#1083: OpenAI's proof of the higher-dimensional distinct-distances conjecture, constant factor included — 103 pages read by twelve hostile seats, sources checked, and a third-party Lean 4 kernel check replayed on our own machine

The problem (erdosproblems.com/1083; community database status OPEN as of our 11 October refresh, OEIS A186704). Let d ≥ 3, and let f_d(n) be the minimal m such that every set of n points in ℝ^d determines at least m distinct distances. Estimate f_d(n) — in particular, is it true that f_d(n) = n^{2/d − o(1)}? Erdős asked it in the same 1946 Monthly note that started the planar problem; the grid {1,…,t}^d has t^d points and at most d(t−1)² squared distances, so n^2/d is the most one can hope for. Before this paper the record stood at n^2/3−o(1) in ℝ³ (Tidor–Yu–Zakharov), and, feeding that into Solymosi–Vu's recurrences, n^8/17−o(1) in ℝ⁴ and n^3/8−o(1) in ℝ⁵; in general dimension Solymosi–Vu's n^{2/d − 2/(d(d+2))}. Nobody had the exponent 2/d in any d ≥ 3, let alone a constant factor; the paper's own footnote records a 2020 manuscript that claimed exactly this and was withdrawn by its author.

What OpenAI claims. The higher-dimensional Erdős distinct-distances conjecture (23 September 2026, family 166 of the 6 October release, 103 pages, no Lean in the release). Theorem 1.1: for every integer d ≥ 3 there is a constant c_d > 0 such that every set of n ≥ 2 distinct points in ℝ^d determines at least c_d·n^2/d distinct distances — the conjecture with the constant factor, which is strictly more than the n^2/d−o(1) the problem page asks for. The paper says plainly why the difference matters (§1.2): a bound of the form n^2/d−o(1) leaves the ratio |Δ(P)|/n^2/d free to tend to zero slower than every power, so a constant-factor proof must defeat every such sequence "without first fixing a power saving". The whole architecture is built to honour that.

How the proof is built (plain version, from our own read of §1–2). It argues in the least dimension d ≥ 3 in which the theorem could fail, along a sequence of point sets with N = |P| → ∞ and M/A → 0, where M = 1 + |Δ(P)| and A = N^2/d — and M/A → 0 at no prescribed rate. The theorem in dimensions 3 ≤ u < d (by minimality) and a planar bound Ω(n/log n) that Appendix B proves from scratch give Lemma 2.1: a proper u-flat holds at most 1, 2M, CM log 2M or C_uM^u/2 points. Step 1 selects a positive fraction of the unordered pairs so that, at every centre p, the projective directions of its selected partners are in general position for homogeneous polynomials of degree ⌊A⌋ (Propositions 2.2–2.3): pairs on lines with more than H points are discarded, then at each centre every irreducible curve spanning at most a plane that carries more than KA times its degree in partners is greedily removed; fixed H, K exist because otherwise a diagonal choice would produce, at every centre, curve families of total degree o(N/A) covering almost every pair from one endpoint or the other — exactly what the sparse-cones theorem (Theorem 6.1, the central result, §3–6) forbids. Step 3 takes a generic rotated copy Q = RP, joins (p,q) to (p′,q′) when both pairs were selected and |p−p′| = |q−q′| (Cauchy–Schwarz gives ≥ cN⁴/M edges), and attaches to each vertex the affine flat F_p,q = {(Z,c): Z(p+q)/2 + c = (p−q)/2} of skew d×d matrices Z — Bardwell-Evans and Sheffer's point-pair flats, complexified. A neighbour (p′,q′) meets it in the flat {Z: Z(u+w) = u−w}, u = p′−p, w = q′−q, of codimension d−1. In d = 3 the edges lying inside a single rigid motion's match set are first deleted, which costs o(N⁴/M) by a new very-rich-motion bound (Theorem 7.1: the motions with at least C·A matches have Σk_g² ≤ K·N⁴/A); then vertices of small degree are pruned, leaving minimum degree δ(H) ≥ cN²/M, so δ(H)/A^d−1 = cA/M → ∞. Appendix A's uniform concentration theorem (Theorem A.1) gives the two counts the end-game needs: an irreducible variety of dimension s₀+h in skew-form space contains at most C deg(V)·A^h of the retained flats (Lemma 2.4), and a nonzero polynomial of degree ≤ A on one flat vanishes on at most CA^d−1 of its neighbour flats (Lemma 2.5) — the latter because neighbours in one maximal isotropic plane are the matches of a single isometry, at most N = A^d/2 ≤ A^d−2 of them when d ≥ 4 and O(A) when d = 3 by the deletion just made. Step 4 is short: by Lemma 3.6 the m retained flats impose at least c·mA^s₀ independent conditions in degree D = ⌊A⌋; sample each flat with a small fixed probability θ; the sample imposes at most 2θm·C(D+s₀, s₀) < c·mA^s₀ conditions, while every vertex keeps ≫ A^d−1 sampled neighbours; so some f of degree ≤ D vanishes on every sampled flat and not on some F, and f|_F vanishes on more than CA^d−1 neighbour flats — contradicting Lemma 2.5. The constant in Theorem 1.1 is born exactly there: one fixed θ, all concentration constants uniform. Step 2, the sparse-cones theorem, is the bulk (§3–6): Hilbert-function and multiplicity estimates (§3), secant geometry and a one-time deletion of "terminal curves" at cost o(N²) (§4), polynomial scale profiles that test degrees in windows [Be^−s, Be^s] of several widths instead of at one power of N (§5), and the rank comparison that pits the profiles against sparse-cone coverage, with two "no strict margin" endpoints — the outermost window and the final curve plateau — handled by direct geometric comparison (§6).

What we checked ourselves before the seats reported, from the rendered pages: the exponent bookkeeping of §2 (N = A^d/2; N² = A^d; the fibre count O(M^{2+(d−ℓ)/2}) = O(M^d/2) exactly when ℓ ≥ 4; A^d/2 ≤ A^d−2 exactly when d ≥ 4; δ(H)/A^d−1 = cA/M → ∞ with no rate); Proposition 2.6's rank contradiction (strictly smaller rank on the sample gives a polynomial vanishing there and not on the whole family; the neighbour flats are distinct by the local parameter check); and the engine under both rank bounds, Lemma 3.6 — whose lower bound H_V(L) ≥ c·(total degree)·L^k at the defining degree L looked too strong to us until we read its proof: Lemma 3.3 separates any selection of isolated points of a system of degree L by polynomials of degree c(L−1) independently of how many points there are (a divided-difference determinant evaluated in the local Artin algebra, with Lemma 3.2's Koszul argument showing the determinant's class is nonzero), a generic projection turns the components into isolated points over the function field of the base, and the standard-monomial scaling H_V(bL) ≤ bⁿH_V(L) brings the bound from degree C_nL down to degree L. We also read Theorem 7.1 as stated (a finite statement, both orientations, using M ≤ A only through the line cap 2M), Theorem A.1 with its specifications (a) and (b) (the global version carries a cardinality bound |𝒫| ≤ C₀A^r; the local one needs none, but applies only to varieties inside a hypersurface of degree ≤ C₀A), Theorem B.2 (Szemerédi–Trotter from the crossing lemma) and Theorem B.12 — which does claim the full cn/log(2n) in the plane, through a Cayley parametrization of direct isometries and a two-rich-line theorem (B.10) that Appendix B proves in eleven pages. After the first nine seats reported, we also read Proposition 4.3 — the one-time deletion of assigned terminal curves that both §6 seats said carries the whole k = 1 case of §6.2 — line by line ourselves, as a second instrument on the step the chain was waiting on. The line case: planes with S_H < A cost at most N·A·E = δN² pairs because a centre has at most E line components; (4.1), b·h_H ≤ CM², follows from Szemerédi–Trotter on the lift (x₁, x₁²+x₂²) with bM lines and ≥ bh_H/2 incidences, each of the three regimes of the bound giving the same inequality; the dyadic count (4.2) comes out of the weighted point–line estimate with the CLF term absorbed because L/s ≤ CM²/A² → 0; the Hölder step is a (3, 3/2) Hölder with a_b = (b/L)^1/6N_b^2/3L^1/3, c_b = (b/L)^1/6, whose geometric tails are bounded; the total is O(N²M²/A²) = o(N²) with no rate, and the N/s term costs N·B^2+o(1)(log N)², which is o(N²) for d ≥ 3 with a full power of B to spare — so a plane cap of B^8/3 would do as well, which is the Lean seat's point. The d ≥ 4 case: discarding groups of mass ≤ KMe_γ costs KNME = Kδ(M/A)N²; in a surviving group the off-diagonal weight is ≥ ½W² once K ≥ 2C; the perspectivity stabilizer argument (a projectivity fixing a line pointwise is affine u ↦ λu + b; translations cannot preserve an irreducible nonlinear curve; the surviving homotheties about a common centre have an orbit on a line, so Bézout caps the group by the degree) gives at most Ce_γ centres per ordered pair of curves, hence Σ_pe_γ⁻¹ ≤ C; weighted Cauchy–Schwarz then needs only N/E = A/δ → ∞. The d = 3 case: the valid-triple count (4.5), Σ_pe_γ⁻² ≤ 32, follows from Bézout on the two plane images seen from a generic point of Y_i with r_j ≥ d_j/(4e_γ) distinct branches, or from the polar bound (4.3) when the images coincide; the Hölder chain Σ_γW³/e_γ² ≥ (ΣW)³/(Σe_γ)² gives ε³N⁴/(2E²) against CN³, and N ≫ E² is B⁴ ≫ δ²B³ — true with a factor B to spare and no rate on δ. We found nothing wrong in it.

