Derivation of the Strong Coupling Constant from an Associative 3-Cycle of a φ-Metric 7-Torus

We derive the strong coupling constant αs from the geometry of a compact associative 3-cycle ofthe φ-metric 7-torus introduced in a companion paper 1. The strong coupling arises from KaluzaKlein reduction of a gauge field on the unique minimum-volume associative 3-cycle of the φ-metrictorus– the Fano triple (1,2,4). This cycle has volume (2π)3 φ−7, giving the tree-level resultαs = φ7(2π)3 = 0.11705.The PDG2022 value is αs(MZ) = 0.1180±0.0009, placing our result 1.05 standard deviations belowthe central value. The formula contains no free parameters. Both αs and the electromagnetic finestructure constant α are derived from the same geometric framework, providing a unified descriptionof the two most precisely measured gauge couplings from a single compact manifold.