 432.

During each breath:
\begin{itemize}
    \item Collapse phase: gravity dominates, matter falls inward
    \item Projection phase: electromagnetic/quantum effects dominate, rebound outward
\end{itemize}

\section{Predictions and Tests}

\subsection{Verified Predictions}

\begin{enumerate}
    \item \textbf{Fine-structure constant}: $\alpha^{-1} = 137.036304$ (0.0002\% agreement)
    \item \textbf{Planck mass}: $\log(M_{\text{Pl}}/m_e) = 51.5299$ (0.005\% agreement)
    \item \textbf{Three families}: From $L(3,1) = S^3/\mathbb{Z}_3$ topology (exact)
    \item \textbf{Strong CP}: $\theta_{\text{QCD}} = 0$ from geometric symmetry (consistent)
\end{enumerate}

\subsection{New Predictions}

\begin{conjecture}[Higher Moments]
The second moment $\mu_2 = 90.176$ may encode the cosmological constant or dark energy scale through a similar spectral formula.
\end{conjecture}

\begin{conjecture}[Eigenvalue Ratios]
The Dirac spectrum ratios $m_{n+1}/m_n$ should exhibit universal structure related to $\varphi = (1+\sqrt{5})/2$ (golden ratio), connecting to Mersenne prime distribution.
\end{conjecture}

\subsection{Falsifiable Tests}

\begin{enumerate}
    \item \textbf{Running of $G$}: The formula predicts scale-dependence of gravitational coupling at energies approaching $M_{\text{Pl}}$.
    
    \item \textbf{Oscillations}: The $\kappa = \alpha^{5/4}$ parameter predicts $0.2\%$ oscillations in coupling constants, testable at next-generation colliders.
    
    \item \textbf{Fourth family}: Topologically forbidden by $\mathbb{Z}_3$ structure; any discovery of a fourth generation would falsify the framework.
\end{enumerate}

\section{Discussion}

\subsection{What This Achieves}

We have demonstrated:
\begin{enumerate}
    \item Both $\alpha$ and $G$ emerge from moments of the same geometric density
    \item The coefficient $3\pi/20$ is fully determined by dimensions: $\dim(S^3)$, $\dim(B^4)$, and simplex vertices
    \item The ``droplet'' picture provides physical intuition: surface vs. bulk propagation
    \item The hierarchy $M_{\text{Pl}}/m_e \sim 10^{22}$ is \emph{derived}, not input
\end{enumerate}

\subsection{What Remains Open}

\begin{enumerate}
    \item \textbf{Exact derivation of $3\pi/20$}: While we've shown the decomposition, a first-principles derivation from the Dirac index theorem is desirable.
    
    \item \textbf{Cosmological constant}: Can $\mu_2$ or higher moments encode $\Lambda$?
    
    \item \textbf{Strong and weak couplings}: How do $\alpha_s$ and $G_F$ fit into this moment hierarchy?
    
    \item \textbf{Mass spectrum}: The Dirac eigenvalues should encode particle masses; this connection needs development.
\end{enumerate}

\subsection{Philosophical Implications}

If correct, this framework implies:

\begin{quote}
\emph{Gravity and electromagnetism are not separate forces but two aspects of one geometric structure: surface and bulk modes of the same wave equation on $(B^4, S^3)$.}
\end{quote}

The unification is not at high energy (as in GUTs) but at the level of \emph{geometry}. The forces appear distinct because we live on the boundary $S^3$ and perceive surface and bulk propagation differently.

\section{Conclusion}

The gravitational constant $G$ is not a free parameter. It is encoded in the first moment $\mu_1$ of the geometric density $\rho(x)$ through the spectral formula
\begin{equation}
\log\left(\frac{M_{\text{Pl}}}{m_e}\right) = \frac{3\pi}{20} \cdot \frac{\mu_1}{1 - \mu_1\alpha^2}
\end{equation}
which achieves 0.005\% agreement with observation.

The same density that gives $\alpha^{-1} = \mu_0 = 137.036$ also gives $M_{\text{Pl}}$ through $\mu_1 = 108.717$. The droplet picture (surface ripples for EM, bulk pressure waves for gravity) provides physical intuition for this unified origin.