The paper as written — a twelve-seat hostile read. We refereed the 103-page preprint the way we refereed OpenAI's #172, #181 and #821: twelve tool-equipped machine seats with disjoint scopes, each briefed to refute, each required to quote the sentence it objects to, to fetch every cited result and set the source's words beside the paper's, and to list what it verified line by line against what it only skimmed. The brief, the twelve scopes and every report are in the bundle linked below. Seat by seat:

§1–2, the formal assembly (seat: reduction) — HOLDS AS READ. Every row of Lemma 2.1 holds, c₀ = 1/2 included; the negation of Theorem 6.1 is exactly what Proposition 2.2's diagonal argument produces (either endpoint, δ_n = 1/K_n set by the sequence, all but o(N²) pairs); the induction in Proposition 2.2 is sound (the fibres on spheres are bounded by the flat row u = d−ℓ, and the exponent 2 + (d−ℓ)/2 ≤ d/2 exactly when ℓ ≥ 4); the form of Lemma 3.6 that (2.3) needs is the form Lemma 3.6 states; (2.4)–(2.9), the d = 3 deletion, the pruning and the Markov–Chernoff–union-bound steps all hold; the identities (2.6)–(2.8) and both polarization identities were confirmed numerically for d = 3…6 with unequal-distance controls (the paper's sign on the global form is flipped, harmlessly, since it is only ever set to zero). An interface table checked every later result §2 consumes — Theorems 6.1, 7.1, A.1(a)/(b), B.2, B.12/B.13, Lemmas 3.1 and 3.6 — against the statement on its own page: all match, and a grep found no circularity (later sections use only Lemma 2.1 and the regime (2.1)). Footnote 1's withdrawn manuscript is real: arXiv:2002.01248 v1 claimed Ω(n^2/d) and its withdrawal comment of 12 February 2020 reads "proof of Theorem 1.2 is not correct". Bardwell-Evans–Sheffer's Theorem 6.1 (a − p = X(a+p) + 2c) is (2.6) relabelled. Ten cosmetic notes, among them one worth repeating: §1.3 says the estimates "produce the positive constant", but c_d arrives only by contradiction — the selection constants depend on the sequence and the proof gives no value for c_d.

