#138: Erdős's $500 question — is W(k)^(1/k) → ∞? — answered yes by OpenAI's superexponential bound W_r(k) > k^(ck⌊log₂ r⌋) for every r ≥ 2, where every earlier two-colour lower bound, from Erdős–Rado (1952) to Campos–Fox–Schildkraut (August 2026), was exponential; its 115-module Lean kernel-replayed here with the page's sentence compiled as a probe, and seven hostile seats read all 25 pages: no gap in the proof of Theorem 1.1

The problem (erdosproblems.com/138, $500; Erdős 1957, 1961, 1973–1981, 1997 and Erdős–Graham 1979 and 1980). Let W(k) be the least N such that every two-colouring of {1, …, N} contains a monochromatic k-term arithmetic progression. "Improve the bounds for W(k) — for example, prove that W(k)^1/k → ∞." Erdős offered the $500 for a proof or disproof of that limit; in the same survey he asked the weaker W(k)/2^k → ∞. The page's record on 15 October: Berlekamp's W(p+1) ≥ p·2^p for primes p; Kozik–Shabanov's W(k) ≫ 2^k as "the best general lower bound"; Gowers's tower upper bound; the DeepMind prover's W(k+1) ≥ W(k) + k; and, for r ≥ 3 colours, Fox–Hunter's W₃(k)^1/k ≥ C^{log* k} (June 2026), which the page calls the resolution of "the analogous question for r ≥ 3". The two-colour case — the case Erdős priced — is unresolved in the page's text (the site has displayed no statuses since 6 October). Since its last edit (2 June) one comment, 24 August, links Campos–Fox–Schildkraut's W(k) ≥ (1 − o(1))k·2^k−1 (arXiv 2608.20824, five pages, colouring found by ChatGPT 5.6 Sol Pro), which settles W(k)/2^k → ∞ and not the k-th root. The proof-claims tab is empty.

The claim. OpenAI's 23 September 2026 preprint "Quantitative Superexponential Bounds for van der Waerden Numbers" (25 pages, family 160 of the 6 October release; on Bloom's 8 October list of 25 and among the twelve he marks as FrontierMath Erdős-list problems). Theorem 1.1: there is an absolute integer K₀ such that for every integer k ≥ K₀ and every integer r ≥ 2, W_r(k) > k^{ck⌊log₂ r⌋} with c = 10⁻⁵; hence W_r(k)^1/k → ∞ for each fixed r ≥ 2 — in the abstract's words, "including the two-color case". The threshold is one number for all r. Two further theorems ride along: W_r(k) > exp((log r)²/(64 log 2)) for every r ≥ 256 and k ≥ 3 (Theorem 7.2, a Behrend digits-and-sphere colouring, superpolynomial growth in the number of colours), and an elementary recursive finiteness bound (Appendix A) with the boundary values W₁(k) = k, W_r(1) = 1, W_r(2) = r + 1.

The architecture is a two-colouring of one cyclic group, lifted to every number of colours by a digit product; everything new lives in §2–§6. Fix c = 10⁻⁵, D = ⌈k^1/10⌉, M = ⌈k^1/2⌉, h₀ = D², p = k^−1/20, H = k⁻², η = 1/(1000D). Take a prime P ∈ (k, 2k²], the least power q of P with q ≥ k^ck/D, and G = ℤ/q^Dℤ, so |G| ≥ k^ck. Each n ∈ G gets two systems of circle coordinates, x_i(n) = n/qⁱ mod 1 and y_i(n) = λx_i(n), where the integer dilation λ = ∏_{ℓ ≤ h0} ℓ^{⌊log(2M)/log ℓ⌋} contains the full ℓ-primary part of every period h ≤ 2M at every prime ℓ ≤ D² — so after dilation a rational period is either exactly 1 or larger than D² (Lemma 2.1), the separation the whole argument turns on. The x-circles are cut into uniform intervals of width H/λ, the y-circles into a geometric mesh that gets finer toward the cut ±½ used to pick centred representatives ỹ ∈ [−½, ½)^D (Lemma 2.3: width ≍ H·(distance to the cut + α), α = q^−(D+1)). A point's label is its 2D-tuple of intervals. §3 counts: the number of distinct label words realized by k-term progressions is exp(O(k^3/10 log k)) (one hyperplane arrangement in the two scalar parameters start and step, per coordinate — Lemma 3.1 counts equality signs too, for the half-open intervals), and the number of "eligible local patterns" of period h through one specified label is at most (CDh)^6D with C absolute and independent of q, λ, and the width of the anchor interval (Lemma 3.4 — the stationary drift is bounded relative to the interval width, which is what makes the count survive arbitrarily fine mesh). §4 proves a finite asymmetric local lemma and uses it to pick one outer colouring c_* of all labels such that every progression with ≥ k/10 light positions (label multiplicity ≤ k/M) sees each bit on a quarter of them, and every eligible pattern is balanced (weights e^−l/32, incident weight per label ≤ 1/500, criterion (1/8 − 1/32 − 2/500)·l ≥ log 2 for l ≥ 100). §5 is the dichotomy: every progression either carries ≥ k/100 of each outer bit, or its centred y-representatives are globally affine in j. A heavy label's closest return fixes a rational period h ≤ 2M and a drift (Lemmas 5.2–5.3; the gcd argument needs the representative box to be a product of half-open intervals); if h ≤ D² the dilation makes the y-motion a pure real step w ∈ q⁻ⁱℤ with total motion < ½, and a cut crossing would shrink every mesh interval in that coordinate below |w_i| and make all labels distinct — contradicting the heavy label, so the lift is affine; if D² < h ≤ 2M, each full period block containing a heavy index is an eligible pattern (regular fraction ≥ 1 − D(4η + 1/h₀) > ½), and the balanced patterns cover four fifths of the progression. §6 perturbs: a key is a y-box together with the integer band ⌊‖qỹ‖₂²⌋; any two distinct terms of a progression are ≥ 1 apart after scaling by q (the coordinate where λ(n_j − n_j′) first fails to vanish mod q), so a norm band meets the affine line through a box's points in two segments of length ≤ 1 and a key occurs at most four times along any progression (Lemma 6.1, Behrend's annulus in a bounded-width form). Flip an independent Bernoulli(p) bit per key. A progression rich in both outer colours needs ≥ k/400 distinct keys flipped: probability ≤ p^k/400, and the union over |G|² progressions has exponent (2c − 1/8000)k log k = −(21/200000)k log k → −∞. An affine progression's signature (outer bits and literal keys) is determined by its label word and a quadratic in j compared with ≤ k(Dq² + 2) integers — exp(O(k^9/10 log k)) signatures, each needing ≥ k/4 distinct keys to all land one way, probability ≤ (1 − p)^k/4 ≤ exp(−k^19/20/4); identical signatures are the same event. Both unions are o(1), so some choice of bits colours G with no monochromatic nonzero-step progression (Theorem 6.3). §7: base-q^D digits with that cyclic colouring in every digit give W_2^m(k) > q^Dm — the lowest digit at which the step is nonzero is a cyclic progression with nonzero step — hence W_r(k) > q^{D⌊log₂ r⌋} ≥ k^{ck⌊log₂ r⌋}. The threshold on k is fixed in §2–§6 before r appears, which is the uniformity. The exponent 10⁻⁵ is where the rich-case union is made to clear (the affine union clears for any fixed c): 2c must sit below γ/80 with γ = 1/100, and nothing in the paper claims it is close to the truth.

The Lean, and what the kernel certifies. This one is catalogued: lean/formalization.yaml lists the paper, ComparatorChallenges/QuantitativeVanDerWaerden.json names the solution module OAI.Combinatorics.ProgressionColoring.Main, the theorem OAI.QuantitativeVanDerWaerden.uniform_lower_bound, permitted axioms propext/Quot.sound/Classical.choice, and lean/docs/160.md says the formalized result is the absolute threshold with k^{k⌊log₂ r⌋/100000}, "the associated growth limits, finiteness, and boundary values", and that "the sharper intermediate estimates used in the paper are not part of the described formalization". The formalization is 115 files and 14,837 lines under OAI/Combinatorics/ProgressionColoring/, importing nothing but Mathlib (and core's omega), unchanged since the repository's initial commit — the commit Google DeepMind's Formal Conjectures already cites. The definitions are four short ones (Model.lean): a colouring is any function ℕ → α; MonoAP c k a d is ∀ j < k, c(a + jd) = c(a); HasMonoAP c k N asks for a, d with 0 < d and a + (k−1)d < N; IsRamsey α k N is "every colouring has one"; and W r k is the least positive N with IsRamsey (Fin r) k N, as an sInf — so 0-based {0, …, N−1} where the paper has {1, …, N}, the same number, and an sInf that would be 0 (making the inequality false, not vacuous) if no finite N worked. Main.lean proves uniform_lower_bound — the comparator's statement, ∃ K, ∀ k ≥ K, ∀ r ≥ 2, (k : ℝ)^{(1/100000)·k·Nat.log 2 r} < W r k — from exists_cyclic_coloring (Theorem 6.3's shape: ∃ K, ∀ k ≥ K, ∃ N ≥ 2 with k^k/100000 ≤ N and a colouring ZMod N → Bool avoiding k-progressions) through UniformTransfer.lean's digit product, and then kthRoot_tendsto: for every r ≥ 2, (W r k)^1/k → ∞ — at r = 2, the sentence on the problem page.

