The Monty Gap: Selection Evidence and Its Limits

A verifier's choice of which answer to reject can carry evidence beyond the rejection itself. We audit whether this selection evidence improves model decisions: no tested prompted critic establishes a positive subject-cluster-robust accuracy gain over re-prompting, and a separate confidence study finds no reliable decision-cost benefit over calibrated controls. To interpret these results, we define the Monty gap, the expected posterior information discarded by masking rejected answers. We prove that every conclusive verifier that always spares the model's commitment at a positive fixed budget is non-ignorable. This guarantees a posterior discrepancy, not a decision gain. Under uniform selection among wrong alternatives, switching has a simple threshold: the surviving rival's prior probability relative to the commitment's must exceed $(N-1-k)/(N-1)$. We characterize when masking is exact and show how intermediate discrepancies can disappear over a fixed budget. Scripted oracles establish the mechanism on model output distributions; prompted critics violate its assumptions, and the only positive four-option pairing is concentrated in one subject block. The resulting contribution is an audit of verifier protocols and their limits: selection evidence can exist without being reliably usable by the decoder.