Why Bayes Emerges: Public Interfaces, Selection and Residual Structure

Bayes reappears across conceptually distant settings. We argue that this recurrence is neither accidental nor evidence that Bayes is the deepest universal law of rationality. Instead, we frame Bayes as the characteristic law of a recurrent public evidential interface. Rich evidential regimes must often pass into publicly accountable form by reflection, descent, quotient, or stabilized completion. When that happens, well-behaved public interfaces are fixed by subtopos modalities, polynomial functor structure fixes the admissible class of compositional update systems on the public side, and Bayesian conditionalization is uniquely determined as the public law in the exact fixed-point case: the only update transformation that respects publicity, preserves relative support on the surviving fragment, normalizes, and composes under base change. We organize this claim by four schemata: interface formation, Bayesian emergence, selection, and residuality. The first explains how a public evidential world is formed at all; the second explains why its exact law is Bayesian; the third explains why one interface or law is privileged among rivals; and the fourth explains what remains outside the public image. The payoff is both explanatory and classificatory: Bayes is exact on the public side, but the public side is generally non-faithful, so different cases can be read as different Bayes-defect types. The synthesis also adds a converse reconstruction problem: given a public Bayesian law, which richer upstream dynamics lift it, and how are those lifts classified by defect?