% Paper 20: Standard Physics Embedding — TOE canon. Léon Fernando Vlegels, CC BY 4.0.
% Status: the paper's electroweak VEV conjecture is refuted — v = M_Pl/sqrt(mu0*mu1) evaluates to 1.0e17 GeV against the measured 246 GeV, off by a factor of 4.1e14 (A293); recorded in Paper 40.
% Registry: P020_1 (refuted) — ledger: Paper 40 and addenda/verify/tbs_registry.json.
% Verifier: verify_P020.py (historical series; includes expected-fail checks that document recorded gaps).

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\title{Standard Physics Embedding:\\
A Complete Dictionary from Geometric TOE to Particle Physics}

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

\date{2025-12}

\begin{document}

\maketitle

\begin{abstract}
We provide a complete translation between the geometric Theory of Everything based on $(B^4, S^3)$ topology and the standard notation of particle physics. Every element of the geometric framework is mapped to its physical counterpart: the density $\rho(x)$ to spectral/energy density, the master operator $\hat{O}$ to the Standard Model Lagrangian, the $\mathbb{Z}_3$ symmetry to fermion families, the division algebra tower to gauge groups, and the moment hierarchy to coupling constants. We present a comprehensive dictionary, worked examples, and a summary of all predictions with their experimental status. This paper serves as the interface between the geometric formalism and the physics community. The electroweak VEV conjecture recorded here fails evaluation by fourteen orders of magnitude and is refuted in later corpus work (Paper 40); the paper stands as a record with its registry status attached.

\medskip
\noindent\textbf{Keywords:} Standard Model, particle physics, geometric unification, translation dictionary

\medskip
\noindent\textbf{PACS:} 12.10.-g, 11.30.-j, 14.60.-z
\end{abstract}

\tableofcontents
\newpage
\section{Introduction}

\subsection{Purpose}

The geometric Theory of Everything (TOE) based on the $(B^4, S^3)$ manifold has been developed across a series of papers establishing:
\begin{itemize}
    \item The geometric density $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$
    \item The master operator $\hat{O}$
    \item The Unified Field Equation $\hat{O}\Phi = J$
    \item Derivations of $\alpha$, $G$, three families, and gauge structure
\end{itemize}

This paper provides the \textbf{translation layer}: a complete dictionary mapping every geometric concept to its Standard Model counterpart. The goal is to make the framework accessible to particle physicists, cosmologists, and experimentalists.

\subsection{Structure}

\begin{enumerate}
    \item \textbf{Section 2}: The Master Dictionary (geometry $\leftrightarrow$ physics)
    \item \textbf{Section 3}: Particle Content (fermions, bosons, Higgs)
    \item \textbf{Section 4}: Coupling Constants (all derived values)
    \item \textbf{Section 5}: Symmetries (gauge groups, families, CP)
    \item \textbf{Section 6}: Mass Spectrum (hierarchy and predictions)
    \item \textbf{Section 7}: Cosmology (dark sector, inflation, $\Lambda$)
    \item \textbf{Section 8}: Experimental Tests (current and future)
    \item \textbf{Section 9}: Complete Summary
\end{enumerate}

\section{The Master Dictionary}

\subsection{Geometric Objects $\to$ Physical Objects}

\begin{longtable}{p{4cm}p{5cm}p{5cm}}
\toprule
\textbf{Geometric Object} & \textbf{Physical Counterpart} & \textbf{Notes} \\
\midrule
\endhead

$B^4$ (4-ball) & Quantum vacuum / bulk spacetime & Where quantum fields propagate \\
\addlinespace

$S^3$ (3-sphere boundary) & Classical spacetime / observable universe & Where we live and measure \\
\addlinespace

$S^1$ (Hopf fiber) & Phase / U(1) gauge / observation & Internal degree of freedom \\
\addlinespace

$\rho(x)$ density & Spectral density / vacuum energy & Encodes all coupling strengths \\
\addlinespace

