CAT's Theory: From Alpha to Omega: Zero-Parameter Derivation of the Standard Model Coupling Constants from Fibonacci Recursion on the Triadic Invariant

This paper derives all three Standard Model coupling constants and the weak mixing angle from the fiber dimension d = 3 using Fibonacci recursion, Lucas duality, Pisano periodicity, and the three counting modes of the Triadic Coherence Invariant. The fine-structure constant is derived as α = (d_Ψ + d_P + p)²π/(d³D²·p·F₁₂²) = 289π/124416, yielding 1/α = 137.034 (0.0014% from measured 137.036), where the leading-order result π/432 is corrected by the first excited harmonic on S³ through the Hopf angle of the T(2,3) knot. The weak mixing angle is derived as sin²θ_W = d/F(d_Ψ) = 3/13 = 0.2308 (0.19% from measured at M_Z). The strong coupling is derived as α_s(M_Z) = π/d³ = π/27 = 0.1164 (1.3% from measured). The ratio α_s/α = D² = 16 is exact at leading order. Additional derived quantities include 1/α_W = 31.73 (0.13% deviation) and M_W/M_Z = √(10/13) = 0.8771 (0.50% deviation). The QCD scale follows from dimensional transmutation with derived β-function coefficients b₀ = d_Ψ = 7 and b₁ = Σ = 26, producing the exponent −2d³/d_Ψ = −54/7. The lattice partition function for the triadic field equations has zero remaining free parameters once one dimensionful anchor is supplied, and the dielectric staircase prediction provides a tabletop experimental shortcut requiring only broadband microwave spectroscopy on supercooled water. Part of the CAT'S Theory corpus Keywords: fine-structure constant, weak mixing angle, strong coupling constant, Fibonacci sequence, Lucas numbers, Pisano period, golden ratio, torus knot, triadic coherence invariant, zero free parameters, CAT'S Theory, dimensional refraction, coupling constant derivation, Weinberg angle, Standard Model