Finite-Query Coherent Projection of Black-Box Conditional Predictions

Local conditional predictions from a black box need not admit a common joint distribution. We study two aspects of their coherent projection: finite-access precision and directional fidelity. For arbitrary deterministic binary conditional tables, we establish matching expected minimax excess conditional-Kullback–Leibler risk $Θ(\min\{R\sqrt{d/B},d^2/B\})$ under independent uniform exact-table queries, with repeated draws charged, for a local bounded pairwise joint class. The result holds for sufficiently large $d$, $R\sqrt d\le M_0$, and $Cd^3\le B\le(d2^{d-1})^{1/3}$, where $M_0$ is fixed. A moment estimator attains the upper bound. The main technical contribution is a matching lower-bound construction that respects shared pairwise interactions and frozen exact-table responses, with comparison to the best projection of the same realized table. We then examine directional fidelity in a computable hard-marginal formulation that preserves direct black-box marginals. A preregistered TabPFN-v2 case study combines a Car anchor with a frozen two-dataset, three-split extension. All $1{,}152$ extension optimization instances pass independent numerical checks at registered tolerances. Directional contrasts are dataset-dependent and asymmetric: optimizing one direction need not improve the other. Together, the analyses distinguish accurate approximation of a selected projection from improvement in conditional directions outside its fitted objective.