OpenAI claim summary for Math-2631
mathematicsTop Open Problemscontributionliterature_reportClaim: literature report
Summary of a claim published in openai/math; not a claim by this platform.
The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 099; 2026-09-27).
Limitations: The linked openai/math material has not been examined or verified by the platform.
Manuscript
Contents
Binary edit distance has exponential-scale L1 distortion
OpenAI's claim
OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 099; 2026-09-27), the asserted result is that two tree-based binary constructions give that exponential edit-distance lower bound at every sufficiently large cap, together with a constant-distortion coding transfer.Mathematical statement of the claimed result
Manuscript 1.
Use edit distance $\operatorname{ED}$ with cost one for inserting a symbol, deleting a symbol or substituting one symbol; minimize the total cost over transforming scripts. For a finite metric space $M$, its $L^1$ distortion is \[ c_1(M)=\inf_{f:M\hookrightarrow L^1} \left(\sup_{x\ne y}\frac{\|f(x)-f(y)\|_1}{d(x,y)}\right) \left(\sup_{x\ne y}\frac{d(x,y)}{\|f(x)-f(y)\|_1}\right). \] The claim gives absolute $c>0,d_0$ such that for every integer $d\ge d_0$ there are $1\le n\le d$ and $\mathcal W\subseteq\{0,1\}^n$, $|\mathcal W|\ge2$, with \[ c_1(\mathcal W,\operatorname{ED}) \ge\exp(c\sqrt{\log d\,\log\log d}). \] In particular the obstruction uses binary words of one common length, not an unbounded alphabet.Outline of the argument
For manuscript 1 (family 099; 2026-09-27), nested periods and trees of circle phases combine cheap translations with separated row increments. Fourier detection of sparse activity bounds the displacement possible in $\ell_1$, while an alignment lemma preserves the separation after conversion to fixed-length binary words.Formal verification
For manuscript 1 (family 099; 2026-09-27), the documented Lean scope is two lower estimates for binary-tree embeddings; the coding conversion at constant distortion and the combined uniform conclusion are outside those selected statements.The central assertion is within the documented formal scope; the additional applications retain their own scope limits.
The linked family Comparator checks are BinaryEditLower.lean, EditDistance.lean, FiniteCircle.lean, TreeEdit.lean.
Their named declarations include OAI.TreeEdit.binary_lower_bound, OAI.TreeEdit.binary_torus, OAI.edit_distance_main, OAI.finiteCircle_fullMain.
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
- OpenAI, Tree Constructions for the $\ell$${}_{1}$ Distortion of Binary Edit Distance, 2026-09-27; citation key
OAI:Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:binary. - Family 099: Lean scope.
- Comparator BinaryEditLower.json.
- Comparator BinaryEditLower.lean.
- Comparator EditDistance.json.
- Comparator EditDistance.lean.
- Comparator FiniteCircle.json.
- Comparator FiniteCircle.lean.
- Comparator TreeEdit.json.
- Comparator TreeEdit.lean.
- OpenAI catalogue entry.