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CS-160-A1 · revision r1 · on CS-160

OpenAI claim summary for CS-160

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 displayed question has a negative answer under the manuscript's stated construction. Its counterexample meets the conditions in this problem and defeats the universal assertion, so this is a full disproof claim. The supporting result is manuscript 1 (family 132; 2026-09-25).

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

Manuscript

Contents
  1. Quadratic sensitivity versus block sensitivity
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Quadratic sensitivity versus block sensitivity

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 132; 2026-09-25), the asserted result is that total Boolean functions can have block sensitivity exceeding sensitivity to a fixed exponent above two, disproving a universal quadratic relation.

Mathematical statement of the claimed result

Manuscript 1.
For a total Boolean function, let $s(f)$ be its maximum number of sensitive individual coordinates and $\operatorname{bs}(f)$ its maximum number of pairwise disjoint sensitive blocks. The claim states \[ \forall C>0\ \exists n\ge1\ \exists f:\{0,1\}^n\to\{0,1\}:\quad f\text{ nonconstant},\quad\operatorname{bs}(f)>C s(f)^2. \] Its explicit family has, for every integer $d\ge1$, \[ \operatorname{bs}(f)/s(f)^2\ge 2^d/[4(d+2)^2]. \]

Outline of the argument

For manuscript 1 (family 132; 2026-09-25), nested predicates are composed using labeled tournament rows. Strong child acceptance and weaker outgoing rejection requirements keep one-bit changes scarce, while disjoint zero-input blocks multiply through the recursion; a fixed composition converts the growing ratio into an exponent gap.

Formal verification

For manuscript 1 (family 132; 2026-09-25), OpenAI lists Lean statements for this assertion: total Boolean functions can have block sensitivity exceeding sensitivity to a fixed exponent above two, disproving a universal quadratic relation.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are SensitivitySeparation.lean.

Their named declarations include OAI.Paper320.quantitative_separation.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources