The count of things seen exactly once — ecology's 'singletons', linguistics' 'hapax legomena' — is the diagnostic of the unseen: Good-Turing sets the unseen mass at roughly f₁/n

Across the fields that share the unseen-species problem (claim-unseen-species-problem-has-dual-origin-ecology-and-cryptanalysis), the quantity that tells you how much you have not seen is the number of things you have seen exactly once. Ecology calls these singletons; linguistics calls them hapax legomena — words appearing once in a corpus. The same statistic, two field-names, is the observable proxy for the invisible tail.

Two estimators formalize it. Good-Turing sets the total probability mass of never-seen items at approximately f₁/n — the fraction of the sample made of once-seen items — the intuition being that if many things have shown up only once, many more are still waiting to show up at all. The Chao1 lower-bound estimator of total richness uses singletons and doubletons: Chao1 = (n−1)/n · f₁²/2f₂, where f₁ is the count of singletons and f₂ of doubletons. When there are no doubletons the community is well-sampled; a large singleton-to-doubleton ratio signals a large hidden tail.

This diagnostic is what powers the sample-coverage machinery in claim-paleobiology-reinvented-coverage-based-rarefaction-as-quorum-subsampling: coverage is estimated from exactly these low-frequency counts. It is also the hook that connects the problem to authorship attribution — Efron and Thisted used the same once-seen statistics to estimate the words Shakespeare knew but never wrote — which sits near the vault's estimation cluster (claim-james-stein-estimator-uniformly-dominates-the-sample-mean, claim-efron-baseball-shrinkage-halved-batting-average-prediction-error).

[unverified-quant — needs primary]. The two formulas are standard but are carried here from a Tier-2 blog rather than their primaries (Good 1953; Chao 1984); verification is routed to question-verify-good-turing-chao1-formulas-primary. The note stays seedling.