$x \in [0,1]$ & Radial coordinate / energy scale & $x=0$: center; $x=1$: boundary \\
\addlinespace

$\hat{O}$ master operator & $\mathcal{L}_{\text{SM}}$ Lagrangian & Generates all dynamics \\
\addlinespace

$\Phi$ universal field & All SM fields combined & Fermions + bosons + Higgs \\
\addlinespace

$J$ source term & Interactions / currents & Self-lensing = self-interaction \\
\addlinespace

$T_{\rm cycle}$ layer-cycle & Generation mixing & Cycles through 3 families \\
\addlinespace

$\mathcal{R}$ RG generator & Renormalization group flow & Scale transformations \\
\addlinespace

$\mathcal{M}$ moment operator & Mass matrix structure & Hierarchies from $\mu_n$ \\
\bottomrule
\caption{Geometric objects and their physical counterparts}
\end{longtable}

\subsection{Geometric Parameters $\to$ Physical Constants}

\begin{longtable}{p{3.5cm}p{3cm}p{3.5cm}p{3.5cm}}
\toprule
\textbf{Geometric} & \textbf{Value} & \textbf{Physical} & \textbf{Measured} \\
\midrule
\endhead

$\mu_0 = \int\rho\,dx$ & $137.036$ & $\alpha^{-1}$ & $137.035999...$  \\
\addlinespace

$\mu_1 = \int x\rho\,dx$ & $108.717$ & Controls $M_{\text{Pl}}$ & Via $G$ \\
\addlinespace

$\mu_2 = \int x^2\rho\,dx$ & $90.176$ & Controls $\Lambda$? & TBD \\
\addlinespace

$\mu_1/\mu_0$ & $0.7933$ & $\beta_{\text{geom}}$ & RG coefficient \\
\addlinespace

$\kappa = \alpha^{5/4}$ & $0.00213$ & Oscillation scale & Testable \\
\addlinespace

$\gamma = 3/4$ & $0.75$ & Boundary/bulk ratio & dim$(S^3)$/dim$(B^4)$ \\
\addlinespace

$\beta = 3\pi/20$ & $0.4712$ & Gravity coefficient & In log formula \\
\addlinespace

$E_{\text{self}}$ & $13.177$ & Self-lensing energy & Bound state scale \\
\bottomrule
\caption{Geometric parameters and physical constants}
\end{longtable}

\subsection{Geometric Structures $\to$ Symmetries}

\begin{longtable}{p{4cm}p{4cm}p{5.5cm}}
\toprule
\textbf{Geometric Structure} & \textbf{Symmetry Group} & \textbf{Physical Role} \\
\midrule
\endhead

$S^3 \cong SU(2)$ & $SU(2)_L$ & Weak isospin \\
\addlinespace

Hopf fiber $S^1$ & $U(1)_Y$ & Hypercharge \\
\addlinespace

$G_2 = \text{Aut}(\mathbb{O})$ & Contains $SU(3)$ & Color symmetry \\
\addlinespace

$L(3,1) = S^3/\mathbb{Z}_3$ & $\mathbb{Z}_3$ & Three families \\
\addlinespace

$C \circ P = I$ & CPT & Anti-collapse symmetry \\
\addlinespace

Division algebras & $\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$ & Force hierarchy \\
\addlinespace

Hopf fibration & $S^1 \to S^3 \to S^2$ & Electroweak structure \\
\bottomrule
\caption{Geometric structures and symmetry groups}
\end{longtable}

\section{Particle Content}

\subsection{Fermions from Bulk Modes}

Fermions are eigenmodes of the bulk Dirac operator $D_{B^4}$ with $\mathbb{Z}_3$ family structure.

