Certify the Event: Target Geometry and Measurement Depth

Suppose a frozen generator is randomized by evidence order and decoding, and let Q denote its ordering-specific probability of passing a declared verifier. A service may require E[Q] > p* while an acquisition heuristic averages a decreasing uncertainty score φ(Q). We characterize when thresholding that mean score certifies the required event. Using one-moment geometry, we show that the largest distribution-free safe cutoff is the lower convex envelope of φ at p*, and that a mean-score gate is lossless for every ordering law if and only if φ is strictly decreasing affine. Support or shape restrictions can recover nonlinear calibration, for which we derive sharp completeness margins. We then turn from population targets to their measurement. With n verifier draws per ordering, the binomial count law identifies exactly the polynomial functionals of degree at most n; thus, increasing the number of orderings at fixed n cannot uniformly recover a nonpolynomial log mean. We derive an unbiased variance decomposition and a count-based jackknife correction, then use homogeneous controls to separate finite-depth bias from genuine ordering heterogeneity. Controlled probability-law experiments exhibit both phenomena without claiming their prevalence in language models. The resulting protocol is target aligned: certify fresh-draw success from fresh verifier bits, and use nonlinear score means only with explicit population calibration and an ordering-clustered measurement audit.