Beyond Estimator Moments: Minimal Information for Predicting Zeroth-Order Adam's Response

Zeroth-order (ZO) query laws are conventionally compared through estimator mean, covariance, and mean-squared error. We show that these statistics need not determine Adam's response: in a local separable exponential-loss model with an annealed stabilizer, equal-cost laws can match the zero-radius mean, full covariance, and MSE yet induce distinct mean normalized updates at their respective slow equilibria. The mechanism is root adjustment: equilibrium motion absorbs part of the direct query-law effect, leaving a generally oblique-projected response. For a sphere-supported class with prescribed data-contracted moments, we characterize exactly how much additional information is required for uniform next-order prediction. The minimum number of exact scalar expectation readings equals the rank of the surviving response operator and, under nondegeneracy and full realizability, is $d-r$, where $d$ is the parameter dimension and $r$ the active-row rank. Indistinguishable admissible laws prove necessity, while explicit polynomial readings attain the bound. On a matched angular-mixture family, a response-aligned reading estimated from only 16 i.i.d. directions reduces held-out prediction MSE to about 1.4% of the reference-only baseline across two disjoint phases, whereas a same-count orthogonal reading misses the response. Finite-radius bounds and a local initialized weak-limit result further delimit the regime in which the response characterization applies.