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Claim. OpenAI, The higher-dimensional Erdős distinct-distances conjecture, OpenAI Math Release preprint, 23 September 2026, linked above at the pinned revision. The release's README says that its manuscripts were produced by an internal OpenAI model and are at different stages of verification, so the author of record is OpenAI and the system is the internal model the release names. The manuscript is carded as the library's intake card, and its Theorem 1.1 page states the main theorem. Theorem 1.1 states that for every integer $d\ge3$ there is a constant $c_d>0$ such that every set $P\subset\mathbb R^d$ of $n\ge2$ distinct points determines at least $c_d\,n^{2/d}$ distinct distances. In the notation of Problem 1083 this is $f_d(n)\gg_d n^{2/d}$; with Erdős's upper bound $f_d(n)\ll_d n^{2/d}$ from the grid $\{1,\dots,t\}^d$, whose squared distances are integers below $dt^2$, it gives $f_d(n)=\Theta_d(n^{2/d})$ for every fixed $d\ge3$. The question whether $f_d(n)=n^{2/d-o(1)}$ is answered yes, in the stronger form with no $o(1)$ loss, and the estimate the problem asks for is settled up to the constants $c_d$. No hypothesis is placed on the position, spacing or concentration of the points.

The proof argues in the least dimension in which the theorem could fail and excludes every sequence of sets along which the ratio of the distance count to $n^{2/d}$ tends to zero, since a bound with a $o(1)$ loss would not imply the constant factor. Its two-dimensional input is the planar bound of Guth and Katz; the repeated distances are counted through rigid motions, in the manner of Elekes and Sharir and of Bardwell-Evans and Sheffer, and the resulting families of flats are controlled through Hilbert function estimates of Chardin and Chardin and Philippon and through approximate complete intersections in the sense of Walsh. The preprint says that the methods of Tidor, Yu and Zakharov, who proved $n^{2/3-o(1)}$ distinct distances in $\mathbb R^3$ in August 2026, provide close antecedents for several parts of its argument (the flat families, the recovery of contained flats from tangent data and the treatment of projected curves); that result, recorded on [[problems/distance_problems/E1083/claims/2026_08_14_tidor_yu_zakharov|its own claim page]] as a pending partial claim for $d=3$, and the Solymosi–Vu recurrences give $n^{8/17-o(1)}$ in dimension four and $n^{3/8-o(1)}$ in dimension five, both below the exponent $2/d$ the claim reaches. The preprint also notes that an earlier manuscript claiming the constant-factor bound in all dimensions was withdrawn by its author.

Depends on. No page of this wiki.

Acceptance. None documented. The result is a theorem statement in a release preprint; no referee, outside reviewer or curator is recorded as having examined it, the site's page labels the problem OPEN, was last edited on 16 October 2025 and carried no proof claim on 2026-10-06, and the proof is not reviewed. The release carries no Lean for this paper, and the release's Lean proof of the Falconer distance conjecture, from its companion paper on the continuum analogue, concerns compact sets of Hausdorff dimension above $d/2$ and the measure of their distance sets, which neither implies nor is implied by a count for finite sets. The claim is therefore claimed, and the problem's standing is claimed through it.