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Math-2701-A1 · revision r1 · on Math-2701

OpenAI claim summary for Math-2701

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.

The displayed question has a negative answer under the manuscript's stated construction. Its counterexample meets the conditions in this problem and defeats the universal assertion, so this is a full disproof claim. The supporting result is manuscript 1 (family 329; 2026-09-24).

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

Manuscript

Contents
  1. Uniform duality for convex-body covering entropy
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Uniform duality for convex-body covering entropy

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 329; 2026-09-24), the asserted result is that the proposed primal--polar entropy comparison fails for any choice of universal constants, even if one covers by translates of a cube.

Mathematical statement of the claimed result

Manuscript 1.
Write $N(K,L)$ for the least number of translates of $L$ covering $K$, and $K^{\circ}$ for the Euclidean polar. The claimed counterexamples satisfy \[ \forall a,b\ge1\ \exists n\ge1\ \exists K=-K\subset\mathbb R^n:\quad \log N(K,[-1,1]^n)>b\log N(([-1,1]^n)^{\circ},a^{-1}K^{\circ}), \] where $K$ is a full-dimensional convex body. Thus no fixed pair of dilation and multiplicative constants supplies the proposed dual covering-number comparison in every dimension.

Outline of the argument

For manuscript 1 (family 329; 2026-09-24), partitions compress convex-hull columns uniformly across coordinates. Finite-field zero estimates supply many separated rows whose dual images remain compressible; choosing the matrix parameters makes the primal entropy arbitrarily larger than the proposed polar comparison.

Formal verification

For manuscript 1 (family 329; 2026-09-24), OpenAI lists Lean statements for this assertion: the proposed primal--polar entropy comparison fails for any choice of universal constants, even if one covers by translates of a cube.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are MetricEntropyDuality.lean.

Their named declarations include OAI.MetricEntropyDuality.exists_entropy_duality_counterexample_with_covers.

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