\begin{center}
\begin{tabular}{llccc}
\toprule
\textbf{Particle} & \textbf{Geometric Mode} & \textbf{$T_{\rm cycle}$ eigenvalue} & \textbf{$SU(3)$} & \textbf{$SU(2)$} \\
\toprule
$e, \mu, \tau$ & $\psi^{(k)}_{\ell}$, $k=0,1,2$ & $1, \omega, \omega^2$ & $\mathbf{1}$ & $\mathbf{2}$ \\
$\nu_e, \nu_\mu, \nu_\tau$ & $\psi^{(k)}_{\nu}$, $k=0,1,2$ & $1, \omega, \omega^2$ & $\mathbf{1}$ & $\mathbf{2}$ \\
$u, c, t$ & $\psi^{(k)}_{u}$, $k=0,1,2$ & $1, \omega, \omega^2$ & $\mathbf{3}$ & $\mathbf{2}$ \\
$d, s, b$ & $\psi^{(k)}_{d}$, $k=0,1,2$ & $1, \omega, \omega^2$ & $\mathbf{3}$ & $\mathbf{2}$ \\
\bottomrule
\end{tabular}
\end{center}

where $\omega = e^{2\pi i/3}$.

\begin{proposition}[Family Origin]
The three families arise from the $\mathbb{Z}_3$ eigenspaces of the layer-cycle operator $T_{\rm cycle}$:
\begin{equation}
T_{\rm cycle}\psi^{(k)} = \omega^k \psi^{(k)}, \quad k = 0, 1, 2
\end{equation}
This is topologically enforced by $\pi_1(L(3,1)) = \mathbb{Z}_3$.
\end{proposition}

\subsection{Gauge Bosons from Boundary Modes}

Gauge bosons are eigenmodes of the boundary Laplacian $\Delta_{S^3}$.

\begin{center}
\begin{tabular}{llll}
\toprule
\textbf{Boson} & \textbf{Geometric Origin} & \textbf{Gauge Group} & \textbf{Mass} \\
\toprule
$\gamma$ (photon) & $S^1$ fiber & $U(1)_{\text{EM}}$ & 0 \\
$W^\pm$ & $S^3$ left modes & $SU(2)_L$ & 80.4 GeV \\
$Z^0$ & $S^3$ mixed mode & $SU(2)_L \times U(1)_Y$ & 91.2 GeV \\
$g$ (gluons) & Octonionic $G_2$ & $SU(3)_c$ & 0 \\
\bottomrule
\end{tabular}
\end{center}

\begin{proposition}[Massless Photon]
The photon remains massless because $U(1)_{\text{EM}}$ is the unbroken subgroup:
\begin{equation}
SU(2)_L \times U(1)_Y \xrightarrow{\text{SSB}} U(1)_{\text{EM}}
\end{equation}
The Hopf fiber $S^1$ encodes this residual symmetry.
\end{proposition}

\subsection{Higgs from Fiber Mode}

The Higgs field arises from the fiber sector $\Delta_{S^1}$.

\begin{proposition}[Higgs Identification]
The Higgs doublet $H$ corresponds to the scalar mode on the $S^1$ fiber:
\begin{equation}
H = \begin{pmatrix} H^+ \\ H^0 \end{pmatrix} \sim \phi_{S^1}
\end{equation}
The vacuum expectation value:
\begin{equation}
\langle H \rangle = \frac{v}{\sqrt{2}} \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \quad v = 246 \text{ GeV}
\end{equation}
\end{proposition}

\begin{conjecture}[Higgs VEV from Geometry]
The electroweak scale is:
\begin{equation}
v = \frac{M_{\text{Pl}}}{\sqrt{\mu_0 \cdot \mu_1}} \approx 246 \text{ GeV}  %% UNRESOLVED: naive evaluation gives $M_\text{Pl}/\sqrt{\mu_0\mu_1} \sim 10^{17}$\,GeV, not 246\,GeV. A missing geometric projection or dimensionless ratio is required.
\end{equation}
\end{conjecture}

\begin{remark}[Open]\label{rmk:open:P020_1}
The Higgs VEV formula derived here evaluates to $\sim 10^{17}\ \mathrm{GeV}$, approximately $15$ orders of magnitude above the physical value of $246\ \mathrm{GeV}$. No addendum addresses this VEV discrepancy; the Higgs mass was fixed separately in P085/P101/P102 without correcting the VEV derivation.
\end{remark}

\paragraph{Status.} This claim is refuted (Addendum 293; Paper 40). The conjecture $v = M_{\text{Pl}}/\sqrt{\mu_0\,\mu_1}$ evaluates to $1.0\times10^{17}\ \mathrm{GeV}$ against the measured $246\ \mathrm{GeV}$, off by a factor of $4.1\times10^{14}$; it is refuted as stated. Two numerical flags recorded in that audit were noted but not promoted to a repair, so no corrected VEV derivation exists in the corpus. The conjecture stands in the text as the record, with this status attached; the remark above predates Addendum 293.

\section{Coupling Constants}

\subsection{Electromagnetic: $\alpha$}

\begin{theorem}[Fine-Structure Constant]
\begin{equation}
\boxed{\alpha^{-1} = \int_0^1 \rho(x)\,dx = 4\pi^3 + \pi^2 + \pi = 137.036304}
\end{equation}
\end{theorem}

\textbf{Comparison:}
\begin{center}
\begin{tabular}{lc}
\toprule
\textbf{Source} & \textbf{Value} \\
\toprule
Geometric (this work) & $137.036304$ \\
CODATA 2018 & $137.035999084(21)$ \\
Difference & $0.0002\%$ \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Gravitational: $G$}

\begin{theorem}[Planck Mass]
\begin{equation}
\boxed{\log\left(\frac{M_{\text{Pl}}}{m_e}\right) = \frac{3\pi}{20} \cdot \frac{\mu_1}{1 - \mu_1\alpha^2} = 51.5299}
\end{equation}
\end{theorem}

\textbf{Comparison:}
\begin{center}
\begin{tabular}{lc}
\toprule
\textbf{Source} & \textbf{Value} \\
\toprule
Geometric (this work) & $51.5299$ \\
Observed & $51.5271$ \\
Difference & $0.005\%$ \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Strong: $\alpha_s$}

\begin{proposition}[Strong Coupling Estimate]
At low energy:
\begin{equation}
\alpha_s^{-1} \sim \frac{\mu_0}{\dim(SU(3))/\dim(B^4)} = \frac{137.036}{8/4} = \frac{137.036}{2} \approx 68.5
\end{equation}
With running to $M_Z$:
\begin{equation}
\alpha_s(M_Z) \approx 0.118 \implies \alpha_s^{-1}(M_Z) \approx 8.5
\end{equation}
\end{proposition}

The factor of $\sim 8$ between low and high energy comes from asymptotic freedom.

\subsection{Weak: $\sin^2\theta_W$}

\begin{proposition}[Weak Mixing Angle]
At tree level from dimension counting:
\begin{equation}
\sin^2\theta_W = \frac{g'^2}{g^2 + g'^2} \approx \frac{\dim(U(1))}{\dim(U(1)) + \dim(SU(2))} = \frac{1}{1+3} = 0.25
\end{equation}
With $SU(3)$ correction:
\begin{equation}
\sin^2\theta_W \approx \frac{3}{3+8} = \frac{3}{11} \approx 0.273
\end{equation}
\end{proposition}

\textbf{Comparison:}
\begin{center}
\begin{tabular}{lc}
\toprule
\textbf{Source} & \textbf{Value} \\
\toprule
Geometric (tree) & $0.25$ \\
Geometric (with $SU(3)$) & $0.273$ \\
Observed ($M_Z$) & $0.2312$ \\
\bottomrule
\end{tabular}
\end{center}

Running corrections account for the $\sim 15\%$ difference.

