What does this research mean for the field?

The proton-to-electron mass ratio is derived as μ = 6π5 + 1/29, matching the experimental value to 0.045 parts per million. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The aim is to derive the proton-to-electron mass ratio using mathematical structures from the Pisano tower.

March 7, 2026Open Access

The Proton-Electron Mass Ratio from Tower Arithmetic: mp/me = 6π5 + 1/29

Key Points

The aim is to derive the proton-to-electron mass ratio using mathematical structures from the Pisano tower.
Derive the formula for the proton-to-electron mass ratio from GF(3k) structures.
Calculate mass differences using specific mathematical constants and prime numbers.
Test other primes (p = 5, 7) to compare potential results.
The derived proton-electron mass ratio matches the experimental value to 0.045 parts per million.
The neutron-proton mass difference approximation aligns closely with established values to 2 ppm.
The tau-to-electron mass ratio demonstrates a 0.00017% accuracy compared to experimental results.

Abstract

We derive the proton-to-electron mass ratio μ = mp/me from the Pisano tower of GF(3k).The formula μ = 6π5 + 1/29 = 1836.15259 matches the experimental value 1836.15267to 0.045 parts per million (ppm). The integer 6 = φ(9) = |(Z/9Z)∗| and 29 is the 10thprime, where 10 = |P1(GF(9))|. We also derive the neutron–proton mass difference Δmnp =mp(1+2α/3)/729, where 729 = 36, matching the PDG value to 2 ppm. The tau-to-electronmass ratio mτ/me = (91/8)π5 − 15/4 = 3477.224 matches experiment to 0.00017%. Allthree formulas use only algebraic objects from p = 3. We test p = 5, 7 and show they fail byorders of magnitude.

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The Proton-Electron Mass Ratio from Tower Arithmetic: mp/me = 6π5 + 1/29

Key Points

Abstract

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