The information limit of consensus detection on bounded ordinal scales

Consensus in Delphi studies is almost always declared by applying a heuristic threshold to the mean and variance of expert votes on a bounded ordinal (Likert) scale. We ask what such per-item summaries can, in principle, decide. Framing consensus detection against latent polarization as a discrimination problem between vote-generating laws, we prove that the error of any per-item test decays exponentially in the panel size N at a rate equal to the Chernoff information C between the competing laws, giving an explicit minimum-panel-size rule N ⋆ = ⌈ − log ( 2 ϵ ) / C ⌉ . The mean/variance heuristic is exponentially sub-optimal: its separating signal is carried solely by the second-order departure of the bounded-scale variance ceiling from a parabola, so its error decays only at the slower rate c K 2 / 8 . Recast as the conditional risk of false consensus, both risks vanish with N , but at the distinct rates C and c K 2 / 8 ; their ratio diverges exponentially, so larger panels make the moment rule increasingly sub-optimal relative to the information limit even as its absolute risk improves. For the composite problem, the floor is non-vacuous once a minimal faction separation δ min is fixed, and the Chernoff information of symmetric soft polarization vanishes quartically, C ∼ κ δ 4 , identifying balanced soft polarization as the hardest case and genuine non-consensus ( C = 0 ) as the irreducible limit of single-item inference. Practically, C converts consensus detection into a power-analysis problem, yielding explicit minimum panel sizes and flagging items whose declared consensus is informationally indistinguishable from a balanced split, where an indeterminacy verdict is warranted rather than forced classification.