Bounded Uncertainty Metrics Cannot Price Unbounded Confabulation: A Non-Identifiability Result

On a certified reasoning task the hidden answer Z is constructed to be statistically independent of every chain-of-thought trace the model can emit. A calibrated reasoner's answer-belief must therefore be invariant to its own trace, and the self-predictive information E has a known ideal value of exactly zero. This makes E an attractive certificate of false epiphany: the manufacture of conviction out of self-generated text that provably carries no information about the truth. E is a sound detector: under logit readout its null is an exact, variance-free zero and its finite-sample bias is negative, so any positive estimate is conservative. Our main result is that it is nonetheless a worthless severity measure, and not by accident. Certification makes the true posterior known rather than estimated, so the regret decomposes exactly, with G >= 0 the regret price of dispersion — the nats a model burns by letting a provably uninformative trace move its belief. We prove that E does not identify G: for every epsilon in (0, epsilon*), the supremum of G given E = epsilon is infinite. We exhibit a family holding E at exactly 0.1000 bits while G ranges over orders of magnitude and diverges; E can also exceed G, so the two are not monotonically related in either direction. The obstruction is boundedness, not E. Semantic entropy, predictive entropy, BALD/epistemic mutual information, max-probability, verbalised confidence and belief variance are all bounded, and the result applies to each.