The replay. We rebuilt the formalization on our own machine, lake-free, one module at a time in import order, against the Mathlib the repository pins (d13f23b7 = v4.34.1, toolchain v4.34.1, openai/math at 1581d4c), one thread at the lowest priority behind live solvers, every module build waiting for 28 GB of free memory. The estimate was written before the build started. All 115 modules compiled with 0 errors in 430 s. #print axioms, in a probe run with --trust=0 over the compiled modules, reports that uniform_lower_bound, uniform_lower_bound_floor, exists_uniform_lower_bound, kthRoot_tendsto, length_rate_tendsto and exists_cyclic_coloring each depend on exactly propext, Classical.choice and Quot.sound. The comparator file compiled on its own against Mathlib (its one theorem shows sorryAx, as it must), and the fully elaborated pp.all statement of uniform_lower_bound and the four definitions it is built from (W, IsRamsey, HasMonoAP, MonoAP) are byte-identical between the solution and the challenge — 1,238, 304, 187, 664 and 355 characters with whitespace collapsed, the match keyed on declaration heads after our first script merged two blocks and keyed the universe-polymorphic definitions wrongly (both failures were toward the alarming direction; v1 and v2 are in the bundle). Three probes: a from-scratch file that proves the page's sentence W(k)^1/k → ∞ for two colours from kthRoot_tendsto, the explicit two-colour bound k^k/100000 < W(2, k) from the theorem (with Nat.log 2 2 = 1 decided), the non-vacuity 0 < W(2, k) for all large k, and Theorem 1.1 in the comparator's exact type — compiles; a wrong-shape file claiming the same with coefficient 1 in place of 1/100000 is rejected by the type checker, the control that the right-shape file can fail; a planted sorry reports sorryAx. The first right-shape run failed on our own simpa, which cannot evaluate Nat.log; decide can, and the v1 log is kept beside the v2. leanchecker --fresh over the whole import closure, Mathlib included, ran over the whole import closure, Mathlib included, and exited 0 after 3,494 s (07:25:05Z; an empty log is success); the missing-module control exited 1 with "Could not find any oleans", so the checker can fail. What the kernel certifies is the theorem as stated in Main.lean; what it does not certify is any sentence of the 25-page prose, which is the seats' job.

What the seats found. Seven tool-equipped referee seats, each briefed to refute and told to quote the sentence it objects to — five on §2, §3, §4, §5 and §6, one on the whole paper as a dependency graph with §1, §7, the appendix and the Lean statements, one on the 25 references and the world — — the six mathematical seats read their sections of the paper's own LaTeX line by line, the sources seat every citing sentence against its source — and every one returned HOLDS AS READ (the sources seat: "for every citation I could fetch"): no FATAL and no GAP in the proof of Theorem 1.1; 45 numbered objections — 36 cosmetic (the compiled PDF prints 34 cross-references, and one range, to lemmas, definitions and the proposition as "Theorem"; unstated large-k hypotheses; a factor 1/q in α never used; Szabó's "Theorem 2" locator unverifiable), six unclear (exposition or paper-vs-Lean correspondence), one exposition GAP outside Theorem 1.1's proof (the quasipolynomial aside below), a provenance note and one correction to our brief. Each seat rebuilt something and made it fail on purpose. The §2 seat built the adaptive mesh exactly as defined for every k from 3 to 32 and in windows up to k = 10¹²: the paper's bounds hold with measured values 4 (mesh endpoints near a cell, Lemma 2.4), 4000D+1 (breakpoints on a 4H arc, Lemma 2.5) and ≈1 (width ratio, Lemma 2.3) where the paper proves 49, 16004D and ¼…2, the dilation's full primary part is genuinely needed (lcm(1..h₀) breaks "s is 1 or > h₀" at h = 16), and q is only a prime power — nothing downstream needs it prime. The §3 seat checked the arrangement lemma on 690 random degenerate arrangements by exact Fourier–Motzkin, the e = 2 cases also by face enumeration and Zaslavsky's count, and showed the load-bearing claim of Lemma 3.4 directly: under the width-relative drift bound the reachable label count on a surrogate mesh stays at 3 as α shrinks from 2⁻²⁰ to 2⁻⁴⁵, where an absolute drift bound grows 335 → 557 → 889 — the dependence on q that the paper's clause (iii) removes; the absolute constant is C = 1322. The §4 seat ran the local-lemma invariant over every subset of 226 random instances, the exact binomial tails for l ≤ 600 against 2e^−l/8 (largest ratio 0.567), and the criterion at l = 100 as the exact certificate (1/8 − 1/32 − 2/500)·100 = 359/40 with e^359/40 > 2; it also named what §4 needs from §3 — log T_glob ≤ √k/320 − O(log k) and a per-label local count below exp(D²/65) — so that a factor λ^Ω(D) or any anchor-width dependence in Lemma 3.4 would have broken (4.6). The §5 seat listed all fourteen "sufficiently large k" inequalities of the dichotomy, checked them exactly to k = 20,000 and analytically beyond — all hold from k = 1056, the binding one 2M/k ≤ 1/16 — then ran a toy cyclic model with the paper's formulas (q chosen freely so that labels repeat) at k = 12, 100 and 1056 (52,388 long-period block checks at k = 1056; no failure in any base run) where each control breaks exactly the step the proof attributes to it (no dilation: 425 of 1,528 short-period (progression, heavy-label) cases lose the affine lift; a coarse α or a uniform mesh: the cut claim fails on nine tenths); the proof of Lemma 5.4 is essentially sharp at k = 1056 (observed ratio 2.072 against the proof's floor of 2; the lemma as stated claims only 1), and the point we flagged — whether the eligibility bound is applied at the right heavy index — dissolves because there is one (h, t, u, v) per progression. The §6 seat checked the separation ‖qỹ(n_j) − qỹ(n_j′)‖ ≥ 1 exactly over every ordered pair of seven cyclic models (13.8 million pairs in all, up to 5.76 million in one model; a non-unit λ fails on 14,406), the quadratic signature identity on 48,910 instances, 2c − γ/80 = −21/200000, and both union bounds below ½ at 745 scanned values of k from 2²⁰ to 10¹⁸, every jump of D included; it found the key-multiplicity bound is really 2, not 4 (the Lean proves affine_norm_band_card_le_two and then weakens), and that (6.8) is the inequality that caps c — below 1/16000 with these γ and p, so the method cannot approach k^k and does not prove too much. The whole-paper seat traced every hypothesis to the place that supplies it (the dichotomy's period blocks are eligible patterns for the same h, t; the cyclic theorem forbids wrapping steps, which the digit product needs — its control, colourings avoiding only non-wrapping progressions, produces a monochromatic one), tabulated about thirty large-k thresholds from the paper's own constants — none reversed, none depending on r, the progression or the prime; K₀ ≈ 10^44.7, the binding step the outer-colouring sum at D ≥ 29,210 — brute-forced the digit product on ℤ/3, ℤ/4, ℤ/11, Theorem 7.2's colouring for ten values of r with an exact sweep of the colour count over every r from 256 to 2·10⁶, and the boundary values, and read the Lean: uniform_lower_bound is Theorem 1.1 with the same quantifier order, W is the paper's number up to the shift by one and cannot be vacuous, kthRoot_tendsto at r = 2 is exactly the problem page's statement (up to the shift by one and monotonicity), exists_cyclic_coloring is Theorem 6.3 with N left existential and a budget line that is literally the paper's two probability bounds. Three findings are more than cosmetic. First, the Lean does not prove the paper's §3 and §6 counts as printed: the arrangement bound is (e+2)²(s+1)^2(e+1) by Sauer–Shelah for (e+1)²(s+1)^e, the per-coordinate word count 16(·)⁶ for 9(·)², the local count (16005Dh)^14D for (CDh)^6D, the affine-signature count with exponent 8 for 3 — weaker forms that still clear §4 and §6 — while the prime lemma is replaced by Mathlib's Bertrand (P ≤ 2k, stronger than the paper's P ≤ 2k², by a different proof); this is consistent with OpenAI's own scope note, "the sharper intermediate estimates used in the paper are not part of the described formalization"; the kernel certifies the theorem; the paper's sharper lemmas rest only on the seats' hand reading and finite checks, and they are two different arguments. Second, the introduction's aside that growth in the number of colours is "at most quasipolynomial" for every k rests on Equation (1.3) of the unrefereed 198-page companion paper, whose own Lean leaves that bound out; the lower half of that display is this paper's Theorem 7.2 and holds, the upper half is secure only at k = 3 via Kelley–Meka. Third, the sizes: k^ck overtakes 2^k only when log₂ k > 10⁵, i.e. k > 2¹⁰⁰⁰⁰⁰ ≈ 10³⁰¹⁰³, and the threshold from the paper's own inequalities sits between 10⁴¹ and 10⁴⁵ by three seats' independent estimates — so no van der Waerden number anyone will ever write down is improved by this theorem, which is a statement about the limit. The brief we gave the seats contained one error of ours, caught by the sources seat: Fox–Hunter's three-colour construction is not a Behrend-type covering (one colour class is a dense progression-free set, the other two are random on the rest); nothing on this page rested on it.