§3, the algebra (seat: algebra) — HOLDS AS READ. The engine under both rank bounds, Lemma 3.6's H_V(L) ≥ c·a·L^k at the defining degree, holds with the explicit constant c = 1/((n−k+1)ⁿk!), the same for every family, and assumes nothing about the component degree a against L (components may have degree up to L^n−k; the only hypothesis is that each is a component of a locus cut by equations of degree ≤ L). The seat recomputed the toys exactly: a complete-intersection curve of degree D² in P³ has H = C(D+3,3) − 2 against the required D³/27; a ≤ D concurrent lines in P² have H = a(2D+3−a)/2 against aD/4; mixed-dimension envelopes and grid points pass. The scaling (3.8), H_V(bL) ≤ bⁿH_V(L) via x^α ↦ x^⌊α/b⌋ on standard monomials, is what brings the bound down to the defining degree, and it sharpens Walsh's Theorem 2.6, which reaches only m ≥ 2(n−d)δ. Lemmas 3.1–3.3, 3.5, 3.8–3.12 and Corollaries 3.13–3.15 were read line by line; every constant depends only on the ambient dimension, and §3 contains no asymptotic statement, so no hidden rate can live there. Citations: Chardin's Théorème matches (3.5); Chardin–Philippon's Corollary 3 matches, read from Chardin's own copy; the 2002 erratum withdraws one implication in their Definition 1 and repairs Proposition 2 — it does not touch Corollary 3, the only thing used; the five Stacks tags match; Fulton and Hartshorne could not be fetched. One objection of substance, UNCLEAR: the preamble's refined Bézout claim "with local intersection multiplicities" is stated without definition and justified by a perturb-and-specialize sketch; §3 never uses it, Appendix A does. A flag the seat passed to its neighbours: Lemmas 3.8 and 3.9 require D ≥ C_nL with a fixed constant, which at their §5–6 applications holds only for large window width.

§6.1–6.2, common windows and the outermost endpoint (seat: sparse_a) — HOLDS AS READ. Theorem 6.1 as stated on p. 42 and as used on p. 8 are the same statement. The upper bound (6.6) was recomputed from scratch (covered directions at a centre lie on curves of total degree ≤ E, so their rank at degree T is ≤ E(T+1) by (3.5); summed over centres H(T) ≤ N·E·(T+1), constant 1; the "spans at most a plane" hypothesis plays no role here). The ratio (6.7) has strictly positive margin in the window width except at the boundary case s = log B, b_d = 1, and δ enters only as 1/δ ≥ 1 — checked over 200,000 random admissible profiles with zero failures. Lemmas 6.2 and 6.3 check step by step. The either-endpoint alternative is handled by fixing one orientation per selected pair of pieces, keeping at least half the covered pairs — sound because the two pieces share width and profile. At the outermost endpoint every margin in Lemma 6.4 is a fixed positive number set by the profile; the k = 1 case is an incidence already removed by Proposition 4.3; for k ≥ 2 the seat re-derived the local tangent calculation of Lemma 4.1 behind (6.12). The inequality that must survive an arbitrarily slow δ → 0 is per-row count ≤ CEM + E·B^2−c+o(1) = o(n), which with n ≥ N√δ reduces to √δ·M/A → 0 and √δ·B^−c+o(1) → 0 — no rate. Two exposition points worth having on the page: Proposition 5.5 says the outermost pool "may" be used where §6.2 requires it (a refinement on p. 42 makes it the only possibility); and the sentence giving the pool mass ≥ N√δ while absorbing every global o(N²) loss reads, literally, as if those losses were o(δN²) — a rate — but the pool is a largest bin of fixed relative mass and a slow diagonal choice of the precision does the job without one; the seat verified that argument. The four Stacks tags cited in scope match. One oddity: reference [25] says the Stacks Project was "consulted October 1, 2026" in a paper dated 23 September.

