The Fractal Coupling Flow of the Minimal Discrete Épure₅/6

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 Φₙ (G■) = cₙ Bⁿ induces a canonical, parameter-free coupling flow αₙ = cₙ²/ (4π). MAIN RESULTS T1) EXACT GAUGE COUPLING g²G■ = 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/αEM = (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.