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Math-2612-A1 · revision r1 · on Math-2612

OpenAI claim summary for Math-2612

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.

Higher-dimensional distinct distances: For each fixed d>=3, every n-point set in R^d has at least c_d n^(2/d) distinct distances; no sharp planar bound is asserted.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

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

Manuscript

Contents
  1. Optimal-order distinct distances in every dimension at least three
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Optimal-order distinct distances in every dimension at least three

OpenAI's claim

Higher-dimensional distinct distances: For fixed $d\ge3$, the claimed distance lower bound for $n$ distinct points in $\mathbb R^d$ is $c_d n^{2/d}$. The constant depends only on dimension, with no vanishing subpower factor.

Mathematical statement of the claimed result

Mathematical claim 1.
In every fixed integer dimension $d\ge3$, a constant $c_d>0$ is claimed to satisfy \[\left|\{|x-y|:x,y\in P,\ x\ne y\}\right|\ge c_d|P|^{2/d}\] for every finite set $P\subset\mathbb R^d$ of at least two distinct points. The constant depends only on the dimension. This is a global distinct-distance count, not a bound on how often one distance occurs.

Outline of the argument

Higher-dimensional distinct distances: Polynomial interpolation and Hilbert-function estimates track concentration at several degree scales. Geometry of skew-form evaluation flats gives sparse-cone and rich-motion bounds, with exceptional algebraic families treated explicitly. These incidence estimates are combined with the distance reduction in a way that retains a fixed multiplicative constant rather than only the exponent up to a small loss.

Formal verification

Higher-dimensional distinct distances: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources