One mixer suffices: a minimal symmetry premise for the Born exponent in coordinate-additive probability rules

Abstract

How much unitary symmetry does the Born exponent actually need? Within coordinate-additive probability rules—each outcome weighted by a function $g$ of its amplitude modulus—invariance under a single genuine two-coordinate mixer already forces the quadratic $g(x)=cx^2$ for every dimension $n\ge 3$, where the standard route (Gleason 1957; Hossenfelder 2021; Gogioso 2023) assumes the full unitary group. A single beam-splitter-like unitary on two coordinates, with phase freedom, reduces the invariance condition to a measurable Jensen (midpoint) functional equation whose only solutions are quadratic; the forced weight functional obeys the parallelogram law, so Jordan–von Neumann polarization reconstructs the inner product. The result is a characterization and a no-go, not a derivation of the Born rule: any axiomatization in this class demanding even one genuine mixing symmetry has already presupposed the inner product, so ‘why the exponent is two’ coincides with ‘why coordinate mixing is unitary’. Generalized permutations alone leave the exponent free; the forcing fails at $n=2$, mirroring Gleason’s dimension threshold; and the intermediate regime is empty for the two symmetry classes treated. Exact-arithmetic and numerical certificates are included. Target venue: Foundations of Physics. v2 (August 2026) — premise-level correction. The symmetry premise is restated at the level at which it holds, the weight functional: invariance of the normalised probability rule does not follow from it, and an explicit counterexample is given. A global probability-level invariance premise is shown to be vacuous for any genuine mixer with non-vanishing quadratic weight, and domain-of-definition conditions are made explicit. A prior-art and novelty assessment accompanies this version. Earlier versions remain in the version history.