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Talagrand's conjecture that fractional and integral expectation thresholds agree up to a constant

Theorem 1.1: $q_f(\mathcal F)\le25\cdot512^4\,q(\mathcal F)$ for every finite nonempty $X$ and nonempty proper increasing $\mathcal F\subseteq2^X$ , independently of $|X|$ , the sizes of minimal members and the support of the fractional cover; equivalently every fractional cover of cost at most $1/2$ at $r$ rounds to an integral cover of cost at most $1/2$ at $r/(25\cdot512^4)$ . Previously the best general loss was of order $\log\log|X|$ (Fischer-Person with Pham). The constant is explicit but not optimized.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Talagrand's conjecture that fractional and integral expectation thresholds agree up to a constant · Fractional vs integral expectation thresholds

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
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Unreleased internal OpenAI model
model · ai model contributor · OpenAI

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Named evidence

VibeMathed record: Talagrand's conjecture that fractional and integral expectation thresholds agree up to a constant

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Talagrand's Conjecture 6.3 as the manuscript states it. It gives $q_f(\mathcal F)\le25\cdot512^4\,q(\mathcal F)$ for every nonempty proper increasing family on a finite nonempty ground set, both with budget $1/2$ . lean/formalization.yaml lists TalagrandExpectationThreshold.json (declaration OAI.TalagrandThreshold.talagrand_expectation_threshold_equivalence, file OAI/Combinatorics/ExpectationThreshold/Main.lean). The statement TalagrandExpectationThreshold.lean was read here; not rebuilt here. It defines integral and fractional smallness with budget 1/2 (fractional weights in [0,1], including the empty set), $q$ and $q_f$ as suprema of feasible $p\in[0,1]$ , and states $q_f\le25\cdot512^4\,q$ for nonempty, proper, increasing families. This states the headline. Permitted axioms: propext, Quot.sound, Classical.choice.

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Family: Integral and fractional expectation thresholds are equivalent

paper · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Input: Talagrand's discrete-convexity conjecture

paper · pending

paper link supplied by the VibeMathed registry.

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The Ascoli-He-Park-Talagrand graph-decomposition conjecture at the expectation threshold

witness · pending

paper link supplied by the VibeMathed registry.

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Lean proof (OAI.TalagrandThreshold.talagrand_expectation_threshold_equivalence)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Talagrand, Are many small sets explicitly small? (STOC 2010)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean proof (OAI.TalagrandThreshold.talagrand_expectation_threshold_equivalence)

lean artifact · pending

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Compute record

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Lean proof (OAI.TalagrandThreshold.talagrand_expectation_threshold_equivalence) evidence for this event

This event attributed to Unreleased internal OpenAI model

VibeMathed record: Talagrand's conjecture that fractional and integral expectation thresholds agree up to a constant evidence for this event

Input: Talagrand's discrete-convexity conjecture evidence for this event

Family: Integral and fractional expectation thresholds are equivalent evidence for this event

Talagrand's conjecture that fractional and integral expectation thresholds agree up to a constant parent of this event

Talagrand, Are many small sets explicitly small? (STOC 2010) evidence for this event

The Ascoli-He-Park-Talagrand graph-decomposition conjecture at the expectation threshold evidence for this event

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