The Beta-Bound: Drift constraints for Gated Quantum Probabilities

Quantum mechanics provides extraordinarily accurate probabilistic predictions, yet the framework remains silent on what distinguishes quantum systems from definite measurement outcomes. This paper develops a measurement-theoretic framework for projective gating. The central object is the β-bound, an inequality that controls how much probability assignments can drift when gating and measurement fail to commute. For a density operator ρ, projector F, and effect E, with gate-passage probability s = {rm Tr}(ρF) and commutator norm varepsilon = \|[F, E]\|, the symmetric partial-gating drift satisfies |Δp_F(E)| leq 2 sqrt{(1 - s)/s} cdot varepsilon. The constant 2 is sharp. We introduce two diagnostic quantities: the coherence witness W(ρ, F) = \|F ρ(I - F)\|₁, measuring cross-boundary coherence, and the record fidelity gap Δ_T\(ρ_F, R\), measuring expectation-value change under symmetrisation. Three experimental vignettes demonstrate falsifiability: Hong--Ou--Mandel interferometry, atomic energy-basis dephasing, and decoherence-induced classicality. The framework is operational and interpretation-neutral, compatible with Everettian, Bohmian, QBist, and collapse approaches. It provides quantitative structure that any interpretation must accommodate, along with a template for experimental tests.