Local Uniqueness of the Born Rule on Categories with Complex-Weighted Morphisms

I prove a local uniqueness theorem for the Born rule in the setting of small categories equipped with complex morphism weights and path-amplitude probability functionals. Given (i) non-negativity, (ii) polynomiality of bounded total degree, (iii) global U(1) invariance, (iv) classical-limit additivity over mutually exclusive paths, and (v) normalization, I show that the probability assignment P : ℂ → ℝ≥₀ is uniquely determined to be P(z) = |z|^2. The notion of mutually exclusive paths is given a precise categorical formulation as the absence of a shared factorization through any common morphism. I relate the result to reconstructions of quantum probability due to Gleason, Hardy, and Chiribella–D’Ariano–Perinotti, and identify the extension to global coherence under morphism composition as an open problem connected to synthetic probability theory in Markov categories. The Born rule emerges as the unique locally consistent probability law on complex amplitudes, fixed solely by phase invariance and classical-limit behavior, independent of any Hilbert-space framework.