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Math-2631 · mathematics · Geometry

Binary edit distance has exponential-scale L1 distortion

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Do absolute $c>0,d_{0}$ exist such that, for every integer $d\geq d_{0}$, some common word length $1\leq n\leq d$ and finite $W\subset {0,1}^{n}$ with at least two words have least $L^{1}$ distortion at least $\exp (c\surd (\log $ d loglog d)) for ordinary unit-cost insertion, deletion and substitution edit distance?

Short English statement

Do absolute $c>0,d_{0}$ exist such that, for every integer $d\geq d_{0}$, some common word length $1\leq n\leq d$ and finite $W\subset {0,1}^{n}$ with at least two words have least $L^{1}$ distortion at least $\exp (c\surd (\log $ d loglog d)) for ordinary unit-cost insertion, deletion and substitution edit distance?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2631-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2631

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).

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

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