Claim. The first question of Problem 604 is answered yes. For a finite set $P\subset\mathbb R^2$ and a point $x\in P$ write $D_x(P)=\{\lVert y-x\rVert:y\in P\setminus\{x\}\}$ for the set of distances from $x$, and for distinct $x,y\in P$ let $k_P(x,y)$ be the number of points $z\in P\setminus\{x\}$ with $\lVert z-x\rVert=\lVert y-x\rVert$, the size of the distance fiber at $x$ through $y$. The manuscript The weak pinned planar distance theorem of the OpenAI mathematics release, dated 23 September 2026 and authored by OpenAI, carded in the library at openai_2026_weak_pinned_planar_distance_theorem with result pages Theorem 1.1 and Corollary 1.2, proves as Theorem 1.1 that for every fixed $s>0$ the largest possible proportion of ordered pairs of distinct points $(x,y)$ of an $n$-point set with $k_P(x,y)\ge n^{s}$ tends to $0$ as $n\to\infty$, uniformly over all configurations and with no hypothesis of separation or general position, and deduces as Corollary 1.2 that for every fixed $\varepsilon>0$

so that every sufficiently large $n$-point planar set has a point $x$ with $|D_x(P)|\ge n^{1-\varepsilon}$. Letting $\varepsilon$ decrease with $n$ gives a point with $n^{1-o(1)}$ distinct distances, which is the bound the problem's first question asks for. The manuscript notes that the question appears in Erdős's 1957 problem list, that an exceptional set of points is necessary (the center of a set of concyclic points sees one distance), and that the theorem supplies no rate of decay. Its introduction outlines the proof as a contradiction along a sequence of configurations with nearly maximal density of pairs in large fibers: a directed graph of such pairs is extracted, each finite configuration is moved into a number field by transfer for real closed fields, squared distances are factored through the coordinate maps $u\pm iv$ so that on a fiber one coordinate is a fractional linear function of the other, the product formula for the number field is turned into an additive overlap identity for randomly shifted nested grid partitions at every absolute value, and a variance estimate for off-diagonal sampling leads to a contradiction in both the bounded and the unbounded overlap-scale regimes. The prose proof (Sections 2 through 7) is unreviewed; acceptance rests on the audited Lean statements.

Covers. Q1 only, answered yes. For every $\varepsilon>0$ and all large $n$, every set of $n$ distinct points in the Euclidean plane has a point $x$ with at least $n^{1-\varepsilon}$ distinct nonzero distances to the other points. This is $\#\{d(x,y)\}\gg n^{1-o(1)}$, uniform over sets. Q2 (whether $\gg n/\sqrt{\log n}$ holds) is not settled.

Depends on. No page of this wiki. The theorem is the manuscript's own, and the Lean development, the modules under lean/OAI/Geometry/PinnedDistances/, imports only Mathlib and its own files.

Formalization. The release's Lean tree at the pinned revision defines, in lean/OAI/Geometry/PinnedDistances/Model.lean, Plane as EuclideanSpace ℝ (Fin 2), k P x y as the size of the distance fiber at x through y within P.erase x, F n s as the supremum over n-point finite sets of the proportion of ordered distinct pairs with fiber size at least n ^ s, distances P x as the image of fun y => dist y x over P.erase x, and B n ε as the supremum over n-point sets of the proportion of points x with fewer than n ^ (1 - ε) distances. The declaration OAI.WeakPinned.main in Main.lean proves that F n s tends to 0 for every s > 0, which is Theorem 1.1; OAI.WeakPinned.pins in Pins.lean proves that B n ε tends to 0 for every ε > 0, which is Corollary 1.2; and OAI.WeakPinned.exists_pin_eventually in the same file proves

theorem exists_pin_eventually (ε : ℝ) (hε : 0 < ε) :
    ∀ᶠ n : ℕ in atTop, ∀ P : Finset Plane, P.card = n →
      ∃ x ∈ P, (P.card : ℝ) ^ (1-ε) ≤ ((distances P x).card : ℝ)

The last statement is the problem's first question in its $\varepsilon$ form: the points of a Finset are distinct, the distance is Euclidean, the distance from x to itself is left out, so the count is of nonzero distances and the bound is slightly stronger than one that includes it, the real power has base at least $1$, and the statement has no supremum, division or natural-number subtraction and no hypothesis. Because the threshold in $n$ comes before the quantifier over sets, the family of these statements over $\varepsilon>0$ is the uniform bound $n^{1-o(1)}$. The comparator challenge lean/ComparatorChallenges/PinnedDistances.lean pins OAI.WeakPinned.main with the definitions of Model.lean, and Pins.lean lies outside the challenge; the release's lean/docs/167.md describes the formalized scope as this theorem and its consequence for pins, and its lean/formalization.yaml lists OAI.WeakPinned.main for the family. The second question, the bound $n/\sqrt{\log n}$, is not stated in the development.

Acceptance. Formalized: this corpus's verification built OAI.WeakPinned.main, OAI.WeakPinned.pins and OAI.WeakPinned.exists_pin_eventually at the pinned revision, found their axiom closures to be exactly propext, Classical.choice and Quot.sound, found the comparator fingerprint for main identical to the pinned challenge, and audited the whole statement of exists_pin_eventually against the problem's first question as set out above. The acceptance is of the Lean statements so audited; the manuscript's prose proof is unreviewed. Not reviewed and not refereed: no outside review of the result is recorded, the manuscript is a release preprint with no journal record and no arXiv version, and the site's page for Problem 604, as accessed on 2026-09-04 and last edited on the site on 23 March 2026, shows OPEN with no mention of the release. The release's own README says that its manuscripts were produced by an internal OpenAI model and are at different stages of verification.