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Math-567-A1 · revision r1 · on Math-567

OpenAI claim summary for Math-567

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 at unsolvedmath.com: 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.

Planar unit-distance power saving: The bound u(n)<=C n^beta with beta<4/3 is an improved planar upper bound, not the conjectured n^(1+o(1)) order, an exact extremal value, or the other bundled dimension/distance questions.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

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

Manuscript

Contents
  1. The Unit Distance Problem
  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

The Unit Distance Problem

OpenAI's claim

Planar unit-distance power saving: The unit-distance manuscript claims an absolute exponent $\beta<4/3$ bounding the number of unordered unit pairs by $C n^\beta$. The point configuration is otherwise arbitrary.

Mathematical statement of the claimed result

Mathematical claim 1.
For $P\subset\mathbb R^2$ of $n$ distinct points let $u(P)$ count unordered pairs at Euclidean distance $1$, and let $u(n)=\max_{|P|=n}u(P)$. The claim is that absolute constants $C<\infty$ and $1\le\beta<4/3$ satisfy \[u(n)\le Cn^\beta\quad(n\ge0).\] The unspecified saving $4/3-\beta>0$ improves the known exponent but does not establish every proposed near-linear bound.

Outline of the argument

Planar unit-distance power saving: A near-extremal incidence sequence is transferred to algebraic configurations and analyzed using distance identities over a negligible-height coefficient field. After differentiating those identities, valuations distinguish the individual radicands and allow independent square-root sign changes. One unused sign change contradicts the differentiated system; a separate exclusion of popular unit circles handles the possible square-class failure, preventing approach to exponent $4/3$.

Formal verification

Family 167 describes Lean coverage for the weak pinned-distance conclusion and the separate fixed-power unit-distance improvement.

Planar unit-distance power saving: The manuscript links PinnedDistances.lean (declaration OAI.WeakPinned.main), PlanarUnitDistances.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

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