Orbital Model Theory of Primordial Particles

Abstract

# Derivation of Core Standard‑Model Structure from Nine Axioms This paper derives the core structure of the Standard Model of particle physics starting from nine axioms. The axioms postulate a fundamental entity—the primordial particle—that undergoes isotropic closed‑orbit motion on a compact three‑dimensional Riemannian manifold $\mathcal{M} = S^2(r_0) \times I_\delta$. Its orbital phase propagates through three‑dimensional space via a scalar field $k(\mathbf{x})$. The $k$-field adopts force‑balanced vacua at integer‑valued points, and its self‑interaction generates a periodic potential $V(k) = V_0\bigl[1-\cos(2\pi k)\bigr]$. The axiomatic framework yields the following results. 1. The geometric classification of many‑body bound states is governed by three‑dimensional point groups. Cascade decay (fragmentation) of fragments produces a decay tree with absolute terminals at $N_v=2$ and $N_v=20$. 2. The $\ell=2$ quadrupole‑deformation moduli space of the $k$-field on the compact $S^2$ manifold gives rise, through differential geometry and Cartan involution, to the emergent gauge symmetry $U(1)\times SU(2)\times SU(3)$. The gauge‑group dimensions and coupling ratios are determined by the purely geometric quantity $f_{\text{geom}}$ of the underlying entity geometry. 3. Terminal entities with $N_v=2$ correspond to leptons. The mass formula $M = 2(1+n)m_0$ ($n$ being a positive integer) yields an equidistant resonance spectrum with spacing $\Delta M = 2m_0 \approx 0.2555$ MeV. The case $n=1$ is topologically absolutely stable and is identified with the electron. The cases $n=409$ and $n=6960$ (muon and tau lepton) lie inside the visibility window set by production cross‑sections and gauge‑decay lifetimes. Gauge decay widths follow an $M^5$ scaling law; the prediction for $\tau\to e\nu\bar{\nu}$ deviates from observation by less than $5\%$. 4. CKM‑matrix parameters and the Weinberg angle are fixed by the $T_d$ geometry and $f_{\text{geom}}$, with deviations from experimental values below $4\%$ and $2\%$, respectively. Neutrino masses arise naturally via a seesaw mechanism from two‑step vacuum tearing along equilateral fragmentation chains. The theory predicts the normal mass hierarchy with three‑generation neutrino mass values $m_2\approx 8.6$ meV and $m_3\approx 48.9$ meV. The theory requires three experimental calibrations: $m_0 = m_e/4$, $f^2 r_0^2$ (fixed by $\alpha_1$), and $f$ (fixed by $m_W$). These correspond to $m_e$, $\alpha_1(M_Z)$, and $m_W$ in the Standard Model. Out of the 19 free parameters of the SM, gauge couplings, CKM parameters, hypercharge assignments and the functional form of the lepton‑mass spectrum reduce to these three calibrations plus purely geometric outputs. The Higgs mass, dark‑energy density and the existence of three generations are not independently locked by the axioms; see the honesty audit in the main text for details. The most direct falsifiable test of the theory consists in searching for equidistant resonance peaks with spacing $\Delta M = 0.2555$ MeV in the $e^+e^-$ invariant‑mass spectrum over the 0.5–100 MeV energy range. This is a rigid prediction with no adjustable free parameters.