A superquadratic separation of sensitivity and block sensitivity
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Constructs total Boolean functions with block sensitivity \(\mathop{\mathrm{bs}}\nolimits (f)\ge s(f)^\alpha\) for a fixed α > 2, disproving the quadratic strengthening of the Sensitivity Conjecture. Here \(s(f)\) counts influential individual-bit flips, while block sensitivity allows disjoint groups of bits to change together. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement.Read the assessments
Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.
For every real C>0, there are a positive integer n and a nonconstant total Boolean function f:\{0,1\}^n\to\{0,1\} such that
\bs(f)>C\,s(f)^2.
More precisely, for every integer d\ge1 the construction below gives a nonconstant total Boolean function f with
\frac{\bs(f)}{s(f)^2}\ge \frac{2^d}{4(d+2)^2}.
The release does not say which paper theorem its catalogue describes. The statement above shares the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.
Reported proof verification
One Lean statement in the release; three independent groups report that Comparator accepted runs for this file at the version shown here.
We disprove the quadratic strengthening of the Sensitivity Conjecture by constructing nonconstant total Boolean functions whose block sensitivity grows faster than any constant multiple of sensitivity squared. In fact, for some fixed α > 2, our examples have unbounded block sensitivity and satisfy \(\mathop{\mathrm{bs}}\nolimits (f)\ge s(f)^\alpha\).
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
SensitivitySeparation.lean, the release lists it as a main result.
theoremquantitative_separation (d : ℕ) (hd : 1 ≤ d) : ∃ (n : ℕ) (f : (Fin n → Bool) → Bool),0 < n ∧ f (fun _ => false) = false ∧ (∃ x y : Fin n → Bool, f x ≠ f y) ∧ (2 : ℝ) ^ d / (4 * ((d : ℝ) + 2) ^ 2) ≤ (blockSensitivity f : ℝ) / (sensitivity f : ℝ) ^ 2 := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express a weaker statement than the advertised result (their term: “weaker-statement”).
Their note:Unbounded ratio bs/s^2 only; the fixed exponent alpha > 2 (bs >= s^alpha) of the summary is not stated in the challenge even though the docs claim it.
Definitions the group asks readers to compare with the paper:blockSensitivityAt f x := Finset.univ.sup (fun blocks => if pairwise disjoint nonempty sensitive blocks then blocks.card else 0) and sensitivity := sup of sensitivityAt are standard; the ratio is real division, so sensitivity f = 0 would make the inequality false rather than vacuous
“weaker-statement”, in their words: The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).
The gap they name:The fixed-exponent separation bs(f) ≥ s(f)^α with α > 2 (and unbounded bs) is claimed by the headline but not stated by any Comparator challenge. The scope note does not declare this omission; it asserts the opposite (‘The formalization also gives a fixed exponent α>2 …’,), which is true of the formal library but not of the Comparator statement.
“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The formalized result disproves a universal quadratic bound of block sensitivity by sensitivity for total Boolean functions. For every integer $d\ge1$, it constructs a nonconstant function with $\mathrm{bs}(f)/s(f)^2\ge2^d/(4(d+2)^2)$, making the ratio unbounded. The formalization also gives a fixed exponent $\alpha>2$ and a sequence with $s(f)^\alpha\le\mathrm{bs}(f,0)$ while the latter tends to infinity; here $\mathrm{bs}(f,0)$ is block sensitivity at the all-zero input.
AI-drafted comparisons, with no recorded human review
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem that bs(f)/s(f)² is unbounded, with the explicit ratio 2^d/(4(d + 2)²), together with a polynomial separation of exponent above 2 at the all-zero input.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
Follow-up work by others
acprk, acprk/openai-math-followups: follow-up results on three OpenAI families, 9 October 2026. Source
Any explicit seed for a superquadratic separation of block sensitivity from sensitivity needs sensitivity at least 4 and at least 31 variables (using Amano 2017 and SAT refutations checked by an independent RUP checker); the release’s displayed parameters guarantee an exponent of only about 2.000252.
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.
Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source
A Formal Conjectures maintainer, after source inspection, links the superquadratic sensitivity separation to the strong sensitivity conjecture, stated as a negation.
tparker, Philip Weiss, Will Sawin, Nik Weaver, Pace Nielsen, mathworker21, Joshua P. Swanson, Timothy Chow, Daniel Soltesz, MathOverflow: how many of the released results went against the conventional wisdom?, 8 October 2026. Source
An answerer reads the sensitivity result as disproving the stronger quadratic conjecture proposed in the 1990s while leaving the known fourth-power bound untouched.
tparker with nine answerers, How many of the results released by OpenAI went against the conventional mathematical wisdom?, 8 October 2026. Source
The bound bs(f) <= s(f)^2+epsilon construction refutes the Nisan-Szegedy-type quadratic bound proposed in 1994, without contradicting Huang’s s^4 bound.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The doc and headline claim a power separation, but the theorem Comparator checks gives only a weaker bound that implies none.
Questions awaiting review
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 132No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded.Open the form
Address a question raised in an independent assessment
Family 132Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In A superquadratic separation between sensitivity and block sensitivity, 25 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
boolean, total
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The record’s reason
No unique candidate with at least three shared catalogue terms.
Independent assessment
MathVet
Their term
“weaker-statement”
What their term means
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
Re-run through Comparator with Lean’s kernel: accepted (the check took 45 s; peak memory 6.1 GiB).
Date
11 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.