Result 132, Theoretical computer science

A superquadratic separation of sensitivity and block sensitivity

Constructs total Boolean functions with block sensitivity bs(f)≥s(f)α\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)s(f) counts influential individual-bit flips, while block sensitivity allows disjoint groups of bits to change together.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

A yes-or-no rule can respond very differently to changing one input bit and changing several together. This result challenges a proposed limit on that difference, sharpening our understanding of how Boolean functions depend on their inputs.

What changes?

The manuscript reports nonconstant total Boolean functions: rules assigning zero or one to every possible bit string of a given length. Sensitivity counts individual bit flips that change the output; block sensitivity counts disjoint groups whose separate flips change it. Both take the maximum over inputs. The examples have unbounded block sensitivity, at least sensitivity raised to a fixed exponent greater than two. Thus no constant times sensitivity squared bounds block sensitivity for all such functions.

What does that help mathematicians do?

The construction rules out a universal quadratic comparison between these two measures of input dependence. A researcher cannot assume that counting influential individual bits controls the number of independently influential groups within a squared bound. Any proposed universal upper bound must accommodate the reported larger exponent. This refutes the quadratic strengthening, rather than the Sensitivity Conjecture as a whole.

Are there practical applications?

The immediate value is foundational for theoretical computer science. These measures describe how a Boolean computation reacts to changes in its input. The examples provide a test for proposed relationships between single-bit and group effects. The reported consequence is a limitation on universal bounds, not a faster algorithm or a demonstrated practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A superquadratic separation between sensitivity and block sensitivity

September 25, 2026 11 pages Main result formalized in Lean

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 bs(f)≥s(f)α\mathop{\mathrm{bs}}\nolimits (f)\ge s(f)^\alpha.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026,
  author = {{OpenAI}},
  title = {{A superquadratic separation between sensitivity and block sensitivity}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026/paper.pdf}{OAI:A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/132.md.

A superquadratic separation of sensitivity and block sensitivity

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalized result disproves a universal quadratic bound of block sensitivity by sensitivity for total Boolean functions. For every integer d≥1d\ge1, it constructs a nonconstant function with bs(f)/s(f)2≥2d/(4(d+2)2)\mathrm{bs}(f)/s(f)^2\ge2^d/(4(d+2)^2), making the ratio unbounded. The formalization also gives a fixed exponent α>2\alpha>2 and a sequence with s(f)α≤bs(f,0)s(f)^\alpha\le\mathrm{bs}(f,0) while the latter tends to infinity; here bs(f,0)\mathrm{bs}(f,0) is block sensitivity at the all-zero input.

Comparator links

Result Comparator statement
Superquadratic sensitivity separation SensitivitySeparation.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.