A speaker-normalized social-listening share is a statement about the units who spoke; interpreting it as the attitude share of an audience requires an assumption about self-selected — and selectively captured — participation: an assumption rarely stated and not identified from listening data alone. (Mention-level percentages are not even that: they must first be converted to a speaker-normalized share under a declared aggregation protocol, a gate this paper makes explicit.) This paper treats the population attitude share as what it is: a partially identified quantity. If a fraction s of the audience speaks and a fraction p of speakers hold the attitude, the population share theta is only bounded, theta in ps, ps + (1 - s) — the classical worst-case bound of the selection-problem literature (Manski 1989, 2003), which we state for the social-listening estimand under explicit eligibility gates. The contribution is the layer above the anchor. We parameterize selection by a single scalar gamma — the response-probability ratio of the missing-binary-outcomes literature (Magder 2003): the factor by which holding the attitude multiplies the probability of speaking — and prove a closed-form identity: given gamma, the population share is point-identified as theta(gamma) = p / (p + gamma(1 - p)), strictly decreasing in gamma, with no dependence on s. Because gamma is known only within a declared rectangle, theta is not point-identified; we show that the set of gamma values compatible with the observed pair (p, s) is itself a closed interval with closed-form endpoints, that those endpoints map exactly onto the worst-case anchor bounds, and that the identified interval under any declared rectangle is attained at the rectangle's corners with no optimization. Misspecification is refuted rather than absorbed: a declared rectangle disjoint from the data-compatible interval produces no estimate. Two decision-grade objects follow in closed form: a breakdown frontier — the exact value of the ratio, gamma*, at which a threshold claim about the population loses its sign, gamma* = p(1 - tau) / (tau(1 - p)) — and an identification budget separating selection-induced width — which additional utterances from the same observed speaker population cannot reduce at fixed p and s — from sampling width that classification effort and direct audience measurement can. We state plainly what the method does not prove. The sensitivity parameter itself is the response-probability ratio of Magder (2003); the contribution claimed is not the parameter but the identification framework built on it for a formally declared social-listening estimand — the exact feasibility region of the ratio given observed participation, its exact correspondence with the Manski bounds, the breakdown frontier as a reporting object, and a refusal-capable instrument. The closest methodological relatives are Magder (2003), the informative-missingness odds-ratio tradition (Higgins, White and Wood 2008), the nonignorable-nonresponse sensitivity literature (Scharfstein, Rotnitzky and Robins 1999) and the survey-nonresponse bounds of Manski.
Alex-George Adam (Sun,) studied this question.