Preprint: Resolution-Limited Distinguishability Selects the Born Rule

This record provides a preprint version of a manuscript submitted for publication.

The origin of the Born rule remains a central open problem in the foundations of quantum theory. 

While previous approaches derive the quadratic probability rule from structural, decision-theoretic, or measure-theoretic assumptions, the physical origin of these assumptions is not always transparent.

In this work, we propose that observational resolution provides a fundamental constraint on distinguishability, which in turn restricts admissible probability measures. 

We introduce a resolution-based observational framework in which finite resolution induces coarse-graining and suppresses phase information below a characteristic scale.

We show that consistency under resolution-limited coarse-graining selects a quadratic probability structure. 

In particular, only the quadratic form remains stable under phase averaging while reproducing additive behavior in the classical limit.

We further demonstrate that this structure is naturally connected to the quadratic mapping from complex state space to observable quantities, as realized in spinorial representations of three-dimensional geometry.

These results suggest that the Born rule can be understood as a structural consequence of resolution-limited observation.

This version corresponds to a manuscript submitted to Physical Review X.

Version 2: Major revision focusing on the emergence of the Born rule under resolution-limited observation. The manuscript has been restructured for journal submission.