OpenAI · 175-03 · 2026-09-23 · Combinatorics
Talagrand’s discrete-convexity conjecture
preprints/Talagrands-discrete-convexity-conjecture-September-23-2026
Family 175. Talagrand's conjectures and graph decompositions at expectation thresholds.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves that integral and fractional expectation thresholds differ by at most a universal factor, and resolves Talagrand's discrete-convexity conjecture. An application proves the Ascoli–He–Park–Talagrand graph-decomposition conjecture: every graph's edges split into a universally bounded number of fixed pieces, each with containment threshold at most a universal constant times the original graph's integral expectation threshold. The pieces' embeddings need not agree on shared vertices.
We prove Talagrand's discrete-convexity conjecture. There is a universal integer k such that, whenever an arbitrary family has Bernoulli product measure at least , the sets not contained in a union of k members admit a cover of total cost at most 1/2 at the same density.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 175: Talagrand’s expectation thresholds, discrete convexity, and graph decompositions. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.