The Fractal Coupling Flow of the Minimal Discrete Épure_[5/6]

We study the natural U(1) gauge coupling generated by the Minimal Discrete Épure G■ = (Z², d■), the unique lattice metric with π■ = 1. The unfolding sequence Φ_n(G■) = c_n B^n induces a canonical, parameter-free coupling flow α_n = c_n²/(4π). MAIN RESULTS T1) EXACT GAUGE COUPLING g²_G■ = 2 exactly (Theorem 1) Derived by Cauchy formula — zero free parameters. (T2) ASYMPTOTIC BETA FUNCTION β-function = 8π²/5 (Theorem 2) Intrinsic to the épure geometry. (T3) METALLIC RATIOS The golden ratio φ = (1+√5)/2 and silver ratio δ = 1+√2 jointly encode all geometric data of the épure. MAIN CONJECTURE — FINE STRUCTURE CONSTANT Under the discrete tangent-fiber hypothesis H₉, the exact coupling at n=9 is: 1/α₉ = 262144π/6125 ≈ 134.457 (within 1.92% of experimental) The fractal resummation of anisotropy corrections yields: 1/α_EM = (262144π/6125) / (1 − (2/3)|ξ₂|(1 − (π/8)|ξ₂|)) ≈ 137.036043 ERROR: 0.000032% — zero free parameters Where |ξ₂| = 0.028549 is the LIV coefficient of the épure (Article 4 of this series), and all coefficients (2/3, π/8 = c₃/(2c₂)) are intrinsic to the épure geometry. Comparison: k=0 (raw geometry) : 1/α = 134.457 error = 1.92% k=1 (first LIV) : 1/α = 137.066 error = 0.022% k=2 (second LIV) : 1/α = 137.036 error = 0.000032% ✓ NOTE The formula is presented as a rigorously supported numerical conjecture. Its theoretical derivation is identified as the central open problem of the π■ = 1 program. All results verified in Python 3 (scipy, numpy). Code available upon request.