The world, 15 October. The problem page records none of this: Kozik–Shabanov is still "the best general lower bound", the proof-claims tab is empty, comments are suspended, and the one post-June comment (24 August) links Campos–Fox–Schildkraut, whose k·2^k−1 answers the W(k)/2^k question and whose abstract says nothing about the k-th root. The teorth database says open (last updated 31 August 2025). Formal Conjectures' ErdosProblems/138.lean already tags erdos_138 research solved, "proved by an internal OpenAI model in September 2026 [OAI26], which shows W(k) > k^k/100000 for all large k; see the linked Lean proof" — the file we replayed (at 1581d4c, unchanged since the commit adc7f124 that it links) — alongside the difference variant (DeepMind's prover, April) and the W(k)/2^k variant (a Lean derived from Meta's Atlas proofs, citing Campos–Fox–Schildkraut). Bloom's post lists #138 with the rest and says he has not yet looked at any of them in any serious way. The sources seat checked all 25 references against Crossref and the cited arXiv versions and found no mismatch; five could not be read at the primary source — Szabó 1990 (content matches the publisher abstract; the "Theorem 2" locator is unverified), Behrend 1946 and Rankin 1961 (confirmed through Green–Wolf, Kelley–Meka and the publisher synopsis), Schmidt 1962 (through Berlekamp) and Shelah 1988 (by title and Gowers's account) — Kozik–Shabanov's Theorem 2 is exactly W_r(k) ≥ βr^k−1, Erdős 1980 p. 90 (2) is the $500 sentence, Fox–Hunter's Theorems 1 and 2 read as cited. Its priority search (23 web queries, arXiv, Hacker News, MathOverflow, erdosproblems.com, Bloom's blog; X, Reddit, the live Zulip, Google Scholar and non-English venues not covered) found no human-authored claim that W(k)^1/k → ∞ for two colours — Fox–Hunter in June and Campos–Fox–Schildkraut in August both call it open — so, as far as that search reaches, the claim is first, and positive; Formal Conjectures marked it solved on 11 October (commit 3dfd999d) by linking OpenAI's Lean, with its own statement still a sorry and no lemma bridging its W to OpenAI's; two independent comparator runs (mathvet, 8 October; nealmcb, 11 October) report the catalogued comparator passing. The paper was first public on 6 October (the dated "23 September" is the preprint's own date) and is unchanged since. One honest magnitude the paper does not print: k^ck exceeds 2^k only once log₂ k > 10⁵, i.e. k > 2¹⁰⁰⁰⁰⁰ ≈ 10³⁰¹⁰³, and the §4 seat's reading of the paper's own inequalities puts the proof's threshold K₀ beyond 10⁴¹ — the theorem is a limit statement, and no finite W(k) is improved by it. What is OpenAI's: the theorem, the construction, and its Lean. What is ours: this audit, the replay on a machine that is not theirs, and the statement of exactly what the kernel did and did not certify.

Labels. Theorem 1.1 and the page's question W(k)^1/k → ∞ (every r ≥ 2): (d) computer-assisted — OpenAI's own Lean, kernel-replayed here with the statement and its definitions matched to the comparator byte for byte and the page's sentence compiled as a probe, standard axioms throughout, controls behaving (leanchecker over the whole closure exited 0 after 3,494 s, missing-module control exited 1). The 25-page human-readable proof: (c) plausible, unverified by a human — seven hostile machine seats found no gap in the proof of Theorem 1.1 and mark what they could not check; the caveat that matters is that its §3 and §6 constants as printed are not what the kernel certified (the Lean proves weaker forms), so those lemmas carry the seats' reading, not the kernel's. Theorem 7.2 and Appendix A: (a) elementary — read line by line and brute-forced by the whole-paper seat (ten values of r, an exact sweep of the colour count to 2·10⁶, the boundary values); the Lean's LargeColorConsequences.lean carries the corresponding limit statements, which we did not probe separately. The problem's open-ended half — "improve the bounds" — stays open; nothing here says how fast W(k)^1/k grows beyond k¹⁰⁻⁵, and the page's remaining questions (W(k+1)/W(k) → ∞?) are untouched. Nothing here was reviewed by a human; everything here is reproducible from the bundle. Not ours.