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Math-2624-A1 · revision r1 · on Math-2624

OpenAI claim summary for Math-2624

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The explicit $k=2^{75}$ theorem supplies the requested containment cover with budget one half for every product-measure parameter and arbitrary family, allowing repeated tuple members. Relabelling handles all finite nonempty ground sets; for the empty ground set the exceptional family is empty.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Talagrand discrete convexity conjecture
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Talagrand discrete convexity conjecture

OpenAI's claim

Talagrand discrete convexity: The paper claims a universal number of family members whose unions leave only a small containment-cover obstruction, provided the original Bernoulli measure is sufficiently close to one. The cover cost is bounded at the original density.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $k=2^{75}$ and $\mu_p$ be product Bernoulli measure on $2^{[N]}$. For every $N\ge1$, $0<p<1$, and arbitrary family $\mathcal D$, define $E_k(\mathcal D)=\{S:S\nsubseteq D_1\cup\cdots\cup D_k\text{ for every }D_1,\ldots,D_k\in\mathcal D\}$. Repetitions among the $D_i$ are allowed. The theorem claims \[\mu_p(\mathcal D)\ge1-1/k\quad\Longrightarrow\quad\exists\mathcal G:\ E_k(\mathcal D)\subseteq\{S:\exists I\in\mathcal G,\ I\subseteq S\},\quad\sum_{I\in\mathcal G}p^{|I|}\le1/2.\] No increasingness or disjointness hypothesis is imposed on $\mathcal D$.

Outline of the argument

Talagrand discrete convexity: A biased orthogonal expansion produces nonnegative subset weights with a global budget. A signed many-row identity turns failure of a union cover into a large higher-power sum of those weights. Large weights give cheap generators directly, while weighted-degree selection and random unions cover the remaining classes; a final density transfer gives the universal union count.

Formal verification

Family 175 describes Lean coverage for the integral/fractional threshold comparison and the discrete-convexity statement; graph decomposition is a later separate consequence.

Talagrand discrete convexity: The manuscript links TalagrandDiscreteConvexity.lean (declaration OAI.TalagrandDiscreteConvexity.talagrand_discrete_convexity), TalagrandExpectationThreshold.lean (declaration OAI.TalagrandThreshold.talagrand_expectation_threshold_equivalence).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources