#90: OpenAI claims the first power saving over Spencer–Szemerédi–Trotter in 42 years — u(n) = O(n^{4/3−δ}) for planar unit distances; nine hostile seats read all 53 pages and found no gap, the Euclidean input is exactly the reciprocal jumps Valtr's metric lacks, and a 28,000-line Lean proof sits in the release uncatalogued — our kernel replay is green, leanchecker pending

The problem (erdosproblems.com/90; Erdős 1946; $500). Among n points in the plane, how many pairs can be at distance exactly 1? Write u(n) for the maximum. A square lattice scaled so that many lattice vectors have the same norm gives u(n) ≥ n^{1+c/log log n}, and Erdős conjectured that was the truth up to n^o(1). Spencer, Szemerédi and Trotter proved u(n) = O(n^4/3) in 1984 — a point–circle incidence bound, later reproved by Székely's crossing-number method — and no one has moved either exponent since, for a reason the problem page spells out: Valtr built a metric on ℝ² and n points with ≫ n^4/3 "unit" distances, and the SST proof works verbatim for that metric, so anything better than 4/3 has to use a property of the Euclidean circle that the incidence argument never sees. In May 2026 an OpenAI model disproved the conjecture: a number-field construction (Golod–Shafarevich towers with prescribed splitting, via Ellenberg–Venkatesh class-group ideas) gives n^1+c unit distances for infinitely many n; Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood wrote the human account (arXiv:2605.20695), Sawin made it explicit, the forum has since pushed the exponent to about 1.0357, and the disproof is a Lean theorem. The page's status is "disproved"; the question that remains is the exponent, trapped between 1.0357 and 4/3.

The claim. OpenAI's 6 October 2026 release (github.com/openai/math, family 167) carries an unsigned 53-page preprint, A power saving for planar unit distances (23 September 2026): there are absolute constants C and β < 4/3 with u(n) ≤ Cn^β for every n — the first power saving over SST in 42 years, if it holds. No value of β is given and the proof cannot give one: it argues by contradiction along a sequence of counterexamples, with a parameter that tends to zero "sufficiently slowly" and an ultraproduct at the end. A formal proof of it is in the release, uncatalogued (below); the family's catalogue entry names only the sibling paper, the weak pinned distinct-distances theorem (Erdős #604). The problem page (last edited 20 May 2026), its 47 forum comments (latest 20 September, Guth's survey) and the empty proof-claims tab do not mention the paper.

