Many conventional dashboards report the people who spoke and call it the audience. The two are equal only if speaking is unrelated to attitude — an assumption such reports rarely state. Echo Penumbra treats the audience share as what it is: a partially identified quantity. It reports the interval the truth must live in, refuses when your assumptions are impossible, and prints the exact participation ratio at which your conclusion breaks.
The estimand. θ is the share of a declared audience holding an attitude fixed by a declared measurement protocol — for speakers it is recovered from text; for the silent it is the value the protocol would elicit. The unit is the captured speaker, not the mention: S = 1 means the unit produced an eligible utterance that was captured by a frozen retrieval protocol (sources, query, languages, window — private accounts, deleted posts and API omissions separate speech from captured speech, and γ absorbs the combined effect of speaking and being captured). Utterances must first be mapped to deduplicated units, and a predeclared aggregation protocol must assign at most one attitude value per speaker for the window — mention-level sentiment shares are not valid inputs. The observables are the speaker-normalized share p and the participation rate s (speakers ÷ audience). Three unknowns stand behind them — θ and the speaking probabilities of holders and non-holders — and two equations. The missing degree of freedom is exactly one number: the participation ratio γ. The aggregate pair (p, s) alone cannot supply it, however precisely estimated.
The anchor (Proposition 1; the classical selection-problem bound, Manski 1989/2003). With no assumptions at all,
and both endpoints are attainable. The width, 1 − s, is the identification deficit: at 12% participation, the assumption-free truth occupies an interval of width 0.88, and everything narrower is purchased with assumptions. Penumbra prints this width before it prints anything else.
The identity (Lemma 1). γ is the response-probability ratio of the missing-binary-outcomes literature (Magder 2003): the factor by which holding the attitude multiplies the probability of speaking — a ratio of probabilities, not an odds ratio. Given γ, the population share is point-identified in closed form, independent of s:
strictly decreasing in γ, strictly increasing in p, with θ(1) = p — the dashboard number is the special case "speaking is unrelated to attitude". At census boundaries two readings are printed side by side: the assumption-free anchor, and — under the finite-population empirical-measure interpretation with strictly positive propensities — the model-dependent point θ = p, for every γ.
The compatible set (Proposition 2). The γ values consistent with the observed pair (p, s) form a closed interval with closed-form endpoints, and those endpoints map exactly onto the anchor bounds: the γ-parameterization sweeps precisely the Manski interval. A declared rectangle intersected with this set gives corner-attained bounds. A declared rectangle disjoint from it is rejected with regime-honest wording: under a census this is a logical refutation (REFUTED-CALIBRATION); under sampled classification it is a statistical exclusion conditional on the design and the Wilson procedure (OUTSIDE THE 95% WILSON PROJECTION); CALIBRATION TENSION when disjoint only at the point estimate.
The breakdown frontier (Proposition 3). For a claim θ > τ, the exact participation ratio at which the claim loses its sign is
A report that carries its γ* carries its own conditions of failure: "our majority verdict survives any participation process in which a holder is less than γ* times as likely to speak as a non-holder." The assumption-free version: the claim survives with no assumptions iff s > s* = τ / p.
Monotone selection (Proposition 4). If holders are at least as likely to speak (γ ≥ 1), then θ ∈ [p·s, p]: the dashboard number becomes a ceiling on the audience share, not an estimate of it.
Comparisons (Proposition 5, Corollary 1, Proposition 6). Under a common declared γ, the ordering of two audiences is identified by the ordering of their speaker shares alone — an immediate consequence of θ being increasing in p (Corollary 1): the sign of the gap equals the sign of p_A − p_B for every γ > 0. Only the magnitude varies, with closed-form extrema at the range edges or at γ₀ = √(p_A·p_B / ((1−p_A)(1−p_B))) (Proposition 6). Under independent rectangles, valid corner bounds apply and the sign can genuinely be lost.
Uncertainty. With a census of speakers, p is exact for the declared audience and window and no sampling band is shown. With sampled classification, a 95% Wilson interval for p is projected through the identification map in two exact closed-form steps: rectangle compatibility inverts to an analytic interval of feasible speaker shares, and on it the feasible endpoints are max(θ(γhi, p), s·p) below and min(θ(γlo, p), s·p + 1 − s) above, both increasing in p — so the band is precisely the union of identified sets over the speaker shares that are simultaneously Wilson-plausible and rectangle-compatible — [L(inf), U(sup)] (Proposition 7 of the paper — stated for any input interval for p, so the Wilson interval can later be replaced by a hypergeometric one without touching the projection). This is the union of the identified intervals, not the set of endpoint values, which can have a gap; the union does not. The structural verdict (identified under the declaration?) is printed separately from the projection qualifier (PRESERVES / DOES NOT PRESERVE / INCONCLUSIVE); the claim θ > τ is strict, so an interval whose upper endpoint equals τ exactly is ruled out. The band is conditional on the declared rectangle and is deliberately not called a confidence interval for the identified set. The sampled layer is gated: it requires the classified speakers to be declared a simple random sample, without replacement and with equal inclusion probability, of the observed speakers — otherwise no population projection is reported. Because the n speakers are drawn without replacement from a finite M, p̂ carries design-based (hypergeometric) sampling uncertainty; the current instrument uses a binomial Wilson approximation notwithstanding, which generally ignores the information in a large sampling fraction and will often be wider than an FPC-adjusted analogue, and claims no finite-population coverage. Regimes are derived from the counts: classifying all observed speakers is a census whatever is ticked, and zero observed speakers is its own verdict — no population information, θ ∈ [0, 1].
Boundary. Attitude labels are taken as given: this instrument bounds selection, not classifier error. The audience declaration is an input, never a certified fact. One claim at a time; aggregates only, always.
Full statement, proofs and a seeded numerical companion: "The Silent Denominator: Population Attitudes as a Partially Identified Quantity under Self-Selected Participation" (Adam, 2026, working paper v0.8) — part of the Coordination–Consensus Continuum programme.