What question did this study set out to answer?

The aim is to derive the proton-to-electron mass ratio from a theory involving geometric properties of a compact S².

April 21, 2026Open Access

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

Key Points

The aim is to derive the proton-to-electron mass ratio from a theory involving geometric properties of a compact S².
Utilized the three time dimensions (3+3) spacetime model with a compact S².
Derived the mass ratio from integer inputs representing physical constants involving colour charges and Higgs boson.
Applied the Euler-Poincaré theorem and explored implications of the winding number.
The calculation yielded a proton-to-electron mass ratio of 1836, closely aligning with the established value (1836.152673).
Establishes that QCD confinement scale is fixed by the same integers used in the mass ratio derivation.
Proves that protons are stable under this framework, suggesting no proton decay observed in experiments.

Abstract

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.

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

Key Points

Abstract

Cite This Study