Fractal Mechanics: A Unified Framework

We present Fractal Mechanics (FM), a geometric framework deriving particle masses, force coupling constants, and cosmological parameters from a single organizing principle: the universe as a 291-level cascade of Fibonacci temporal harmonics connecting the Planck scale to the cosmic horizon. Exact coupling constants (FM v2.1, zero free parameters): The fine structure constant is derived as 𝛼⁻¹ = 2𝑅𝐾/𝐹_14 + 4 ln 2/(9𝜋) = 137.036031$ (0.23 ppm precision). The strong coupling constant is derived by the same universal correction: 𝛼⁻¹_MZ = 8𝜋/3 + 4 ln 2/(9𝜋) = 0.11799 (0.013%). The Weinberg angle at the GUT scale derives from the Fibonacci ratio 𝜃𝑊 = 𝐹4/𝐹6 = 3/8, running to 0.2312 at 𝑀𝑍 (0.05%). The SU(3) cascade Landau pole at cascade level 𝑛 ≈ 10 (𝐸 ≈ 1017 GeV) predicts GUT unification without postulating it. Particle masses. The 17 fundamental particle masses are derived with average precision 1.08% and median 0.30%, using two calibration references (electron, quark up) and zero free parameters for mass ratios. Fifteen of 16 derived quantities fall below 2% error in FM v2.1 (vs 9/15 in FM v2.0). New geometric foundations. The Einstein Equivalence Principle is derived as a cascade identity: the parameter Im(𝜑^𝑛) simultaneously controls gravitational coupling and local time rate, with coefficient 𝛽 = 𝜑 from the FM Lagrangian. Antimatter is identified as the reverse-cascade direction 𝜉− (𝐷3 → 𝐷𝑛), replacing CPT with the more fundamental CPG symmetry (C, P, cascade inversion G). This gives M-AM gravitational repulsion, naturally explaining the matter-antimatter asymmetry and predicting dark energy 𝑤 ∈ [−1, −1/3] — consistent with DESI 2024 (𝑤0 ≈ −0.7) without fine-tuning. FM v2 results retained: Hubble tension resolution at 4.36𝜎; vacuum impedance 𝑍0 = 𝐹14 = 377 Ω (0.07%); dark matter as fractal solitons; Big Bang as unique quantum white fountain; gravity as Hopfield cascade relaxation; Weinberg (1976) asymptotic safety completion. Appendix D (FCFT) develops the complete quantum field theory: Feynman rules, soliton quantization (Q-ball, DHN, Pöschl-Teller), BRST gauge consistency (𝑄2 = 0 ⇔ 𝑁 = 291 — three independent derivations), and SU(3) predictions from the D₅ icosahedral sector. Ten near-term falsification criteria are maintained, with primary tests: neutrino normal hierarchy (JUNO 2033), gravitational free-fall of antimatter (ALPHA-g/GBAR 2025), and 𝑤(𝑧) evolution toward -1/3 at low redshift (Euclid 2026–2030). Keywords: Fibonacci cascade, temporal harmonics, fine structure constant, strong coupling constant, Weinberg angle, GUT unification, Pöschl-Teller soliton, BRST symmetry, Einstein equivalence principle, antimatter cascade, dark energy, Fractal Mechanics