We establish the mathematical foundation of Energy Continuum Theory (ECT) through Hamiltonian mechanics on the symplectic phase space T*ℝ³ ≅ ℂ³. The Lorentzian signature emerges from the Kähler structure: the almost complex operator J with J² = -𝕀 induces the metric g(u,v) = ω(u,Jv), where the minus sign arises algebraically rather than kinematically. Time is not a pre-existing dimension but emerges from velocity conservation |v_ext|² + |v_int|² = c² in 6D phase space, with proper time τ measuring internal rotation: c dτ = |ds|. The vacuum geometry discretizes into an AGR/ORG quasicrystal lattice (Rhombic Triacontahedron unit) with stiffness constants derived from symplectic action integrals over the Hopf fibration S¹ ↪ S³ → S²: A = 4π³ + π² + π ≈ 137.036 (bulk/shell/fiber contributions) and B = 30π(π³+π) + 2π³+π ≈ 3283.515 (flux quantization on 30 RT faces). The fine structure constant follows from classical-quantum action difference: α⁻¹ = A - 1/B ≈ 137.036 (3.4 × 10⁻⁸% precision). Particle masses arise as Hamiltonian eigenvalues: 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). The proton-electron mass ratio emerges from symplectic volume 6π⁵ with higher-order corrections m_p/m_e = 6π⁵ + C_α·α + C_B/B achieving 10⁻⁸ precision. The π⁴ term is the symplectic capacity ∫_{S²×S²} Ω∧Ω, rigorously derived from the electron's double fiber bundle structure. Using only two geometric constants A and B with no free parameters, ECT achieves sub-percent agreement across multiple observables, suggesting the Standard Model's 19 parameters may originate from vacuum topology. KEYWORDS:Energy Continuum Theory, Complex Spacetime ℂ³, Hopf Fibration, AGR/ORG Quasicrystal, Geometric Mass Generation, W/Z Boson Masses, Beta Decay Geometry, Topological Integral, Proton-Electron Mass Ratio 6π⁵, Vacuum Impedance, Fine Structure Constant, Time Dilation, Topological Viscosity