Coverage Is Not Ordering: Ancillary Leakage and Representation Dependence in Covariance-Based Inference
A covariance matrix is often used as a compact surrogate for the statistical model of a measurement. We show that this replacement can change not only a fitted value or its uncertainty, but also the likelihood-ratio ordering of the experiment itself. In a tractable correlated-measurement model, the data separate into an informative component and an ancillary residual disagreement. Exact likelihood inference therefore does not use that disagreement, whereas data-dependent covariance matrices and Gaussian reconstructions after nonlinear transformations can reintroduce it into parameter inference. We quantify the resulting change by the probability mass of the symmetric difference of equally calibrated acceptance regions - the fraction of repeated experiments for which the confidence decision about a tested parameter value changes. In the small-uncertainty regime the ordering discrepancy is generically first order in the total relative uncertainty, whereas conventional and ancillary-conditioned coverage defects begin at second order. At 'coverage-blind' points the leading coverage difference vanishes while the ordering discrepancy remains nonzero. In a representative blind case with 10% total relative uncertainty, about 7.6% of experiments change their confidence decision despite exact calibration of both procedures. The same mechanism connects Peelle's Pertinent Puzzle to the classical literature on ancillarity and relevant subsets: a covariance approximation can manufacture inferential relevance for a goodness-of-fit statistic that is ancillary in the true model. We show directly how this changes the confidence interval reported for the same informative content, and extend the analysis beyond equal statistical uncertainties and beyond the two-measurement case.