The duality conjecture for metric entropy
From OpenAI Aftermath
The duality conjecture for metric entropy asks whether the number of translates of one symmetric convex body needed to cover another is comparable, up to universal constants in the exponent and the scale, to the corresponding covering number for the polar bodies with the roles exchanged. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 24 September 2026, claims a counterexample to the dimension-free form of the conjecture. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- For origin-symmetric convex bodies K and L, let N(K, L) be the least number of translates of L that cover K. The conjecture, originating in Pietsch's work on entropy numbers of operators in 1972 and stated geometrically by Artstein, Milman, Szarek and Tomczak-Jaegermann, asks for absolute constants a, b ≥ 1 with log N(K, L) ≤ b log N(L°, a−1K°) and the reverse inequality, in every dimension.
- König and Milman compared the nth roots of the two covering numbers, and Bourgain, Pajor, Szarek and Tomczak-Jaegermann proved comparisons under additional assumptions. Artstein, Milman and Szarek proved the conjecture when one body is a Euclidean ball, hence when either body is an ellipsoid, and with Tomczak-Jaegermann proved duality for convexified packing and announced it for spaces with bounded K-convexity constant.
- Emanuel Milman obtained a comparison for arbitrary symmetric bodies with logarithmic losses in the scale and the entropy factor, and Hu, Peale and Reingold studied dual covering numbers of function classes in learning theory.
The original result
- Claim. Theorem 1.1 of Counterexamples to the duality conjecture for metric entropy states that for every a, b ≥ 1 there are a dimension n and an origin-symmetric convex body K in Rn such that, with L the cube [−1, 1]n, log N(K, L) > b log N(L°, a−1K°). The manuscript states that this disproves the conjecture for every proposed pair of constants, that the dimension grows with the parameters, and that no assertion about a fixed dimension is intended.
- Further claims. A corollary separates ordinary covering entropy from convexified-packing entropy after any fixed change of scale.
- Method, as described: a finite matrix with entries in [0, 1] whose rows are pairwise separated in the sup norm, while the absolutely convex hull of its columns has a small uniform approximation list, is converted into the two convex bodies by a support-function calculation. The columns are logarithmic distance profiles in graphs defined by families of partitions, compressed by storing partition labels instead of points, and the separated rows come from symmetric multilinear forms over a finite field, using the polynomial zero estimate of Schwartz and Zippel.
- Source (openai/math repository, commit
7ef5c0c): Counterexamples to the duality conjecture for metric entropy, dated 24 September 2026. - Formalization, as described by the repository. The scope note lean/docs/329.md states that the formalization disproves the comparison: for every a, b ≥ 1 there is an origin-symmetric convex body K in a positive finite dimension with log N(K, B∞) > b log N(B∞°, a−1K°). It adds that a further construction makes the dual-to-primal logarithmic entropy ratio tend to zero as the dimension grows, and links the comparator statement MetricEntropyDuality.lean.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 28 October 2026) |
|---|---|---|---|---|
| 2026-10-06 | A counterexample to metric-entropy duality, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains the claimed counterexample for every proposed pair of constants, with the cube as covering body: covering numbers and polar bodies, an elementary lemma turning a matrix with separated rows and compressible columns into convex bodies, compression of columns through partition labels, and row separation together with the order in which the parameters are chosen. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-09 | The duality conjecture for metric entropy, a course on Open Mathematics Courses (kokunoyumeto/open-math-courses-public) | Reading (exposition) | Three lessons: covering numbers and polar bodies; compression by partitions; and symmetric forms over finite fields. | The lessons state that they were written by Claude Opus 5.5 (Anthropic) in October 2026 and self-checked by the writing AI; the course lists its status as author self-checked. |
| 2026-10-11 | Formal Conjectures pull request #7152 | Statement (proposed) | Proposes adding Theorem 1.1 of the manuscript, pinned at commit adc7f12, to the repository, with explicit finite-cover hypotheses for both covering numbers. The description states that no external proof build or Comparator check was run, so no formal-proof attribute is added. |
Open pull request, approved by a reviewer on 12 October 2026 and not merged when checked on 28 October 2026; the description discloses AI assistance. |
| 2026-10-19 | AI Proved What?, episode 329 Covering is not self-dual?, remade version of 23 October 2026 | Reading (exposition) | Video explainer of the claim that for any proposed constants a and b there are a dimension and a symmetric convex body K such that covering K by the unit cube costs more than b times the entropy on the dual side, even when the dual covering body is shrunk by the factor a, so that Pietsch's duality conjecture is false. The episode states that no counterexample in a fixed dimension, no explicit body and nothing against weaker versions with logarithmic losses is claimed, and that the ellipsoid case stays proved. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 15 pages, dated September 24, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: the main theorem. The page also states that the Lean formalization of the main theorem has 52 files and about 6,500 lines, not run by the site, that the corollary is not formalized, that three outside projects re-ran the check and accepted it, that the site found no public comment from specialists and no reported error, and that OSVerify has not reviewed the paper. |
Notes
- The background summarizes the manuscript's own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
