From the Pilot Wave $\Psi$ to the Schrödinger Wave Function $\psi$: Dimensional Aliasing as the Bridge Between $\mathcal{A}$ and $\mathcal{M}$ in the Absolute Frame Theory

We establish a rigorous bridge between the deterministic pilot wave $\Psi$ defined on the Absolute Frame $\mathcal{A}$ and the Schrödinger wave function $\psi$ on the observable sub-manifold $\mathcal{M}$ within the Absolute Frame Theory (AFT) framework. Starting from the Klein-Gordon dynamics of $\Psi$ on $\mathcal{A}$ and the perpendicular projection of $\Psi$ along the directions transverse to the embedding $X: \mathcal{M} \to \mathcal{A}$, we derive a four-dimensional Klein-Gordon equation for $\psi$ 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 $|\psi(x)|^2$ as the unique probability density consistent with $\sigma$-additivity and the empirical observation of quantum interference, applying Gleason's theorem to the projected Hilbert space $\mathcal{H}_\mathcal{M} = L^2(\mathcal{M})$. 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 $\lambda\Psi$ and the transverse spectrum $\{\mu_n^2\}_{n=1}^{K_Y}$ enter observable predictions, with neither $\lambda$, $\Psi$, nor the specific transverse topology being separately identifiable from observations restricted to $\mathcal{M}$.