What Does a Predictive Gain Certify? Auditing Information Claims under Approximate Inference

A better predictor need not use new information: it may simply make better use of information already available. We study when a held-out predictive gain supports a claim of additional conditional information. Our audit connects three requirements: attributing the gain, validating the baseline's approximation error, and accounting for uncertainty in the final decision. We prove that probability reports and labels alone cannot distinguish some informative and non-informative problems. For a known categorical baseline, we derive an exact reverse-to-forward KL conversion requiring only its smallest probability and one scalar root, avoiding exponential subset enumeration. A failure-aware audit incorporates uncertain error bounds and supports both betting and fixed-sample confidence statements. For sampled categorical reports, simultaneous intervals and a simplex constraint avoid the universal-bound failure caused by unseen classes. On natural WDBC, Digits, and Wine labels, an independent witness sample yields positive simultaneous information bounds in six of nine fixed analyses; all three Wine analyses remain inconclusive. A learned-representation audit checks attribution on continuous 64-dimensional inputs. A deterministic deeper layer passes the predictive-gain test in every 8,000-case repetition despite having zero additional information; the corrected audit makes no positive information claim. With an added informative signal, the betting audit detects it in 98% of repetitions, compared with 100% for conditional randomization using its exact feature sampler.