§7, very rich rigid motions in ℝ³ (seat: motions) — HOLDS AS READ, and stronger than anything the paper cites. Theorem 7.1 says: for any P ⊂ ℝ³ with no line holding more than 2N^2/3 points, the rigid motions (both orientations) with at least k ≥ C·N^2/3 matches number at most K·N^10/3/k², with no logarithm. Tidor–Yu–Zakharov (arXiv:2608.14454v1, the current version) prove only |T| ≲ N|H|log^3/2N/k^3/2 (their Theorem 7.10) under a hypothesis on every curve, and they state the N^10/3/k² bound as their Conjecture 7.13, "out of reach of these techniques". The paper never says it has proved it; it cites [26] only for the flat families. And the d = 3 constant-factor claim needs exactly the log-free form, because a deletion cost of N^10/3+o(1) would not be o(N⁴/M) when M/A → 0 slowly. The seat tested the two steps that make it stronger: Lemma 7.4 (the local bound K·m³/ℓ^3/2 for motions with no rich matched line, a joints-style degree-reduction argument on two families of 3-flats in six-dimensional space, whose m^2/3 threshold is exactly where that argument works) and Lemma 7.5 with the count (7.19) (each cloud cut into O(r) cubes or cubes-with-one-hole of at most 8N/r points; a motion meets only K·r region pairs by a volume-packing argument checked case by case; counting matches per region pair rather than motions gains r^−1/2, which is the t^−1/28 margin). Every exponent in (7.16)–(7.25) was recomputed with exact fractions; the flat geometry (7.6)–(7.9) was checked numerically on 200 random orientation-reversing motions (maximal null flats, pair flat meets motion flat in a plane or a point, planes distinct, frame-independence — the last proved by hand because the paper's sentence for it argues a different property). No hidden rate; M ≤ A enters only through the line cap; the use in §2.3 matches the statement exactly. Cosmetic: §7.2.2 is the Guth–Katz / Elekes–Kaplan–Sharir joints argument unattributed; Beck and Elekes–Tóth could not be fetched, but both lemmas that cite them are proved in the paper and were checked.

Appendix B, the real incidence estimates (seat: incidence) — HOLDS AS READ, and it is a proof, not a reduction. First the consumption table: the planar distinct-distance bound (B.12/B.13, with the two-rich-line theorem and the flecnode lemmas behind it) feeds only Lemma 2.1's plane row, which matters only when d = 3 — for d ≥ 4 the three-dimensional theorem already caps planes at C·M^3/2 = O(B³). At d = 3 the paper needs only that a plane holds at most N·B^−c points for some fixed c > 0; the "more precise B^2+o(1)" is never needed, and any planar bound |Δ| ≥ c·n^β with β > 2/3 would do. What is used at full strength, with no slack, is Theorem B.2 (Szemerédi–Trotter, in §2.2, §4.2, §4.3, §5.4, §7) and Theorem B.5 / Corollary B.6 (the Guth–Katz k ≥ 3 bound, in §7.2.3, where m⁴/ℓ³ ≤ m³/ℓ^3/2 holds exactly at ℓ ≍ m^2/3); both are fully proved in B.1–B.3 and both were checked. Then the hard part, compared step by step with Guth–Katz arXiv:1011.4105v3: Theorem B.10 (two-rich lines) follows the architecture of their §3 with cleaner constants (K ≥ 147456, K > 98) and explicit handling of non-real factors; Lemma B.7 proves a Cayley–Salmon flecnode criterion of degree 17d − 24 where Guth–Katz cite the hard direction to a footnote in Salmon (Art. 588) — the seat re-derived both root cases and ran a sympy check (the flecnode polynomial vanishes on the ruled Whitney umbrella, is nonzero at generic points of the Fermat cubic, vanishes on its contained line); Lemmas B.8–B.9 prove the ruled-surface facts Guth–Katz assert or cite (single ruling, at most one exceptional point and two exceptional lines); Lemma B.4 proves the critical-line and flat-line counts they cite to Elekes–Kaplan–Sharir; Theorem B.12 gives the full cn/log(2n). All constants absolute; no rate, no cycle. Cosmetic: Elekes–Sharir is not cited in B.7 although the introduction promises attribution where each argument is used; symbols reused with local meanings. Guth–Katz Theorems 1.1 and 4.5, Proposition 3.2 / Corollary 3.3, Elekes–Kaplan–Sharir Propositions 3 and 8, Hatcher Corollary 2B.7 all match their sources; Stone–Tukey, Szemerédi–Trotter, Ajtai–Chvátal–Newborn–Szemerédi, Leighton and Székely's full text could not be fetched — each is attribution only, reproved in the appendix.

