Admissibility and the Quadratic Probability Rule

This work examines the quadratic probability rule (Born rule) from the perspective of structural admissibility constraints on state evolution. Rather than relying on interpretational or decision-theoretic arguments, the analysis proceeds from symmetry, geometric, and compositional principles. Assuming bounded norm-preserving symmetries, invariance under admissible isometries, tensor-product consistency for independent systems, and normalization, the quadratic rule emerges as the probability assignment compatible with inner-product geometry. Full unitary invariance implies the parallelogram identity and therefore inner-product structure via the Jordan–von Neumann theorem. Under these structural constraints, the quadratic rule is uniquely selected. The paper further analyzes controlled deviations from exact isometric evolution. Allowing bounded, contractive perturbations consistent with completely positive trace-preserving (CPTP) dynamics, the first-order structure of admissible departures from the quadratic rule is derived. Deviations are shown to correspond to contractive departures from strict isometry and must remain operationally realizable within standard quantum dynamical frameworks. This work situates the Born rule within a broader admissibility framework previously applied to dynamical and geometric sectors. The argument is structural rather than interpretational, and emphasizes compatibility with symmetry, composition, and operational consistency.