A one-parameter relation for the fine-structure constant from the proton–electron mass ratio

Superseded by version 1.7 (October 21, 2026). Version 1.5 narrowed the premise that carries the physics (in the long-wavelength limit the plasma's mean binding depends on the mass ratio alone, and of eight ways of weighting its modes only equal weight per degree of freedom returns the measured α), reported a look-elsewhere test fixed in advance and reordered Section 3 to run from the relation's input to its output. Version 1.6 showed that the relation needs the electron–proton pair at rest, to within about 5 km/s, identified the plasma's two waves as the pair's relative and centre-of-mass motions, showed that the relation picks out no preferred frame, fitted the lock at recombination directly with a Boltzmann code, and read the gravitational relation the other way, returning both particle masses from e, ħ, c and G. Version 1.7 identifies the two states of the relation's last step as hydrogen's second-shell states 2s and 2p₀ (an electric field along one axis turns the shell's two pseudo-spins by ±θ), derives what the push that turns the shell has to be, gives Brandenburg's exponential a reading as the weight of the proton's zero-point oscillation that reaches one quantum of the electron's, and counts the binding over the ten numbers that fix the vacuum's constitutive law, which gives the same 2/5 in three dimensions if they are degrees of freedom of the vacuum and the registration counts by action. No number in the relation or its predictions has changed.

The fine-structure constant α is an input to quantum electrodynamics, not something the theory calculates. This paper reports a closed-form relation that returns α from one measured number, the electron–proton mass ratio q = mₑ/mₚ, with no adjustable parameters: α = (π/4)·P·cos√P, where P = (2/5)√q·(1 + (4/5)ρ²) and ρ = (2/π)q. It is a product of three factors: sound (2/5, the share of a collisional plasma's slow fluctuation energy that travels as sound), light ((π/4)·cos√P, an amplitude readout and its symmetric return) & frequency (√q, the ratio of the proton's and the electron's plasma frequencies, with the proton's size read on the electron's Compton clock). With the CODATA 2022 mass ratio it gives 1/α = 137.035 999 164(1), which is 0.14σ from the electron g−2 determination as published and 0.09σ from it once the two theory terms revised since 2023 are applied, the tenth-order QED coefficient and the hadronic vacuum polarization. Because electrons and protons feel one electric field, the plasma's two waves, the electron's Langmuir wave and the proton's ion wave, each carry the other species with a mass-weighted amplitude √q times their own, and in a two-fluid model the two waves hold two-fifths of the fluctuation energy; the premise that P is their energy-weighted mean binding makes the sound and frequency factors two properties of the same two waves. The same pieces predict quantities that do not use α's own measurement: - the proton charge radius, 0.841 19(50) fm, against 0.8406(15) fm from 2026 hydrogen spectroscopy and 0.840 60(39) fm from muonic hydrogen; - the mechanical radius of the proton's charged face, √(3/5)·r_p = 0.652 fm, against 0.634(57) fm from Jefferson Lab, with the whole proton, gluons included, at 0.72–0.78 fm against 0.72–0.75 fm measured; - a lock between α and the mass ratio, d ln α / d ln(mₚ/mₑ) = −0.4977, so that α can change only if the mass ratio does, which differs from grand-unified expectations by more than an order of magnitude and is consistent with atomic clocks, quasar spectra and the cosmic microwave background; - if Brandenburg's postulated gravitational relation is granted, Newton's constant to 0.12σ, using the relation's own 4/5. Each prediction is listed in the paper with the measurement that would refute it.

Changes in version 1.4 (October 19, 2026): Section 3.3 is new. Electrons and protons feel one electric field, which couples them through a matrix of rank one whose cross term is the geometric mean of the two species' own terms. At long wavelength the plasma's two branches, the electron's Langmuir wave and the proton's ion wave, are therefore turned off the particle axes by the same angle, and each carries the other species with a mass-weighted amplitude √q times its own, in antiphase in the first and in phase in the second. In the two-fluid model the two waves hold 4/10 of the fluctuation energy, the same fraction as the 2/5 of Sec. 3.2, so the two premises of version 1.3 for the midpoint and the travelling share are replaced by one: that P is the plasma's mean binding, the field's binding averaged over its modes with energy weights, in the long-wavelength limit. A calculation in linear kinetic theory, under a protocol fixed before it was run, reproduces both bindings and the vanishing of the total equal-time correlation between the two species' velocities. The section explains why the information calculations of version 1.3 found the 2/5 and not √q, and gives the exact ratio with which a body at rest weighs the two species. A second test, also fixed in advance, asks how the field variance a proton feels at long wavelength divides among the plasma's modes. The Langmuir wave carries nearly all of it, so a face that registered it would weight the modes by field energy and give 1/α = 55.2; of the weightings tried, only the fluctuation energy each mode holds returns the measured α. Seven premises remain. The abstract is shorter and is followed by a box that states the claim, the evidence and what would refute it. The table of predictions now opens the introduction, with the measurement that would refute each (Table 1), Figure 2 draws the two waves of one field, and Section 4.3 states the value the measured α requires of each coefficient (Table 6) in place of the largest rejection margins. The gravity section notes that the split the measured G asks for, 0.799(5), has the value of the relation's own surface coefficient, which α alone fixes at 0.799 91(95), and corrects the second-order coefficient the split gives from 0.8600 to 0.8599. The distance from the g−2 value as published is now given as 0.14σ, which earlier versions rounded to 0.15σ. The source files add the protocols, scripts and output of both new calculations, the coefficient table and the new figure. No number in the relation or its predictions has changed.

Use of AI: The text, derivations, numerical scripts and figures in this paper were drafted with Claude (Anthropic), working from the author's idea and under his direction. The relation, its interpretation and the claims are the author's, and the author takes full responsibility for the content.