Browse › Functional analysis › 329 A counterexample to metric-entropy duality
Counterexamples to the duality conjecture for metric entropy
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Abstract
We disprove Pietsch's dimension-free duality conjecture for metric entropy. For every proposed pair of universal constants, we construct origin-symmetric convex bodies that violate the corresponding covering-entropy inequality, already when the covering body is a cube.
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Notes
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Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- bocowgill (review on google-deepmind/formal-conjectures PR #7152) (GitHub, 2026-10-12). Approving review saying the proposed Lean statement matches Theorem 1.1 and the manuscript's conventions; it explicitly approves only the statement translation, not a formal proof.
Formalization
- OpenAI math release, Lean scope note for family 329 (lean/docs/329.md, 2026-10-06). The release's scope note: the formalized result covers Theorem 1.1 with $L=B_\infty$, plus a further construction in which the dual-to-primal logarithmic entropy ratio tends to zero as the dimension grows. [family-level]
- OpenAI math release, MetricEntropyDuality.lean (comparator statement) (lean/ComparatorChallenges). Comparator challenge statement
exists_entropy_duality_counterexample_with_coversfor Theorem 1.1, with explicit finite-cover hypotheses. The proof field in this file issorry, as is usual for a comparator target. - anxkhn, feat(Paper): add the metric-entropy duality counterexample (google-deepmind/formal-conjectures PR #7152, opened 2026-10-11, open when checked). Lean 4 statement of Theorem 1.1, pinned to a commit of openai/math; no
formal_proofattribute is added because no external proof build or Comparator check was run.
Related prior literature mentioned by others
- Hu, Peale, Reingold, Metric Entropy Duality and the Sample Complexity of Outcome Indistinguishability (arXiv:2203.04536, ALT 2022). Already cited by the paper. Its arXiv text describes the Pietsch conjecture as open, with weaker versions proved, and links it to learning-theoretic sample complexity.
- S. Artstein, V. Milman and S. J. Szarek, Duality of metric entropy (Annals of Mathematics 159, 2004; announced in C. R. Math. Acad. Sci. Paris 337 (2003), 711–714). Cited in the AI-generated Open Mathematics Courses lesson, this establishes the dimension-free comparison when one body is a Euclidean ball, a special case distinguished from the manuscript's unrestricted conjecture.
- S. Artstein, V. Milman, S. J. Szarek and N. Tomczak-Jaegermann, On convexified packing and entropy duality (Geometric and Functional Analysis 14, 2004). The AI-generated Open Mathematics Courses lesson cites Conjecture 1 and Theorem 2 and presents convexified-packing duality alongside the manuscript's counterexample for ordinary covering numbers.
Possibly relevant literature (found by the agent, not checked)
- Artstein, Milman, Szarek, More on the duality conjecture for entropy numbers (C. R. Math. Acad. Sci. Paris 336 (2003), 479–482). Proves duality up to a logarithmic factor when one body is an ellipsoid, a precursor to the Euclidean-ball case that the paper discusses in its related-work section; not in the bibliography.
- Szarek, Convexity, complexity, and high dimensions (Proc. ICM 2006, vol. II, 1599–1621, doi:10.4171/022-2/76). ICM survey that lists "metric entropy and its duality" among its topics, which may help place the paper's claim against the status of the conjecture as of 2006; not in the bibliography.
Digests and expositions
- Open Mathematics Courses / Claude Opus 5.5, The duality conjecture for metric entropy (Open Mathematics Courses, undated). Three lessons with proofs and exercises explain the manuscript's matrix-to-body conversion, partition compression and finite-field construction; the source states that independent review is incomplete. [AI-generated per source]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 329
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026,
author = {{OpenAI}},
title = {{Counterexamples to the duality conjecture for metric entropy}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026/main.pdf}{OAI:Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/329.md. The Lean statements have not been checked by people against the paper's claims.
Family 329: A counterexample to metric-entropy duality
Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is a cube.
This is the only paper in this family.