Orbital Model Theory of Primordial Particles

Abstract

This paper constructs a unified field theory where the fundamental constituent of the universe is the primordial particle, a point particle executing isotropic closed-loop motion on the 3-dimensional compact Riemannian manifold $\mathcal{M} = S^2(r_0) \times I_\delta$. Interparticle interactions are mediated by a real scalar $k$-field equipped with the circle topology $\mathbb{R}/\mathbb{Z}$. The periodic potential $V(k) = V_0[1-\cos(2\pi k)]$ is rigorously derived via the spectral truncation theorem, and the Yukawa coupling strength $f^2 = c_0^2/(4\pi\alpha_0)$ is uniquely fixed by the self-consistency condition of field energy variation and particle equations of motion. Core theoretical outputs are summarized as follows. Eight types of geometric entities (vertex-transitive point configurations) from Definition 1.8 serve as microscopic building blocks of matter. The entity scaling factor $f(N,G)$ in Definition 1.9 contains seven closed analytical expressions for the eight entity species, uniquely determining entity radii and mass spectra under $k$-field constraints. Theorem 14 derives the gauge group $U(1)\times SU(2)\times SU(3)$ from the moduli space of entity quadrupole deformations: the Cart involution $(A_i,S_a)\mapsto(A_i,iS_a)$ combined with the compact $\mathbb{R}/\mathbb{Z}$ topology lifts real Lie algebra $\mathfrak{sl}(n,\mathbb{R})$ to $\mathfrak{su}(n)$, giving a total dimension of $1+3+8=12$. Renormalization group analysis in Theorem 15 pins down the unification scale $\Lambda \approx 2\times 10^{16}$ GeV, yielding $\sin^2\theta_W(M_Z)=0.231$ and $\alpha_s(M_Z)\approx 0.12$, consistent with experimental measurements. The finite number of fermion flavors (3 lepton generations + 3 quark flavors) emerges as a rigid mathematical outcome of the Diophantine chain $N=4\rightarrow N=3\rightarrow N=2$. We introduce the fragmentation hypercharge mechanism: fermion $U(1)_Y$ hypercharge arises geometrically from fragmentation tree classification, written as $Y = s(N)\cdot Y_0(N)\cdot t + \delta_{\text{frag}}$. The $N$-parity factor $s(N)=(-1)^N$ sets the sign of base hypercharge; even-$N$ configurations carry $Y_0=1/4$ derived from geometric factor $f_{\text{geom}}(2)$, odd-$N$ configurations carry $Y_0=1/18$ from $N_3^2$. The fragmentation shift $\delta_{\text{frag}}$ only takes non-zero values $\pm 1/2$ for right-handed $u_R/d_R$ quarks removed from $N=3$ entities. This setup reproduces all seven independent Standard Model hypercharges and six electric charges precisely. A transverse Goldstone mechanism is proposed: transverse vibrational modes of the $S^2\times I$ orbital manifold produce three Goldstone bosons absorbed by $SU(2)$ gauge bosons, predicting $m_W\approx 82$ GeV and $m_Z\approx 92$ with less than 2% error and no free parameters. The integers labeling lepton radial excitations are solved via joint Diophantine optimization, giving $n=1,409,6960$. The Standard Model empirical values $n=412,6953$ deviate from this geometric optimum. The coefficients $\gamma_{\text{near}}(n)$ with $C=27,\ b=27\sqrt{6}$ are derived purely from first geometric principles. The bare mass $m_0 \approx 0.1278$ MeV is calibrated against electron mass, and the composite constant $f^2 r_0^2 \approx 4.696$ is fixed by the fine-structure constant $\alpha_1$. All key testable observables including generation mass ratios, gauge coupling ratios, weak mixing angle and hypercharge assignments depend solely on the geometric combination $f^2 r_0^2 + m_0$ and fragmentation topology, without any fitted free parameters. Lepton mass predictions deviate by less than 1% ($\mu$: 0.86%, $\tau$: 0.09%). The CKM parameter $\lambda = (n_1/n_2)^{1/4} \approx 0.222$ (1.2% deviation) and $A = \cos(\arccos(1/3)/2) \approx 0.816$ (1.6% deviation) follow entirely from geometric constraints. Nine quantitative predictions span cosmology, astrophysics and high-energy physics: DESI dark energy equation of state $w(z)$, CMB birefringence, early JWST galaxies, the $f^3$ gravitational wave spectrum observed by NANOGrav, PTA anisotropies, narrow equidistant $N=2$ resonances at $\Delta M=0.2555$ MeV, muon $g-2$, and the $C_9$ anomaly in $B$-meson decays. Prediction 7 for cosmic CMB birefringence $\beta\approx 0.3^\circ$ receives preliminary $3.6\sigma$ statistical support from Planck satellite data.