The Born Rule as the Unique Refinement-Stable Induced Weight on Robust Record Sectors

TL;DR

This paper establishes the uniqueness of the Born rule as the only refinement-stable induced weight on robust record sectors within a specialized quantum framework, under certain structural conditions.

Contribution

It introduces a new structural uniqueness theorem for induced weights on record sectors, distinct from Gleason's theorem, emphasizing bundle additivity and refinement stability.

Findings

Quadratic assignment is uniquely refinement-stable under specified conditions.

Admissible binary saturation ensures the structural conditions for uniqueness.

Under normalization, the result reduces to the standard Born rule.

Abstract

This paper proves a conditional structural uniqueness theorem for induced weight on robust record sectors within an admissible Hilbert record layer. Its theorem target and additive carrier differ from those of the standard Born-rule routes: additivity is not placed on the full projector lattice, but on disjoint admissible continuation bundles through an extensive bundle valuation, from which the sector-level additive law is inherited under admissible refinement. Accordingly, the result is not a Gleason-type representation theorem in different language, but a distinct uniqueness theorem about induced sector weight inherited from bundle additivity on admissible continuation structure. Under two explicit structural conditions, internal equivalence of admissible binary refinement profiles and sufficient admissible refinement richness, the quadratic assignment is the only non-negative refinement-stable induced weight on robust record sectors. In the main theorem, refinement richness is secured by admissible binary saturation. A supplementary proposition shows that dense admissible saturation already suffices if continuity of the profile function is added. Under normalization, the result reduces to the standard Born assignment.