Derivation of the Born Rule for Real Amplitudes, in a Discrete Substrate with a Single, Exclusionary Dynamics

The Born rule, P(i) = |ψᵢ|²/Z, is posited as a postulate in the standard formulation of quantum mechanics, where ψ remains a primitive object and the exponent 2 a given fact with no internal justification. This work proposes a derivation of it, decomposing the structure of ψ, within a class of homogeneous discrete substrates defined by a single dynamical postulate of locality, P0. In particular, the amplitude becomes a propagation distance, an identifiable and measurable quantity, whose quadratic character is justified by the lattice geometry that the primitives impose. P0 is the axiom that openly fixes what is permitted and, through its uniqueness, determines what is excluded. It is this dual reading that forms the core of the entire construction that follows. The work operates in a finite-dimensional real vector space, for a homogeneous three-dimensional Euclidean lattice and an isolated object, without presupposing the linear Hilbert space structure (complex or real) that derivations such as Gleason's [1] require. Contact: [email protected]