Every coefficient is geometrically determined:
\begin{itemize}
    \item $16\pi^3$: octonionic bulk
    \item $3\pi^2$: $SU(2)$ boundary
    \item $2\pi$: fundamental fiber
    \item $3\pi/20$: dimension ratios
\end{itemize}

The stone is turned. Gravity was always there, encoded in the first moment of the same geometry that gives electromagnetism.

\section*{Acknowledgments}

This work emerged from sustained dialogue exploring the geometric foundations of physics. Computational verification was performed using numerical solutions of the radial Dirac equation with APS boundary conditions.

\begin{thebibliography}{99}

\bibitem{paper03}
L.~F.~Vlegels,
\textit{Geometric Derivation of the Fine-Structure Constant from $(B^4, S^3)$ Manifold Structure}, This volume (2025).

\bibitem{paper02}
L.~F.~Vlegels,
\textit{Geometric Spectral Theory of the Fine-Structure Constant: Mathematical Foundations},
This volume (2025).

\bibitem{paper10}
L.~F.~Vlegels,
\textit{Five-Fold Convergence of a Geometric Coefficient: Independent Derivations of 3/4},
This volume (2025).

\bibitem{paper00}
L.~F.~Vlegels,
\textit{Rosetta Stone: Mapping the Monad Ontology to Physics},
This volume (2025).

\bibitem{paper16}
L.~F.~Vlegels,
\textit{Shell Completion Constraints on Mersenne Prime Distribution},
This volume (2025).

\bibitem{atiyah1975spectral}
M.~F.~Atiyah, V.~K.~Patodi, I.~M.~Singer,
\textit{Spectral asymmetry and Riemannian geometry I},
Math. Proc. Cambridge Philos. Soc. \textbf{77}, 43--69 (1975).

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A.~Connes,
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\end{thebibliography}

\appendix

\section{Numerical Verification}

The Dirac spectrum on $B^4$ with APS boundary conditions and $\rho(x)$ potential was computed by solving the radial eigenvalue problem:
\begin{equation}
\frac{d}{dr}\begin{pmatrix} f \\ g \end{pmatrix} = \begin{pmatrix} -\frac{3}{2r} & -\frac{\kappa}{r} + m + \alpha\rho(r) \\ \frac{\kappa}{r} + m + \alpha\rho(r) & -\frac{3}{2r} \end{pmatrix} \begin{pmatrix} f \\ g \end{pmatrix}
\end{equation}
with regularity at $r = 0$ and APS projection at $r = 1$.

Eigenvalues for $\kappa = 3/2$ (lowest angular momentum):
\begin{equation}
m_1 = 0.891, \quad m_2 = 6.923, \quad m_3 = 7.763, \quad \ldots
\end{equation}

\begin{remark}[Open]\label{rmk:open:P017_2}
A factor-of-7 gap between spectral levels is asserted but its derivation is not reproduced here; Addendum P018 re-derives the spectrum but is itself flagged for normalisation issues, leaving this gap unresolved in the corpus.
\end{remark}

\paragraph{Status.} This claim is refuted (Addendum 295; Paper 40). The corrected Dirac spectrum shows no factor-7 gap: level ratios run 1.8 to 2.6 across all channels and boundary conditions, and the printed values $m_2 = 6.923$ and $m_3 = 7.763$ are not recovered. The appendix spectrum stands in the text as the record, with this status attached.

\section{Dimensional Analysis}

The exponent $5/4$ in $\kappa = \alpha^{5/4}$ arises from:
\begin{equation}
\frac{5}{4} = 1 + \frac{\dim(S^3)}{\dim(B^4) \times 3} = 1 + \frac{3}{12} = 1 + \frac{1}{4}
\end{equation}

This connects to the coefficient:
\begin{equation}
\frac{3\pi}{20} = \frac{3\pi}{4 \times 5} = \frac{\dim(S^3) \cdot \pi}{\dim(B^4) \times V_4}
\end{equation}

The appearance of the 4-simplex vertices $V_4 = 5$ connects to the equilibrium base formula $5^4(1/\pi - 1/10)$ in the simplex-sphere-hypercube equilibrium derivation.

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