Borns Rule from Reversible Evolution and Irreversible Outcomes

Abstract: We show that the quadratic measure need not be postulated, but follows from the compatibility of two structural features of physical processes: linear reversible evolution prior to the formation of persistent records, and multiplicative composition of outcome weights once such records are established. Reversible evolution combines configurations additively at the level of a compatibility parameter, while the formation of persistent records induces a multiplicative structure on the weights assigned to physically realized outcomes. Requiring consistency between these two regimes constrains the admissible weight assignment to be quadratic in the associated amplitude. The Born rule therefore emerges as the unique measure compatible with reversible linear evolution and irreversible record formation, without assuming a probabilistic interpretation or a specific quantum formalism.