Small samples do not make pooled pricing safe

Unrefereed research candidate. A seller with a small sample can post one price chosen from the whole sample (pooled empirical revenue maximisation) or split buyers into markets by a covariate and price each market separately. Xie, Zhu and Shishkin (Theoretical Economics, 2025) showed that segmentation learns more slowly and asked whether, for small enough samples, pooled pricing always wins under their regularity conditions. This package shows that those conditions alone do not guarantee such a window. For every number of markets K and every finite range of sample sizes, one distribution satisfying the conditions makes K-market pricing strictly more profitable at every sample size in the range; combining it with a distribution on which segmentation wins in large samples gives two fixed distributions in one fixed regularity class that defeat pooled dominance at every sample size, with an explicit threshold for two markets. The mechanism is a rare high-value segment: one high observation pulls a pooled price up and loses ordinary buyers. Empty markets receive the pooled price. The provable constructions use extreme constants and tiny gains; an exploratory simulation shows gains of up to about ten per cent at moderate parameters. The version with moderate constants fixed in advance remains open. The archive contains the manuscript (PDF and Markdown source), an exact-arithmetic certificate program, the simulation, a replay driver with claim checks, semantic negative controls, CI and SHA-256 sums. Assurance boundary: written proofs; exact rational certificates; producer-side replay; internal AI-assisted review (five reviewer perspectives, an independent-model adversarial review and a verification pass with proof audit). Not peer review, independent reproduction or formal verification. Creator: Anonymous. Prose, manuscript and data CC0-1.0; code MIT.