From n samples, the unseen can be predicted only about n·log n observations further, and that horizon is provably the best possible (Orlitsky, Suresh & Wu 2016)

The unseen tail is estimable, but not indefinitely. Orlitsky, Suresh & Wu (PNAS 2016) proved that from a sample of size n, the number of newly appearing elements can be predicted reliably only about n·log n further observations out — and that this range "is the best possible." Valiant & Valiant reached the same limit concurrently, memorably titling the phenomenon "a bird in the hand is worth log n in the bush." Beyond the n·log n horizon the tail is provably unknowable: no estimator, however clever, can extrapolate further from the sample alone.

This is the hard caveat that the diagnostic (claim-singletons-are-the-diagnostic-of-the-unseen) and the coverage machinery (claim-paleobiology-reinvented-coverage-based-rarefaction-as-quorum-subsampling) do not by themselves supply. It bears directly on the vault's No-Change Assumption: treating "no new facts arriving" as proof that a region is complete is licensed only inside a log-factor window. For a heavy-tailed corpus, the absence of new captures certifies completeness only out to ≈ n·log n; past that, the rare tail is formally beyond reach, and silence is not evidence of exhaustion. It is the information-theoretic backstop under the vault's gap-detection thread (claim-obligatory-attributes-as-gap-signal, question-gap-detection).

[unverified-quant — needs primary]. The bound and its optimality are attributed to a Tier-1 PNAS paper but were not read at the primary in this run; verification (and the Valiant & Valiant concurrence) is routed to question-verify-orlitsky-nlogn-unseen-horizon-primary. The note stays seedling.