A preprint posted to arXiv proves that the Born rule, which says the probability of a quantum outcome is the square of the wave function's amplitude, is the only assignment that works under specific structural conditions. The work is purely theoretical: it uses mathematical reasoning about 'robust record sectors' and 'admissible continuation bundles' inside an abstract Hilbert space setup, with no experiments, data, or sample size involved. Instead of the usual Gleason-type approach that assumes additivity across all projectors, this proof derives the quadratic rule from additivity on disjoint bundles that get inherited when the structure is refined, requiring conditions like internal equivalence of binary refinements and enough refinement richness (secured by binary saturation). A supplementary result shows a weaker saturation condition works if continuity is also assumed. Because this is an arXiv preprint (https://arxiv.org/abs/2603.24619) and not yet peer-reviewed, its conclusions remain provisional; the authors themselves note that the result holds only inside the defined abstract framework, which may or may not map perfectly onto physical reality.