From the Pilot Wave Ψ to the Schrödinger Wave Function ψ: Dimensional Aliasing as the Bridge Between A and M in the Absolute Frame Theory

Version of July 9th, 2026, following technical developments within the Absolute Frame Theory programme. We establish a rigorous bridge between the deterministic pilot wave Ψ defined on the Absolute Frame A and the Schrödinger wave function ψ on the observable sub-manifold M within the Absolute Frame Theory (AFT) framework. Starting from the elliptic (Helmholtz-type) dynamics of Ψ on the Euclidean Absolute Frame A and the perpendicular projection of Ψ along the directions transverse to the embedding X: M → A, we derive a four-dimensional Klein–Gordon equation for ψ whose effective mass spectrum is determined by the transverse geometry. In the non-relativistic limit this reduces to the standard Schrödinger equation. We further derive the Born rule |ψ (x) |², for the position measure on M, as the unique probability density consistent with σ-additivity and the empirical observation of quantum interference, applying Gleason's theorem to the projected Hilbert space HM = L² (M) (with the extension of the frame function from the commuting position projectors to the full projection lattice stated as an explicit hypothesis). The derivation identifies an empirical input (interference) as the discriminant between pure states and statistical mixtures, distinguishing this derivation from purely axiomatic formulations of the Born rule. Identifiability constraints in the spirit of Gödel's incompleteness theorems are respected throughout: only the product λΨ and the transverse spectrum μₙ²₍=₁^KY enter observable predictions, with neither λ, Ψ, nor the specific transverse topology being separately identifiable from observations restricted to M.