Why 6π⁵: The Proton–Electron Mass Ratio from Pure Geometry

Every physical constant is a volume or tangency ratio on one geometric object with three aspects: the space of relationship (S⁵ = SU(3)/SU(2)), the structure of tangency (the Apollonian gasket with seed [−1, 2, 2, 3]), and light (the S¹ phase fiber that determines α). The electron is the unit of light. We compute the proton–electron mass ratio as a volume ratio on S⁵ and obtain 6π⁵ = 1836.118... (laboratory value: 1836.15267343, agreement: 99.998%). Twelve further predictions from the same geometry yield 13 independent quantities with 99.86% average accuracy and zero adjustable parameters. A companion note ("The Atomic Recipe") shows that each particle's mass formula reads addresses in the gasket's generation tree, with three geometric constraints (replacement position, sector, depth) collapsing the search space from millions of candidates to single digits. A Reader's Guide provides a plain-language walkthrough for non-specialists.