Electromagnetism, Gravity, and Inertia in One Medium

Abstract

Preprint — not peer reviewed. Two companion papers began from a single continuous medium carrying a complex deformation field Ψ = A·e^(iΘ), calibrated two quantities — the speed of light and the electron mass — and obtained the electron as a topological defect, then the pion and the proton. This paper asks what the same functional says about the long-range forces and about inertia. No new input is introduced — not for electromagnetism, not for gravitation, not for inertia: every quantity the paper draws on was fixed before it began, and a provenance ledger is printed so that this can be checked rather than taken on trust. Nothing in the individual steps is claimed as new. Maxwell's equations from the phase–amplitude split of a scalar condensate have been obtained before; separating a topological charge from the elastic field it sources is standard defect theory; reading gravity's weakness as stiffness rather than weak coupling goes back to Sakharov; effective metrics emerge generically under linearisation. These works are cited and the paper places itself beside them. What is reported is a route: one functional, with no parameter introduced after the electron, yields the four Maxwell equations in Coulomb gauge, the Lorentz force, an inverse-square law that is exact rather than asymptotic, gravitation that is long-ranged and always attractive, the Newtonian limit with source conservation supplied by Noether's theorem, the complete weak-field correspondence with G_μν = 8πT_μν — the infrared source surviving as the tensor T_μν under the protection of that same conservation law — the equivalence principle, and F = ma, with inertial and gravitational mass turning out to be one quantity rather than two that agree. The weak-field correspondence together with a Lovelock uniqueness argument makes a strong — argued, not proved — case that general relativity is the framework's infrared reduction; the single remaining step is a formalisation rather than new physics. Every step is ordinary, and that is the substance of the account: it is what shows that no mechanism was inserted for the sake of any single result. One quantitative statement is offered as new, and a counterpart was searched for and not found: computed purely geometrically from the same functional, the gravitational and electric charges of the electron agree to within O(1), so that the familiar ratio of 10⁻⁴³ is a ratio of deformations, not of couplings. This yields a falsifiable disagreement with the standard extrapolation of gravity to particle scale; the paper states in the same breath that the smallest source mass yet used in a gravitational measurement lies some 29 orders of magnitude above the electron, and it does not ask anyone to attempt the measurement. What is not done is also on record: no value of G is given, no closed form for the radial coupling, no field equations, and no numerical bridge across the remaining hierarchy. Declarations. The author is an independent researcher without institutional affiliation. Derivations and numerical verification were carried out with substantial assistance from a large language model (Anthropic Claude), whose role was technical execution; conceptual direction, criteria of judgement and all substantive decisions are the author's own, who takes full responsibility for the content. This is declared in full within the manuscript. Research record. The complete working record from which this paper is drawn — including all numerical scripts and a full log of approaches that failed and the reasons they were abandoned — is openly archived at https://doi.org/10.5281/zenodo.21764066. It is offered not as supporting material for review, but because the paper's claims about what was and was not fitted can be checked against it. Keywords: Topological defect · Emergent gravity · General relativity · Winding number · Hierarchy problem