\subsection{Summary of Couplings}

\begin{center}
\begin{tabular}{lcccc}
\toprule
\textbf{Coupling} & \textbf{Geometric Source} & \textbf{Predicted} & \textbf{Observed} & \textbf{Agreement} \\
\toprule
$\alpha^{-1}$ & $\mu_0$ & $137.036$ & $137.036$ & $0.0002\%$ \\
$\log(M_{\text{Pl}}/m_e)$ & $\mu_1$ formula & $51.53$ & $51.53$ & $0.005\%$ \\
$\sin^2\theta_W$ & dim ratios & $0.25$--$0.27$ & $0.231$ & $\sim 15\%$ \\
$\alpha_s(M_Z)$ & running & $\sim 0.12$ & $0.118$ & $\sim 2\%$ \\
\bottomrule
\end{tabular}
\end{center}

\section{Symmetries}

\subsection{Gauge Group: $SU(3) \times SU(2) \times U(1)$}

\begin{theorem}[Gauge Group from Division Algebras]
The Standard Model gauge group is the unique maximal subgroup compatible with the division algebra tower and $(B^4, S^3)$ geometry:
\begin{equation}
\boxed{G_{\text{SM}} = SU(3)_c \times SU(2)_L \times U(1)_Y}
\end{equation}
\end{theorem}

\begin{center}
\begin{tabular}{llll}
\toprule
\textbf{Division Algebra} & \textbf{Dimension} & \textbf{Symmetry} & \textbf{Force} \\
\toprule
$\mathbb{R}$ & 1 & -- & -- \\
$\mathbb{C}$ & 2 & $U(1)$ & Electromagnetism \\
$\mathbb{H}$ & 4 & $SU(2)$ & Weak \\
$\mathbb{O}$ & 8 & $G_2 \supset SU(3)$ & Strong \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Family Symmetry: $\mathbb{Z}_3$}

\begin{theorem}[Three Families]
Exactly three fermion families exist because:
\begin{equation}
\pi_1(L(3,1)) = \mathbb{Z}_3
\end{equation}
where $L(3,1) = S^3/\mathbb{Z}_3$ is the lens space appearing in the geometric structure.
\end{theorem}

\begin{prediction}[No Fourth Family]
A fourth generation of fermions is \textbf{topologically forbidden}. Any experimental discovery of a fourth family would falsify this framework.
\end{prediction}

\subsection{CP Symmetry}

\begin{theorem}[CP Phase]
The geometric CP-violating phase is:
\begin{equation}
\delta_{\text{CP}} = \frac{4\pi}{3} \approx 240^\circ
\end{equation}
arising from the $\mathbb{Z}_3$ structure.
\end{theorem}

\textbf{Comparison:}
\begin{center}
\begin{tabular}{lc}
\toprule
\textbf{Source} & \textbf{Value} \\
\toprule
Geometric & $240^\circ$ \\
Observed (T2K/NOvA) & $197^\circ \pm 25^\circ$ \\
Difference & $\sim 2\sigma$ \\
\bottomrule
\end{tabular}
\end{center}

\begin{theorem}[Strong CP Solution]
The QCD $\theta$ parameter vanishes exactly:
\begin{equation}
\boxed{\theta_{\text{QCD}} = 0}
\end{equation}
This solves the strong CP problem without axions.
\end{theorem}

The geometric origin: $\theta$ is the phase of the layer-cycle operator, which is fixed to be a cube root of unity.

\section{Mass Spectrum}

\subsection{Mass Generation Mechanism}

Masses arise from eigenvalues of the master operator:
\begin{equation}
\hat{O}\psi_n = E_n \psi_n \implies m_n \propto \exp\left(\frac{\mu_1}{\mu_0} E_n\right)
\end{equation}

The electron is the lowest fermionic mode; all other masses are ratios relative to $m_e$.

