The Fibonacci Cascade Field Theory (FCFT): Feynman Rules, Exact Predictions for 𝛼 and 𝛼𝑠, and BRST Gauge Structure

We present the formal Feynman rules, tree-level scattering amplitudes, and UV structure of the Fibonacci Cascade Field Theory (FCFT) — the quantum field-theoretic formulation of Fractal Mechanics (FM). The cascade field 𝜓𝑛(𝑥, 𝑡), organized at 291 levels with frequencies 𝜔𝑛 = 𝜔𝑃 𝜑^−𝑛, is shown to be equivalent (in the long-range statistical limit) to the self-gravitating gas field theory of De Vega, Sánchez & Combes (1996–2002), with the golden ratio ln(𝜑) playing the role of the gravitational temperature parameter. The tree-level mass spectrum exhibits spontaneous symmetry breaking: the free-field vacuum 𝜓𝑛 = 0 is unstable (tachyonic modes at high 𝑛), and stable particles emerge as nonperturbative soliton solutions — a FM-native Higgs mechanism requiring no external scalar field. The theory is UV-finite by construction: the Planck frequency 𝜔𝑃 = 𝜔0 provides a physical cutoff, making all loop integrals bounded. We derive the one-loop self-energy correction and show it contributes only finite renormalization. We identify a 𝑍_𝑁 discrete symmetry (𝑛 ↔ 𝑁 −𝑛) that governs anomaly cancellation. The fine structure constant is derived exactly: 𝛼⁻¹ = 2𝑅𝐾/𝐹14 + 4 ln 2/(9𝜋) = 137.03603 (0.23 ppm, one loop); the DHN two-loop correction 𝛿DHN ≈ −8 × 10⁻⁶ improves this to 0.17 ppm with zero free parameters. A new FM-specific energy scale 𝐸_FM ≈ 4.5 × 107 GeV (𝑛 ≈ 55 in the cascade) is predicted, corresponding to the strong-coupling transition of the cascade field — with no Standard Model counterpart. The strong coupling constant is derived by the same mechanism: 𝛼⁻¹(𝑀𝑍) = 8𝜋/3 + 4 ln 2/(9𝜋) = 8.4756, giving 𝛼𝑠(𝑀𝑍) = 0.11799 (observed: 0.1180 ± 0.0009; within experimental uncertainty). The SU(3) FM reference is 𝐹6 = 8 (number of Gell-Mann matrices), while EM uses 𝐹14 = 377 — the two Fibonacci keys are separated by exactly 8 cascade levels, consistent with 𝐹14/𝐹6 ≈ 𝜑8 . A GUT unification scale at 𝑛 ≈ 10, 𝐸 ≈ 1017 GeV is predicted from the SU(3) cascade Landau pole. Eight additional predictions, testable within 5 years, are identified: (1) 𝜑-periodic modulation of quantum revival periods in Rydberg atoms; (2) gravitational coupling correction near compact objects; (3) ISM fractal dimension modulation; (4) VP running scale 𝑞FM = 41/3𝑚𝑒 via electron scattering form factor; (5) FM 𝛽-function at 𝐸FM ≈ 4.5 × 107 GeV; (6) 𝛼𝑠 (𝑀𝑍) from SU(3) D5 cascade [this work]; (7) GUT unification scale from SU(3) Landau pole; (8) boomerang return probability 𝜑-periodically modulated.v1.1: Appendix A (Born rule derived)