Prevalence Determines Precision:Silent Contamination in Detector-Defined Datasets

Behavioral contamination detectors can return "no evidence" either because a benchmark is clean or because the audit has little power. We formalize this distinction for a benchmark in which an unknown fraction alpha of items was seen during training. With matched clean and seen controls, the behavioral channel is the sparse mixture Q_alpha = (1 - alpha) P_0 + alpha P_1, and an exact second-moment argument shows that detectability is governed by alpha * rho * sqrt(m), where rho^2 = chi^2(P_1 || P_0) measures behavioral separability. Any scalar detector reduces to its efficacy, ef = |E_1 f - E_0 f| / sqrt(Var_0(f)) <= rho, which can be estimated from controls before the audit is run. A separate sample-split certificate lower-bounds alpha distribution-free, without requiring an orientation assumption. Our empirical finding is two-sided. Frozen calibration efficacy predicts held-out power curves, with R^2 = 0.83-0.98 across six exact-permutation channels, but the efficacy-only Gaussian budget is miscalibrated at the small sample sizes it prescribes, failing in 9/9 gate-passing channels even though efficacy itself transports. The failure is in the inversion, not the calibration. A predeclared two-stage planner that simulates the deployed test repairs the budgets, is uniformly conservative, and abstains when its probe does not transport. The certificate is valid but vacuous at audit scale, and a five-seed paired injection study recovers the mechanism ordering verbatim > paraphrase > surface, in which the apparent answer-only signal is explained by baseline drift. We report the audit contract and its failures together: a non-rejection is interpretable only alongside the efficacy, budget, and validity gates that produced it.

Estimating the prevalence of a category in a population using imperfect measurement devices (diagnostic tests, classifiers, or large language models) is fundamental to science, public health, and online trust and safety. Standard approaches correct for known device error rates but assume these rates remain stable across populations. We show this assumption fails under covariate shift and that multicalibration, which enforces calibration conditional on the input features rather than just on average, is sufficient for unbiased prevalence estimation under such shift. Standard calibration and quantification methods fail to provide this guarantee. Our work connects recent theoretical work on fairness to a longstanding measurement problem spanning nearly all academic disciplines. A simulation confirms that standard methods exhibit bias growing with shift magnitude, while a multicalibrated estimator maintains near-zero bias. While we focus the discussion mostly on LLMs, our theoretical results apply to any classification model. Two empirical applications -- estimating employment prevalence across U.S. states using the American Community Survey, and classifying political texts across four countries using an LLM -- demonstrate that multicalibration substantially reduces bias in practice, while highlighting that calibration data should cover the key feature dimensions along which target populations may differ.

Benchmark contamination is diagnosed today with n-gram overlap, with likelihood-based membership inference, or with canary strings, and each needs something usually unavailable: the training corpus, a well-chosen test statistic, or foresight at dataset release. A recent alternative reads contamination off a linear probe on internal activations. We show that the natural way to do this does not work, and specify one that survives measurement. The protocol reports a zero-sum contrast on the depth profile of probe accuracy, recentred on a level-matched placebo baseline, tested against a label-permutation null, with the reference set twice the size of the suspect set. Each choice replaces a simpler alternative we measured and rejected. Reporting the level of excess separability rather than its shape makes the false positive rate track the size of the analyst's own control set, from 0.03 to 0.99 under a true null. Contrasting against a flat depth profile fails in both directions, rejecting a true null 0.72 of the time when surface decodability rises with depth and losing all power when it falls. An item bootstrap holds the fitted probe fixed and rejects up to 0.09 of the time where a permutation null that refits it holds 0.02. A half-size baseline triples the error rate. On real transformers, baseline depth profiles are measurably not flat, spanning up to 29.1 accuracy points on a temporal split, and their non-flatness tracks the surface difference between the item sets (correlation 0.87 over 6 audits), so the correction is largest exactly where it is needed. All 4 well-matched Pile arms return null, and the protocol refuses a verdict on the temporal split rather than reporting one. What this does not establish is whether transformers carry a familiarity direction at all: the only positive sits on the split where exchangeability fails. Implementation, tests and audits are released.