Spectral Geometry of the Poincaré Homology Sphere: Particle Structure, Gauge Symmetry, and the Proton–Electron Mass Ratio from S³/2I

The Poincaré homology sphere S³/2I — the quotient of the 3-sphere by the binary icosahedral group of order 120 — is shown to encode the structure of the four stable particles, the SU(3) gauge group, and the proton–electron mass ratio within a single geometric framework. The Zoll S³ commitment, made independently for cosmological reasons, produces the following results without additional assumptions: (1) de Rham cohomology H^k(S³) assigns four stable particles to the four form degrees, with Poincaré duality enforcing exact charge equality between proton and electron; (2) ℤ-valued cohomology groups guarantee baryon and lepton number conservation as topological invariants; (3) the proton (0-form) is necessarily composite while the electron (3-form) is necessarily elementary; (4) Eisenstein three-phase closure in the quaternion structure of S³ produces the A₂ root system, uniquely determining su(3) and its 8 gluon modes; (5) the Laplacian spectrum of S³/2I has its first excited state at l = 12 with eigenvalue 168 = |PSL(2,7)|, and the mode at l = 42 = V + E has eigenvalue 43² − 1 = 1848, with eigenvalue ratio 11; (6) the physical mass ratio μ = 1848 − 12 = 1836 agrees with 6π⁵ to 0.006%. Each result derives from a different branch of standard mathematics, all converging on one manifold with zero adjustable parameters.