Deriving the Born Rule from Discrete-Time Substrate Dynamics

We derive the Born rule of quantum mechanics, P(ε_i) = |⟨ε_i|ψ⟩|², from the substrate framework — a candidate ontology in which time is discrete at the Planck scale, reality has two ontological frames (continuous Frame 1 and discrete Frame 2), and a complex scalar substrate field provides a universal environment for all quantum systems. We present two complementary derivations that converge on the same result.