Marginal and Pairwise Beliefs Do Not Settle a Threshold Verdict

This preprint shows that ten marginal probabilities—and even all 45 pairwise distributions—do not identify the ten-way conjunction used in a published AI-welfare threshold calculation. Holding the published marginals and all pairwise distributions at independence, the sharp range is [0, 6.5%]. The record includes exact endpoint constructions, a 46-check exact-arithmetic verification program, a fixed-seed robustness sweep comprising 900 linear programs, pinned dependencies, and machine-readable results. Claude (Anthropic) and OpenAI Codex (OpenAI), accessed November 2026, assisted with source checking, formalization, programming, drafting, and document preparation under the author’s direction. Tony T. Nguyen reviewed and approved all text, proofs, and code and is responsible for the work.