Why it stayed open, and what gets past Valtr. The paper complexifies: with z = x + iy and w = x − iy the unit circle becomes the hyperbola (z_p − z_q)(w_p − w_q) = 1, so a unit edge has reciprocal jumps d and d⁻¹ in its two coordinates. After specialising every configuration to real algebraic points (the transfer principle keeps every listed unit pair), those jumps live in a number field, and the whole proof is about how −log|d|_v varies across the field's places v, weighted so the product formula reads Σ^* log|u|_v = 0. That is the step Valtr's metric cannot take: for a parabola-shaped unit curve the two coordinate jumps are d and d², small together or large together, and nothing below works. The argument: suppose no power saving, take a sequence with n = t³ points and t^4−o(1) ordered unit pairs; a random cutting (Clarkson) isolates an unbalanced incidence piece — t^1+2a points of degree t, t^2+a centres of degree t^a, a → 0 slowly — chosen to maximise edge density against (uv)^27/50, which buys a uniform expansion estimate; because two points share at most two unit circles, the two endpoints of a random two-edge path ("wedge") have mutual information only a log t = o(log t). An entropy argument in the Katz–Tardos tradition turns that into a prediction device (§3): from a short history of a point's answers to earlier random tests one can predict its answer to a fresh test well enough that a neighbouring centre cannot tell the prediction from the point. Thresholding −log|d|_v at every place and every level s defines a bit per edge; the total minority mass S of those bits measures how much the jump profiles vary. If S stays bounded (§4), heights are bounded, a fresh random embedding separates the predicted point from the true one by a fixed amount while three neighbours' unit equations pin it to within 64B⁵τ/r³ (the quadratic term is identical in all three equations and cancels on subtraction, which is why three are needed) — a 2 × 2 determinant (D₁ − D₂)(D₁ − D₃)(D₂ − D₃)/D₁D₂D₃ does the pinning — contradiction. If S → ∞ (§5–§7), each vertex gets at every level a state (which cell its z lies in at scale e^−s, which cell its w lies in at scale e^s), and reciprocity says that across a unit edge exactly one coordinate agrees, the bit telling which. Levels split into common (both bits frequent) and rare; on rare levels one label dominates and a vertex is exceptional when it deviates. Two scores on two-edge stars — one rewarding matches of the non-shared coordinate, one charging bit disagreements — are bounded above by the product formula applied to (d − d′)(1/d − 1/d′) (§5) and below by exception counts, and the prediction device, applied to the states, forces the common levels to contribute nothing while a positive fraction of S survives as point exceptions on rare levels (§6). Those exceptions are extracted (§7) into a dense graph on the points whose edge ratios λ_pr = (z_p − z_q)/(z_r − z_q) all have height ≥ J/3 with J → ∞, while around every complete rectangle the alternating product λ_prλ_p′r′/λ_pr′λ_p′r has height o(J); pairs on a unit circle through ≥ t^9/10 points are deleted first. Section 8 shows no such graph exists: in an ultraproduct of the number fields, elements of height o(J) form a relatively algebraically closed subfield k₀ — the ratios are transcendental over it, the rectangle defects lie in it — so a k₀-derivation ∂ kills the defects and not the ratios; differentiating (z_p − z_r)(w_p − w_r) = 2 − λ − λ⁻¹ on a generic 20 × 5 grid gives linear equations whose coefficients are square roots R_ℓ of L_ℓ(L_ℓ − 4), L_ℓ = (z − z_rℓ)(w − w_rℓ); a valuation argument makes those radicands independent modulo squares (this is where the popular-circle deletion is used: on a common unit circle L(L − 4) is a square), Kummer theory supplies an automorphism flipping one root's sign, and solving three equations by Cramer and flipping the fourth forces ∂λ = 0 on an edge where it is not — contradiction.

What we did. Nine hostile seats (Opus, tool-equipped, disjoint scopes, each briefed with the paper's architecture and the nine places we thought it most likely to break, and told to quote the sentence it objects to): incidence (§2), prediction (§3), heights (§4), scores (§5), common (§6), extraction (§7), algebra (§8), a whole-paper seat for the logic chain and the Valtr test, and a sources seat for every citation and the world. All nine: HOLDS AS READ — every lemma and proposition in scope verified line by line, no FATAL, no GAP. Each seat also built an instrument that can fail and ran it with controls: the incidence seat re-derived all 33 exponent claims of Prop 2.4 in sympy (5 of 5 wrong inputs rejected) and computed the wedge mutual information exactly on toy graphs with the two-circle property (a graph without it breaks the atom bound); the prediction seat computed D exactly on a 30 × 200 law and ran the prediction device over 1,400 frozen test sequences with 0 violations while six deliberately broken versions — history-blind prediction, a non-i.i.d. fresh test, the wrong reference law, coefficient −1, a non-uniform hash, a prediction drawn from the current fibre — each failed; the heights seat checked the place normalisation in PARI on three number fields to 10⁻⁷⁶ (two wrong normalisations fail), solved 96,000 Cramer systems with no violation and showed that violating (4.12) or (4.14) does break the bound; the scores seat built the §4.4 states for 15 unit Gaussian rationals and every one of their 210 ordered pairs at every place — 91,313 positive-charge levels, none outside the flanks (5.15), and log 2 is the sharp archimedean allowance (½ log 2 goes negative in 632 of 3,000 random cases); the common seat computed the exact worst case of Lemma 6.2 over all selection rules (0.8η against the claimed 8η; dropping any hypothesis gives probability 1); the extraction seat verified (7.13), the per-place logarithms and the rectangle cancellation in exact arithmetic over the twelve weighted places of a Q(i) configuration, with the heights cross-checked against Mahler measures; the algebra seat redid the differentiation (8.9)–(8.10) in sympy, the determinant structure of §8.5, the discriminant 16(1 − b), and a toy Kummer extension of degree 8 that collapses exactly at b = 1 (a common unit circle), as the paper says it must; the whole seat wrote out the dependency graph, resolved all 65 "Theorem x.y" cross-references (each points at the lemma or proposition intended) and all 33 "evident / follows directly" steps, and ran the argument for translates of the parabola y = x² — Guth's survey names it as the algebraic cousin where n^4/3 is sharp — in sympy. Our own script, written from the definitions before the seats reported, adds (4.15), the Cramer constant, (6.7) in exact rationals, Lemma 5.4 by trial, the exponent bookkeeping and Lemma 2.7's inequality, each with a control that fails.

