Result 190, Combinatorics

Polynomial removal fails for ordered binary matrices

Disproves polynomial ordered binary matrix removal with one fixed 66×6666\times66 zero–one pattern. Matrices can require many binary-entry changes to become pattern-free while their copy density is smaller than every proposed polynomial bound in that distance. Copies preserve row and column orders and match both zeros and ones.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

A binary matrix can require many edits to eliminate a fixed pattern while containing very few copies of it. The manuscript reports a counterexample to a proposed polynomial relationship between edit distance and pattern frequency.

What changes?

The manuscript constructs one fixed 66-by-66 zero-one pattern. An ordered copy selects rows and columns in their original relative orders and matches every entry, including zeros. Distance from being pattern-free means the minimum proportion of entries that must change, allowing both zero-to-one and one-to-zero changes. For this pattern, the reported examples have copy densities too small for any polynomial lower bound in that distance. This disproves the polynomial ordered binary matrix-removal conjecture.

What does that help mathematicians do?

The result separates a global repair requirement from the frequency of local evidence: needing many changes does not force polynomially many ordered copies. Researchers therefore cannot use the conjectured polynomial estimate as a general bridge between edit distance and pattern counts. One fixed counterexample defeats that general guarantee, but it does not show that polynomial bounds fail for every ordered binary pattern.

Are there practical applications?

The immediate value is foundational: it identifies a limit on quantitative estimates in ordered matrix combinatorics. It also matters for approaches that detect violations by sampling possible pattern locations, since rare copies can be difficult to encounter despite substantial repair costs. That sampling obstacle is not, by itself, a lower bound for every possible testing algorithm.

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

Polynomial removal fails for ordered binary matrices

September 25, 2026 12 pages Main result formalized in Lean

We construct a fixed 66×6666\times66 binary matrix for which ordered matrix removal has no polynomial bound. This disproves the polynomial ordered binary matrix-removal conjecture. Ordered copies preserve the separate row and column orders and match both zeros and ones; removal permits changing entries in either direction.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026,
  author = {{OpenAI}},
  title = {{Polynomial removal fails for ordered binary matrices}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026/paper.pdf}{OAI:Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/190.md.

Polynomial removal fails for ordered binary matrices

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

Scope

The formalized result disproves a polynomial removal bound for one explicit 66×6666\times66 binary pattern. For every c,C>0c,C>0, there is an n×nn\times n binary matrix at normalized edit distance at least ε>0\varepsilon>0 from being pattern-free but with fewer than cεCn132c\varepsilon^C n^{132} induced ordered copies. Row and column indices are independently increasing, and every zero and one entry must match. The construction gives an explicit sequence of such matrices, allowing edits in both directions.

Comparator links

Result Comparator statement
Failure of polynomial removal for ordered binary matrices MatrixRemoval.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 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.