Paper V: Electromagnetic Constants and Spacetime Geometry A Maxwell–Einstein Ledger of Units, Couplings, and Vacuum Response

The common pedagogical identity c = 1/√ϵ0µ0 invites a tempting but imprecise inference:that ϵ0 and µ0 are direct material-like properties of spacetime. This paper argues that such astatement is not wrong in every loose sense, but is too ambiguous to support serious theoreticalwork. In modern language, ϵ0 and µ0 are dimensional constants tied to a unit convention, a3 +1 observer split of the electromagnetic field, and the vacuum constitutive relation betweenfield strength and excitation. The metric supplies the Hodge dual and therefore the null cone;the scalar vacuum admittance Y0 = 1/Z0 = ϵ0/µ0 fixes the normalization of electromagneticexcitation relative to field strength; the dimensionless fine-structure constant α = e2/(4πϵ0ℏc)fixes the observable quantum coupling. These layers are often compressed into a single sentenceabout “the permittivity and permeability of free space,” but they are not the same physicalstatement.We formulate a Maxwell–Einstein operational ledger for electromagnetic constants. First,we separate Maxwell’s premetric equations dF = 0 and dH = J from the constitutive lawH =Y0⋆F that introduces the spacetime metric and vacuum admittance. Second, we showthat the light cone is determined by the conformal structure of the metric, while ϵ0 and µ0 ariseonly after a temporal-spatial split, a unit convention, and a normalization of charge/currentare chosen. Third, we derive the post-2019 SI relations ϵ0 = e2/(2αhc) and µ0 = 2αh/(e2c),emphasizing that after the exact definition of e, h, and c, both ϵ0 and µ0 inherit their empiricaluncertainty from the measured fine-structure constant. Fourth, we give criteria for meaningfulvariation claims: a variation of ϵ0 or µ0 alone is not an invariant physical claim unless embeddedin a theory of rods, clocks, charge normalization, and dimensionless observables such as α,birefringence parameters, or impedance ratios. The conclusion is a disciplined negative resultwith positive consequences: spacetime geometry participates in vacuum electrodynamics, but ϵ0and µ0 are not standalone geometric observables. The physically sharp quantities are the lightcone, the electromagnetic constitutive tensor, the vacuum admittance in a specified unit system,and dimensionless couplings.