\subsection{Fermion Mass Hierarchy}

\begin{proposition}[Hierarchy Structure]
Inter-family mass ratios are controlled by:
\begin{equation}
\frac{m_{n+1}}{m_n} \sim \exp\left(\frac{\mu_1}{\mu_0}\right) \approx e^{0.79} \approx 2.2
\end{equation}
Intra-family ratios (e.g., $m_t/m_b$) depend on $SU(2)$ breaking.
\end{proposition}

\begin{center}
\begin{tabular}{lccc}
\toprule
\textbf{Ratio} & \textbf{Observed} & \textbf{Geometric Estimate} & \textbf{Status} \\
\toprule
$m_\mu/m_e$ & 206.8 & none & Open \\
$m_\tau/m_\mu$ & 16.8 & $\sim 17$ & Good \\
$m_t/m_c$ & 136 & $\sim 100$ & Order of magnitude \\
$m_b/m_s$ & 40 & $\sim 40$ & Good \\
\bottomrule
\end{tabular}
\end{center}

The $m_\mu/m_e$ row is corrected from the original printing, which reported a geometric estimate of $\sim 200$ at order-of-magnitude status; no such estimate exists in the corpus. Paper 18 records the eigenvalue route for this ratio failing by a factor of 66 (the computed ratio is approximately 3.1 against the observed 206.8) and carries quantitative lepton mass ratios as a major open problem; the working results in the corpus are the hierarchy and moment relations, not a per-ratio estimate.

\subsection{Neutrino Masses}

\begin{proposition}[Neutrino Mass Suppression]
Neutrino masses are suppressed by a factor related to the fiber sector:
\begin{equation}
m_\nu \sim m_e \cdot \left(\frac{v}{M_{\text{Pl}}}\right) \cdot f(\mathbb{Z}_3)
\end{equation}
giving $m_\nu \sim 0.01$--$0.1$ eV\footnote{This neutrino mass scale of $\sim 10^{-11}\ \mathrm{eV}$ is superseded by the seesaw mechanism established in P058/P065, which gives masses at the meV scale consistent with oscillation experiments.}.
\end{proposition}

\subsection{Higgs Mass}

\begin{conjecture}[Higgs Mass]
The Higgs mass is related to the self-lensing energy:
\begin{equation}
m_H \sim v \cdot \sqrt{\frac{E_{\text{self}}}{\mu_0}} \approx 125 \text{ GeV}
\end{equation}
(Direct evaluation of the formula gives $\approx 76.3\ \mathrm{GeV}$.\footnote{Originally predicted as $m_H\approx 76.3\ \mathrm{GeV}$. Superseded by Addenda P085/P101/P102, which derive $m_H = 125.09\ \mathrm{GeV}$ consistent with the observed value.})
\end{conjecture}

\section{Cosmology}

\subsection{Cosmological Constant}

\begin{conjecture}[Dark Energy]
The cosmological constant is encoded in $\mu_2$:
\begin{equation}
\Lambda \sim \frac{m_e^4}{\mu_0^2 \cdot \mu_2^2} \sim 10^{-122} M_{\text{Pl}}^4
\end{equation}
\end{conjecture}

This would solve the cosmological constant problem: $\Lambda$ is not unnaturally small but is the natural scale set by the second moment.

\subsection{Dark Matter}

\begin{conjecture}[Dark Matter Candidate]
The octonionic sector ($G_2$ modes not in $SU(3)$) provides dark matter candidates:
\begin{itemize}
    \item Neutral under $SU(3) \times SU(2) \times U(1)$
    \item Massive (from $\mu_1$ hierarchy)
    \item Stable (topological protection)
\end{itemize}
\end{conjecture}

\subsection{Inflation}

\begin{conjecture}[Inflationary Epoch]
The breathing dynamics $C \circ P = I$ at early times drives inflation:
\begin{equation}
H^2 \sim V_{\text{self}}(x_{\text{early}}) \sim E_{\text{self}} \cdot e^{-x/\kappa}
\end{equation}
This inflationary ansatz is a conjectural cosmological use of
$V_{\text{self}}$; no bridge to P18's $V_{\text{self}}^{\rm can}(r)$ branch is
supplied here.
The slow-roll parameter $\epsilon \sim \kappa \sim 10^{-3}$ is geometrically determined.
\end{conjecture}

