Error mitigation and fidelity certification share a fixed point

Tomography-free certification converts a single measured energy Tr(Hρ) of an effective parent Hamiltonian into a fidelity lower bound F ≥ 1−Tr(Hρ)/Δ, and error mitigation is commonly enabled by default in vendor primitives. These two procedures share an estimand, and therefore a fixed point: because Tr(Hρideal) = 0 holds by construction for any parent Hamiltonian, a mitigator that recovers the zero-noise limit returns an energy estimate approaching zero, and the certificate approaches F ≥ 1 independently of the prepared state. The discrepancy grows as the extrapolation improves and is not reduced by increasing the shot budget. In exact density-matrix simulation the mitigated bound exceeds the true fidelity in 517 of 520 configurations, of which 516 show no internal inconsistency in the reported data; at n = 10 it reports F ≥ 0.9785 against an entanglement threshold of 0.9 and a true fidelity of 0.8602. On hardware the mitigated bound (F ≥ 0.956) exceeds the tomographic fidelity (0.858) at 95% confidence, while the unmitigated bound stays valid. We give an identity separating the two error sources, show that a certifier's tolerance to bias equals its slack, and derive a diagnostic, computable from measured data alone, that flagged every over-statement in our sweep.