Three Fundamental Constants from the Fibonacci Cascade: 𝛼⁻¹, 𝑍₀, and the Tau Lepton Mass

We identify three fundamental constants of nature whose values are reproduced by the Fibonacci golden-ratio cascade with no adjustable parameters: (1) The inverse fine structure constant: 𝛼⁻¹ ≈ (𝜑9 + 𝜑10 + 𝜑11) /3 + 𝜑3 = 136. 903, within 0. 097% of the measured value 137. 036. (2) The vacuum impedance: 𝑍₀ ≈ 144𝜑² = 𝐹14 = 377. 0 Ω, within 0. 070% of the measured value 376. 730 Ω. (3) The tau lepton mass: 𝑚𝜏 = 𝑚𝑒 × (𝛼⁻¹) ⁵/3 = 1858 MeV within 4. 5% of 1776. 86 MeV using the cascade-motivated exponent 𝐷5/𝐷3 = 5/3; the empirical exponent 𝑥 = 1. 6584 reproduces 𝑚𝜏 to < 0. 01%. The convergence of three independent constants at the ∼ 0. 1% precision floor from a single geometric structure — with no free parameters — provides strong evidence that the Fibonacci cascade encodes real physical content. We provide physical interpretations and outline testable implications, including a predicted radiative correction to 𝑍₀.