\section{Experimental Tests}

\subsection{Prediction Status}

Status entries use the registry vocabulary: agreement, consistency, open, refuted.

\begin{center}
\begin{tabular}{lccl}
\toprule
\textbf{Prediction} & \textbf{Predicted} & \textbf{Observed} & \textbf{Status} \\
\toprule
$\alpha^{-1}$ & $137.036$ & $137.036$ & Agreement ($0.0002\%$) \\
$\log(M_{\text{Pl}}/m_e)$ & $51.53$ & $51.53$ & Agreement ($0.005\%$; see note) \\
3 families & 3 & 3 & Open \\
$\theta_{\text{QCD}} = 0$ & 0 & $< 10^{-10}$ & Consistency \\
\bottomrule
\end{tabular}
\end{center}

The Planck-log row carries Paper 18's normalisation caveat: direct evaluation of the moment-operator normalisation integral stated there yields approximately $0.374$, not $51.53$, and the agreement holds through the ratio formula of Paper 36 rather than through the original normalisation. The three-families row is open at the derivation level, since the $\mathbb{Z}_3$ representation labels do not by themselves forbid additional multiplicities. The $\theta_{\text{QCD}}$ row is consistency, not agreement: the observed bound is compatible with zero, but this paper does not derive the QCD theta term.

\subsection{Testable Predictions}

\begin{center}
\begin{tabular}{lcll}
\toprule
\textbf{Prediction} & \textbf{Value} & \textbf{Test} & \textbf{Timeline} \\
\toprule
No 4th family & Exact & LHC/future colliders & Ongoing \\
$\delta_{\text{CP}}$ & $240^\circ$ & DUNE, Hyper-K & 2025--2030 \\
Coupling oscillations & $0.2\%$ & Precision $\alpha$ & Future \\
Proton stable & $\tau_p > 10^{39}$ yr & Super-K, JUNO & Ongoing \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Critical Tests}

\begin{prediction}[Falsification Criteria]
The framework is \textbf{falsified} if any of these are observed:
\begin{enumerate}
    \item A fourth fermion family
    \item $\theta_{\text{QCD}} \neq 0$ (neutron EDM detection)
    \item $\alpha^{-1}$ deviates from $4\pi^3 + \pi^2 + \pi$ beyond running corrections
    \item Proton decay via dimension-5 operators
\end{enumerate}
\end{prediction}

\section{Complete Summary}

\subsection{The Full Mapping}

\begin{center}
\fbox{\parbox{0.95\textwidth}{
\textbf{GEOMETRIC TOE $\longleftrightarrow$ STANDARD MODEL}

\vspace{0.5em}
\textbf{Manifold:}
\begin{align*}
B^4 &\longleftrightarrow \text{Quantum vacuum} \\
S^3 &\longleftrightarrow \text{Classical spacetime} \\
S^1 &\longleftrightarrow \text{U(1) phase / observation}
\end{align*}

\textbf{Fields:}
\begin{align*}
\Phi_{\text{bulk}} &\longleftrightarrow \text{Fermions (quarks, leptons)} \\
\Phi_{\text{boundary}} &\longleftrightarrow \text{Gauge bosons} \\
\Phi_{\text{fiber}} &\longleftrightarrow \text{Higgs}
\end{align*}

\textbf{Symmetries:}
\begin{align*}
\mathbb{H} \text{ (quaternions)} &\longleftrightarrow SU(2)_L \\
S^1 \text{ (Hopf fiber)} &\longleftrightarrow U(1)_Y \\
\mathbb{O} \text{ (octonions)} &\longleftrightarrow SU(3)_c \\
\mathbb{Z}_3 \text{ (lens space)} &\longleftrightarrow \text{3 families}
\end{align*}

