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Math-2616-A2 · revision r1 · on Math-2616

OpenAI claim summary for Math-2616

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.

Ordered binary matrix removal obstruction: The primary fixed 66-by-66 pattern and examples have n_h=(386h+2)2^h, epsilon_h=(386h+2)^(-2), distance>=epsilon_h and ordered-copy density<=epsilon_h 2^(-h). This is below C epsilon_h^c for every fixed positive c,C for large h, refuting the universal polynomial-removal assertion for at least one H. Copies match both zeros and ones with increasing row and column indices.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

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

Manuscript

Contents
  1. Polynomial induced removal for ordered binary matrices
  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

Polynomial induced removal for ordered binary matrices

OpenAI's claim

Ordered binary matrix removal obstruction: The manuscript claims that one fixed $66\times66$ binary pattern has no polynomial ordered removal bound. Copies preserve row and column order and match zero entries as well as ones; either type of entry may be edited.

Mathematical statement of the claimed result

Mathematical claim 1.
The paper specifies one binary $66\times66$ pattern $H$. For every integer $h\ge1$ let $n_h=(386h+2)2^h$ and $\epsilon_h=(386h+2)^{-2}$. It claims a binary $n_h\times n_h$ matrix $A_h$ with \[\operatorname{dist}_H(A_h)\ge\epsilon_h,\qquad N_H(A_h)/n_h^{132}\le\epsilon_h2^{-h}.\] Here $N_H$ counts choices of increasing row and column indices matching every zero and one of $H$, and $\operatorname{dist}_H$ is the minimum fraction of cells changed, in either direction, to remove all such copies. Consequently no constants $c,C>0$ provide $N_H(A)\ge c\epsilon^Cn^{132}$ for every $\epsilon$-far matrix.

Outline of the argument

Ordered binary matrix removal obstruction: A 64-by-64 anchor restricts the roles in a two-by-two pattern at the final rows and columns. Binary-tree host blocks propagate incompatible root values down the same path if all designated patterns are avoided. Cellwise sampling of anchors and auxiliary positions shows that too few edits leave this contradiction intact, while the copy count is much smaller than any proposed polynomial bound.

Formal verification

Family 190 describes Lean coverage for an explicit ordered-pattern sequence contradicting every polynomial removal estimate.

Ordered binary matrix removal obstruction: The manuscript links MatrixRemoval.lean (declaration OAI.Problem348.no_polynomial_removal_bound).

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