Hidden Structure and Explanation: Explanatory Descent and Minimal Hidden Support

Explanatory adequacy descends to the public side of an admissible evidential architecture only under specific structural conditions. The core result is an explanatory descent criterion: explanatory structure is fully public exactly when the invariants required for explanatory adequacy factor through admissible public quotients. Public explanation is therefore a special success case rather than the default form of explanatory order. Three main consequences follow. First, the descent criterion yields a minimal-support theory of explanation: explanatory adequacy depends not only on invariants, but on the least hidden structure needed to preserve them. Second, this support logic induces a role-relative compressibility order: prediction tolerates the most compression, explanation less, and hidden-mediated Bayesian recovery less again. Third, failure of public descent yields both an anti-flattening trilemma and an Occam-disciplined account of Bayesian recovery: if hidden structure can restore Bayes on a designated fragment, there is a least hidden support sufficient to do so. A further operator-level result shows that even where term-level hiddenness is too coarse, explanatory or inferential machinery may remain essentially hidden.