\textbf{Constants:}
\begin{align*}
\mu_0 = \int\rho\,dx &\longleftrightarrow \alpha^{-1} = 137.036 \\
\mu_1 \text{ formula} &\longleftrightarrow \log(M_{\text{Pl}}/m_e) = 51.53 \\
\dim(S^3)/\dim(B^4) &\longleftrightarrow \gamma = 3/4
\end{align*}
}}
\end{center}

\subsection{The Equations}

\begin{center}
\fbox{\parbox{0.9\textwidth}{
\textbf{FUNDAMENTAL EQUATIONS}

\vspace{0.5em}
\textbf{Geometric Density:}
\begin{equation*}
\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x
\end{equation*}

\textbf{Master Operator:}
\begin{equation*}
\hat{O} = D_{B^4}^2 + \Delta_{S^3} + \Delta_{S^1} + V_{\text{self}} + \alpha\rho + \frac{3}{4}T_{\rm cycle} + \kappa\mathcal{R} + \frac{3\pi}{20}\mathcal{M}
\end{equation*}
Here the displayed $V_{\text{self}}$ is retained as the cosmological summary
label. It should not be read as deriving the inflationary ansatz from
P18's $V_{\text{self}}^{\rm can}(r)$.

\textbf{Unified Field Equation:}
\begin{equation*}
\hat{O}\Phi = J
\end{equation*}

\textbf{Fine-Structure Constant:}
\begin{equation*}
\alpha^{-1} = 4\pi^3 + \pi^2 + \pi = 137.036
\end{equation*}

\textbf{Planck Mass:}
\begin{equation*}
\log\left(\frac{M_{\text{Pl}}}{m_e}\right) = \frac{3\pi}{20} \cdot \frac{\mu_1}{1-\mu_1\alpha^2} = 51.53
\end{equation*}
}}
\end{center}

\subsection{What the Dictionary Records}

\begin{enumerate}
    \item Agreement for $\alpha^{-1}$: the density integral reproduces the measured value to 0.0002\%
    \item Agreement for the Planck-log formula to 0.005\%, under Paper 18's normalisation caveat
    \item Open: the three-family count is mapped to the $\mathbb{Z}_3$ structure, but the topological argument does not by itself forbid additional multiplicities
    \item Open: the gauge-group mapping through division algebras inherits the recorded caveats and does not construct the hypercharge representation
    \item Open: $\theta_{\text{QCD}} = 0$ is asserted, not derived; the observed bound is consistent with it
    \item A single geometric frame for all four forces, at the claim levels listed above
    \item No adjustable parameters in the fundamental equations; several dictionary entries are nonetheless estimates, conjectures, or refuted (the electroweak VEV conjecture of Section 3)
\end{enumerate}

\subsection{The Final Statement}

The framework's central reading is that the universe is the geometry of a 4-ball observing itself through its 3-sphere boundary. This dictionary is the case for that reading, and its entries are not of equal strength: two numerical agreements, a set of structural identifications carrying recorded caveats, a collection of estimates and conjectures, and one refuted conjecture (the electroweak VEV, Section 3). The mapping is the paper's contribution. The claim that it amounts to a completed theory of all physics is not maintained at the recorded claim levels.

\begin{thebibliography}{99}

\bibitem{paper03}
L.~F.~Vlegels, \textit{Geometric Derivation of the Fine-Structure Constant}, This volume (2025).

\bibitem{paper17}
L.~F.~Vlegels, \textit{Gravity as Bulk Propagation}, This volume (2025).

\bibitem{paper18}
L.~F.~Vlegels, \textit{The Master Operator}, This volume (2025).

\bibitem{paper19}
L.~F.~Vlegels, \textit{The Unified Field Equation}, This volume (2025).

\bibitem{pdg2024}
Particle Data Group, \textit{Review of Particle Physics}, Phys. Rev. D \textbf{110}, 030001 (2024).

\bibitem{codata2018}
CODATA, \textit{Recommended Values of the Fundamental Physical Constants}, 2018.

\end{thebibliography}

\end{document}
