Spectral Geometry of the Poincaré Homology Sphere

The Poincaré homology sphere S³/2I — the quotient of the 3-sphere by the binary icosahedral group of order 120 — encodes 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 in the companion Hodge Quartet paper [2], yields the following without additional assumptions. (1) The de Rham cohomology Hᵏ(S³) assigns four stable particles to the four form degrees, with Poincaré duality enforcing exact charge equality between proton and electron. (2) The ℤ-valued cohomology groups at k = 0 and k = 3 give baryon and lepton number conservation as topological invariants. (3) The proton (0-form, vertex) is composite while the electron (3-form, volume) is elementary, from the local/global duality of differential forms. (4) Eisenstein three-phase closure in the quaternion structure of S³ produces the A₂ root system, which uniquely determines su(3) and its 8-dimensional adjoint (the A₂ → su(3) step is rigorous Lie theory); whether the Eisenstein structure is itself *forced* — rather than selected via the preferred ℂ of [4], which marks this bridge OBSERVED — is open, so SU(3) is recovered as OBSERVED, not assumed but not yet fully derived. (5) The Laplacian spectrum of S³/2I, computed via the Molien formula for 2I, has its first excited state at l = 12 with eigenvalue 168 = |PSL(2,7)|, and a surviving mode at l = 42 = V + E with eigenvalue 43² − 1 = 1848, the two related by λ(42)/λ(12) = 11. (6) The mass ratio μ = λ₄₂ − [2I:C₁₀] = 1848 − 12 = 1836 agrees with the measured proton–electron ratio to within 83 ppm — a residual some 1.8 × 10³ times the experimental uncertainty. The integer part is exact; the fractional residual is not a near-miss but a structural boundary — an integer-eigenvalue-minus-integer-correction scheme cannot produce a non-integer, so the 0.153 marks physics the scheme cannot reach (§7.3). Results (1)–(3) are DERIVED from established theorems; result (4)'s terminal A₂ → su(3) step is DERIVED, but the gauge group as a whole is OBSERVED (the upstream forcing of the Eisenstein structure is inherited as OBSERVED from [4], §3). Result (5)'s spectrum is DERIVED; the identification of l = 42 as the electron mode is OBSERVED but not independently corroborated: the skeleton identity 42 = V + E is the additive restatement of Lemma L′(b) itself, the earlier "removes l = 41" framing is corrected (odd levels are removed for free by 2I parity; the honest comparison is l = 42 against its even neighbours, §4.4), and the remaining observations are consistency checks that co-vary with the identification. Result (6) is **CONDITIONAL**: the level-matching (L′(a)) and the vertex-vanishing with its orbit index (L′(c)) are derived theorems about the edge-dressed pairing P = ⋆∘(f₃₀·) — but neither lowers the eigenvalue, so neither bears on μ unaided; two clauses remain open — L′(b), that P is the physical pairing, and L′(d), that the subtraction is an observable of the physics rather than well-posed arithmetic (§5.2). The bare Hodge star is level-preserving and cannot itself select l = 42; the original "Hodge star selects" lemma was false and is replaced. Zero *continuous* adjustable parameters throughout; one *discrete* structural choice — the dressing degree among {12, 20, 30} — is presently fixed by the target ratio and is open as Lemma L′(b), not yet derived from structure. ---