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

Version of October 13th, 2026, following technical developments within the Absolute Frame Theory programme. We establish a bridge between the pilot wave Ψ on the Absolute Frame 𝒜 and the wave function ψ on the observable sub-manifold ℳ within the Absolute Frame Theory. We start from the perpendicular projection of Ψ along the directions transverse to the embedding. The pilot wave is not continued to the Lorentzian signature. The dynamics on ℳ is taken to be the Osterwalder–Schrader reconstruction of the measure of the embedding field. We treat the free field with a declared mass, about a flat background, for the homogeneous plane-wave pilot wave whose wavevector is normal to the image, and at fixed regularization. There the vacuum-to-one-particle amplitude of ψ, defined as an Osterwalder–Schrader pairing, obeys the Klein–Gordon equation and, in the non-relativistic limit, the Schrödinger equation. The relation between ψ and the one-particle states is not established here. We derive the Born rule |ψ(x)|² for the position measure, given the frame-function hypothesis on the full projection lattice, and we exclude the fibre-marginalized density by the empirical observation of quantum interference. The derivation applies Gleason's theorem and assumes the pure-state regime.