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Math-2641-A1 · revision r1 · on Math-2641

OpenAI claim summary for Math-2641

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 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 094; 2026-09-23).

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

Manuscript

Contents
  1. Subpolynomial nonlinear dimension reduction in Lp
  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

Subpolynomial nonlinear dimension reduction in Lp

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 094; 2026-09-23), the asserted result is that fixed-distortion embeddings of finite real $L_p$ subsets into the same-exponent finite-dimensional space require only $n^{o(1)}$ dimensions for $1<p<\infty$, with nonlinear maps allowed.

Mathematical statement of the claimed result

Manuscript 1.
Fix $1<p<\infty$, $p\ne2$. Let $d_p(n,D)$ be the minimum dimension sufficient for embedding every $n$-point subset of an $L^p$ space into a finite-dimensional $L^p$ subspace with distortion at most $D$. Put $\gamma(p)=2-p$ for $p<2$ and $\gamma(p)=1-2/p$ for $p>2$. For each $D>1$, the claim gives $C_{p,D}<\infty$ with \[ \frac{\log n}{\log(1+2D)}\le d_p(n,D) \le\exp(C_{p,D}(\log n)^{\gamma(p)})\quad(n\ge2). \] For exact embeddings it instead asserts \[ \lfloor(n-1)/4\rfloor^2\le d_p(n,1)\le\binom n2\quad(n\ge9). \] Consequently the limit of $\log d_p(n,D)/\log n$ is zero for $D>1$ and two for $D=1$. The embeddings need not be linear on the ambient space.

Outline of the argument

For manuscript 1 (family 094; 2026-09-23), homogeneous increments are represented by signed Poisson samples with a shared heavy-tailed marginal. A ramp of clipping thresholds and a controlled projection bound the error before bounded-gradient sampling reduces the dimension; packing supplies the lower comparison.

Formal verification

For manuscript 1 (family 094; 2026-09-23), the documented Lean scope is finite-dimensional approximation bounds for $p>1$ with $p\ne2$, including the fixed-distortion subpolynomial upper estimate and the logarithmic lower estimate; a quadratic comparison is listed for distortion one.

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

The linked family Comparator checks are SubpolynomialLp.lean.

Their named declarations include OAI.SubpolynomialLp.source_main.

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