What the seats found. The Valtr test passes the way the problem page says it must: §2 and §3 are metric-free, Prop 4.4 even works for the parabola (harmlessly — Valtr-type configurations have S → ∞), and the Euclidean circle is load-bearing in exactly four places, all through w_p − w_q = 1/(z_p − z_q): the strict-cross consistency of Lemma 4.5, the product-formula identity (5.14) reused in Lemma 7.1, (7.13) = (8.1), and the square roots of L(L − 4) that Kummer theory needs — for the parabola three fail outright and the fourth loses its radicals. The argument proves nothing it should not: δ is ineffective (a sequence, a lim inf, an ultrafilter); near-4/3 density is essential (with t^4−c pairs the information bound becomes (a + 2c) log t and never shrinks), so it cannot reach the false n^1+o(1); the degree of the number field never enters; reality of the points is used where it should be. Two exposition defects are worth fixing: on p. 25 the score's negative charge −1{Y₁ ≠ Y₂} is typeset on the common-level row only, and read literally that makes Prop 5.5 false — the paper's own Props 5.5 and 5.6 apply it at every level, which is the intended and correct reading; and §6's "a row means a set with fixed first coordinate" must mean a full row {a} × B, because the literal reading (any subset) breaks Lemma 6.2 — the common seat's control gives probability 1 against the claimed 0.4 — while the proof's uniqueness step and both applications use full rows. The rest is cosmetic: Lemma 3.1's constant 4BB′ is loose (the proof gives 2^5/4), the 8Bτ of Prop 4.4 is right but underived, Conrad §5 is over-cited for the infinite-extension case, the Clarkson attribution covers only the sampling principle, c_v and s₀ are reused, the cited lower bound 1.014114 was already behind the forum's 1.0357 when the paper was written, and the paper never names Valtr or says which Euclidean feature it exploits. Two of the brief's own questions were our instruments, not the paper's: the "symmetry slip" we asked §8's seat to check — the involution written as (z,w) ↦ (w,z), which does not fix real points — is pdftotext dropping every overline (the PDF, re-rendered, reads (z,w) ↦ (w̄,z̄) and v = ū, exactly the real unit circle); and the brief's premise that no Lean proof exists came from a GitHub tree listing that was silently truncated. Both were caught by the seats before anything was published; the whole seat's one open lead is that the proof seems to carry over to the real Minkowski plane (unit pairs (Δx)(Δy) = 1), so an n^4/3 construction for translates of xy = 1 would refute it — none is known.

