False Confidence in Generative Samplers: Exact Physical References Expose Hidden Errors and Diagnostic Blindness

Learned samplers are increasingly proposed for equilibrium sampling and free energies in chemistry, and they are usually judged by self-normalized diagnostics such as the effective sample size (ESS). We show with an elementary proposition that these diagnostics are structurally invariant to the Boltzmann mass α of unvisited basins: conditioned on not proposing a basin, the importance weights follow the law of the truncated system, so the ESS stays high while the free energy shifts by −kBT log(1 − α) and the nominal deltamethod interval contracts around the biased value, yielding asymptotically zero coverage. In a toy double well where one basin holds 49.9% of the mass, a proposal putting mass 10−6 on that basin achieves a median ESS of 98% but covers the exact logZ in 0% of 400 runs. On exactly enumerated Sherrington-Kirkpatrick instances (n = 24), nominal 95% intervals from masked diffusion and a GFlowNet cover the truth in only 8% and 47% of runs at β = 2, while classical estimators and an autoregressive network (VAN-MADE) stay near 95%. A reverse check that scores the proposal on draws from an independent reference sampler flags the dropped basin where the forward Jensen gap is blind. We demonstrate this on a continuous molecular model: a RealNVP Boltzmann generator trained on the alanine dipeptide Ramachandran landscape drops the C7ax basin (8.1% mass), reporting a median ESS of 79% with only 10% coverage, while the reverse check detects the omission (Grev = 11.7 nats). Finally, we size the bound on macromolecular systems (chignolin, Trpcage, LJ38), contrasting single-trajectory kinetic trapping with converged Anton and NMR equilibrium baselines to show that dropped macroscopic states shift free energies by 0.68 to 1.02 kcal/mol, exceeding lead-optimization thresholds.