Accuracy-first arguments derive Bayesian updating by assuming that evidential content is already representable as a scalar quantity evaluated by a proper scoring rule. This paper instead identifies structural conditions under which such scalarization is legitimate. We model evidence as interactive witness–challenge refinement and internalize updating in a Dialectica category over credal states. In this setting, feasibility of an update is equivalent to nonnegativity of a minimax security value, and diachronic coherence corresponds to compositional preservation of no-book status. Admissible policies admit a canonical factorization into transport, interactive refinement, and optional selection. Within this interactive universe we isolate the public-score fragment, the reflective subcategory of evidence types indistinguishable by strictly proper scoring rules. On this fragment, equilibrium semantics commutes with reflection: computing equilibria before or after collapse yields the same public reports. Bayesian updating appears exactly as the reporting equilibrium of the reflected evidence game. Outside this fragment the collapse is non-faithful: evidence types that coincide at the scalar level can remain strategically distinct in the interactive universe. A finite normal-form representation theorem and nonclassical LP stress tests make this boundary explicit.