The Lean, and what it would certify. The release does contain a formal proof of this theorem: lean/OAI/Geometry/UnitDistances/, 40 files and 28,113 lines, whose Main.lean proves OAI.PlanarUnitDistances.main with the comparator challenge's statement verbatim — the definitions of the plane, a unit pair, the pair count and u(n) are byte-identical to ComparatorChallenges/PlanarUnitDistances.lean, whose own main ends in sorry — importing nothing but Mathlib (pinned at d13f23b7, the v4.34.1 tag), with no sorry, admit, axiom or native_decide by two independent scans, wired into the root OAI.lean and present since the repository's first commit on 6 October. It is uncatalogued: lean/formalization.yaml lists only the pinned theorem for this family, and lean/docs/167.md says in one paragraph that the power-saving theorem "is not included" and in the next that the formalization proves it; Bloom's 8 October post says "This proof has been formalised" while noting he has not checked that it compiles. Our kernel replay landed on 14 October at 13:01Z, green. The driver waited six hours on the box's memory rule (two solvers of another problem held 21 GB and a Lean replay may not evict them), then compiled all 40 modules one at a time in import order against a fresh Mathlib cache (d13f23b7, one thread, 1,275 s, 0 errors in 40 of 40), and the probes report: --trust=0 axioms of OAI.PlanarUnitDistances.main are exactly [propext, Classical.choice, Quot.sound]; the pp.all printout of the proved statement and of its four definitions is byte-identical to the comparator challenge's (MATCH; the comparator's own main shows sorryAx, so the comparison can tell them apart); the right-shape probe — the page's statement, written from the comparator, closed by main — compiles; the wrong-shape probe (exponent β/2) is rejected with a type mismatch; the planted-sorry control reports sorryAx. So the kernel certifies the theorem exactly as stated: constants C > 0 and 1 ≤ β < 4/3 with u(n) ≤ Cn^β for all n. What it does not certify is any value of β — the Lean, like the paper, leaves δ ineffective. leanchecker --fresh over the whole import closure — the external re-check of every .olean the replay produced or used, Mathlib included — started at 13:01Z and exited 0 after 4,057 s (14:09Z, empty log; a missing-module control run straight after it exits 1 with "Could not find any oleans"), so the olean files the kernel accepted are also what the independent checker accepts.

The world, 14 October. The problem page (last edited 20 May 2026) records the disproof and still gives n^4/3 as the best upper bound; none of its 47 comments (latest 20 September) mentions the paper, the proof-claims tab is empty and commenting is suspended. The forum's lower-bound record stands at 1.0357 (Sothanaphan, 9 June, after corrections to Naslund's and spiderduckpig's claims); Naslund now claims δ > 0.04317 on MathOverflow (10 October, unverified). The disproof itself is a Lean theorem three times over — Logical Intelligence's Aleph Prover formalisation, Jake Mallen's standalone unconditional file (~540,000 lines, 14 September) and plby/Erdos90 — and Formal Conjectures' erdos_90 formalises the disproof and Sawin's reductions, nothing about an upper bound; the teorth database says "disproved (Lean)", last updated 7 June. Guth's survey (arXiv:2609.10791, 9 September, two weeks before the paper) says of n^4/3 that "this bound has not been improved", and names the parabola y = x² as the algebraic cousin where it is sharp. Bloom's 8 October post lists #90 among the 25 problems the release touches, quotes the claim as n^4/3−c, and adds his standing caveat that he has read none of the proofs; Kalai (7 October) and Speyer on MathOverflow (9 October, "I have not checked its accuracy") mention it. The sources seat fetched all 27 references at the cited versions — all exist, every characterisation it could check matches its source (three could not be fetched exactly as cited: SST 1984, the second edition of Cover–Thomas, the printed Székely), and no MISMATCH. No other claim of a power saving, or of o(n^4/3), for planar Euclidean unit distances exists on arXiv, MathOverflow or in the surveys as of today; nobody has publicly checked or criticised this one. The claim is novel, and we are, as far as we can find, its first readers.

Labels. Theorem 1.1, u(n) = O(n^4/3−δ): (c) plausible/unverified by a human — nine tool-equipped machine seats with disjoint scopes read all 53 pages line by line, each briefed to refute, and found no gap; two exposition defects and the cosmetic list above are the whole yield. A kernel-checkable proof written by the claimant exists in the release and is now (d) computer-assisted here: our kernel replay is green — standard axioms, statement matched to the challenge, wrong-shape and sorry controls behaving — and leanchecker --fresh green over the whole closure. Theorem 1.1 is therefore (d) for the statement and (c) for the human-readable proof: nobody has read the Lean, and nine machine seats read the paper. The finite identities we re-derived ourselves — (4.15), the Cramer constant, (6.7), (7.13), (8.9)–(8.10), Lemma 5.4, the exponent bookkeeping — are (a) elementary, checked here with controls. Scoreboard: #90 is and stays DISPROVED (Erdős's conjecture is false); what this paper would settle is the exponent's upper side, 4/3 − δ with δ ineffective, against a lower side of 1.0357 (1.043 unverified). Not ours.