Energy Continuum Theory: Mathematical Foundation of Complex Spacetime and Geometric Mass Generation

We establish the rigorous mathematical foundation of Energy Continuum Theory (ECT) through complex spacetime structure and derive the geometric origin of particle masses from the ℂ³ axiom system.

Building on the fundamental postulate that energy is the sole primary substance, we extend spacetime to a complex 6-dimensional manifold ℂ³ = ℝ³ ⊕ iℝ³, where the imaginary component iℝ³ represents velocity fields and time emerges as accumulated phase along Hopf fibers S¹. The vacuum geometry discretizes into an AGR/ORG quasicrystal lattice with Rhombic Triacontahedron (RT) as the fundamental unit, possessing intrinsic stiffness constants A ≈ 137 (bulk modulus) and B ≈ 3283 (structural rigidity).

Key results include:

1. Precise predictions for W and Z boson masses:
   M_W = (5/8·A + 64/B)·m_p = 80.379 GeV (0.012% error)
   M_Z = (A - 4π² - 19·64/B)·m_p = 91.188 GeV (0.0004% error)

2. The topological integral proof of the proton-electron mass ratio:
   m_p/m_e = 6π⁵ ≈ 1836.15
   achieving 10⁻⁶ precision compared to CODATA 2022.

3. The beta decay mechanism as geometric fracture: neutron decay proceeds via equatorial belt rupture following Fibonacci ratio 5/8, with W boson lifetime τ_W ≈ 3×10⁻²⁵ s arising from 38 damped oscillations of the fractured arc structure.

4. Unified time dilation from vacuum viscosity: gravitational time dilation emerges from local deformation of the AGR/ORG quasicrystal lattice, stretching Hopf fiber S¹ and inducing topological viscosity.

The framework achieves sub-percent precision across multiple independent observables (fine structure constant α⁻¹ = A - 1/B ≈ 137.036 with 10⁻⁸ accuracy, electroweak boson masses, proton-electron ratio) using only two empirically determined geometric constants A and B, suggesting that the Standard Model's 19 free parameters may derive from vacuum topology. We demonstrate that mass arises from topological impedance rather than Higgs field coupling, with particles representing geometric eigenmodes of the same elastic medium.

KEYWORDS:
Energy Continuum Theory, Complex Spacetime, Hopf Fibration, Quasicrystal Lattice, Rhombic Triacontahedron, Geometric Mass Generation, W Boson, Z Boson, Higgs Boson, Proton-Electron Mass Ratio, Fine Structure Constant, Beta Decay, Fibonacci Ratio, Time Dilation, Vacuum Viscosity, Topological Impedance, AGR/ORG Lattice, Shechtman Quasicrystal, Vacuum Constants A and B, Electroweak Unification