The Proton-to-Electron Mass Ratio from S² Topology: A Zero-Parameter Derivation of mₚ / mₑ = 1836 from Two Integer Inputs

The dimensionless ratio mₚ/mₑ = 1836. 152673 is one of the most-studied numbers in physics: Eddington attempted to derive it from pure theory, Dirac included it in his large-numbers hypothesis, and Feynman called it one of the 'great mysteries. ' In the Standard Model it is measured, not derived. We show that in the Three Time Dimensions (3+3) spacetime model — in which the third time dimension t3 is a compact 2-sphere S² of radius R3 = π ℏ / (mₑ c) — the ratio follows as a theorem with zero free parameters from two integer inputs, each of which has an independent geometric derivation within the framework: (i) Nc = 3, the unique positive-integer solution to Ndefects = Nc (Nc+1) = 12 (Euler-Poincaré theorem applied to the S² defect ring, combined with trisection self-consistency), identified physically with the number of QCD colour charges; (ii) lH = 2, the angular-momentum quantum number of the first scalar Kaluza-Klein (KK) excitation on S², identified physically with the Higgs boson. From these two inputs alone, the proton mode number is nₚ = 2 × Nc³ × (Nc lH (lH+1) − 1) = 2 × 27 × 17 = 918, and the winding-mode mass lattice mₙ = 2n mₑ yields mₚ / mₑ = 4 × Nc³ × (Nc lH (lH+1) − 1) = 1836, compared to the PDG 2024 value 1836. 152673 — an error of 0. 008%. The three factors have direct physical meaning: 2 is the Dirac-spinor dimension, 27 = Nc³ is the three-quark / three-colour / three-sector combinatoric count, and 17 = Nc · lH (lH+1) − 1 counts the 18 total Higgs-colour coupling channels minus the one colour-singlet channel — the '−1' being the fingerprint of colour confinement written directly into the proton mode number. Two consequences follow. First, the QCD confinement scale is fixed by the same two integers: ΛQCD = (4/3) × nₚ × E3 = 2 mₚ / (3π), with E3 = mₑ c² / π the S² compactification energy. Second, the proton is absolutely stable: its winding number is a topological invariant of π₂ (S²) = Z, conserved under every renormalisable interaction. DUNE and Hyper-Kamiokande will not observe proton decay. A single observed decay event falsifies the framework.