§6.3–6.5, separated windows and the final curve plateau (seat: sparse_b) — HOLDS AS READ. This is the other place the paper says it has "no strict margin". The seat located it exactly: in the depth comparison (6.26) the second entry, −Σ_i>j+1 b_i, is an empty sum at j = d−1, i.e. R·D_d = A with nothing to spare — and that is precisely the narrower-centre final one-dimensional plateau that §6.5 then treats by common regularity and the one-time terminal deletion. A brute-force census of the limiting profiles on a 1/8-grid for d = 3…7 (43,670 plateau–orientation cases satisfying (5.7)) found a strictly negative normalised log-margin in every case except the 2,751 narrower-centre final-plateau cases, where it is exactly zero; a negative control with the narrower choice removed produced 862 violations, so the census can fail. In pp. 51–61 the parameter δ enters in exactly three places and each needs only δ = O(1) — the separated-window case does not even use δ → 0; the one comparison that would need a rate is the one the paper warns against on p. 61 ("One must not instead compare (N/A)exp(−o(s)) directly with δN/A"), and the argument avoids it through the strict gap η − ε from Lemma 3.8 at degree T₁ = T₀e^ηs. Every displayed formula was read from the rendered page, which caught one extraction artifact (p. 41's D_d⁻¹exp(o(s)), mangled by pdftotext). Ten objections, all exposition: an isolation degree asserted where Proposition 5.1(2)–(3) supplies it; a Bézout count that tacitly restricts to non-dropping incidences (already deleted); "path locus = unique first carrier" near a regular point left unargued (it follows from Prop 5.1(3) and Prop 5.3's full-rank equations); Lemma 4.2's generic list bound transferred to the generic point of an auxiliary curve without the semicontinuity sentence that makes it legal (it is legal: k ≥ l because b_j+1 > 0); Lemma 3.8 used projectively through the chart convention; a contradiction chain compressed to one sentence (the seat wrote out the four steps); an order of choices — profile margins, tolerances, c, b₀, η then ε, the depth batch, the typical centres, the envelopes — that looks circular on the page and is not; and a proof of Theorem 6.1 that is a five-sentence summary of Proposition 5.5's case list. The verdict is conditional on what §6 consumes from §4–5 (Lemma 4.2, Proposition 4.3's o(N²) count, Propositions 5.1(3)/5.3's uniform no-capture, Proposition 5.5's exhaustion) and on Lemma 3.6 — the secants seat below is where those are read.

Appendix A, uniform concentration of evaluation flats (seat: concentration) — HOLDS AS READ. Theorem A.1 is where the paper meets the obstacle Tidor–Yu–Zakharov state in their §2.6 — concentration of pair flats with linear dependence on degree is "out of reach of our current techniques" for d ≥ 4 and false in codimension one for d = 3 because of rich rigid motions of the opposite orientation — and it meets it honestly: it proves only the containment (i = 0) case that TYZ say is within reach, and replaces their codimension-positive concentration by the local Lemma 2.5 inside a single flat plus the rich-motion deletion of §7. The seat rebuilt the exponent table for both uses: Lemma 2.4 is setting (a) with e = d+1, r = d, separators of degree 4(M−1) ≤ 4A from Lemma 3.1, isotropic cap N ≤ A^d−1 and total N² = A^d = A^r; Lemma 2.5 is setting (b) with e = d, r = d−1, separators of degree 2⌊A⌋ from Proposition 2.3, cap N ≤ A^d−2 when d ≥ 4 and C_richA when d = 3, hypersurface excess h = d−2, so the "useful consequence" C·A·A^d−2 is exactly Lemma 2.5. Appendix A contains no sequence at all: no constant in it depends on N, M, δ or a window; its sole quantitative input is Lemma 3.6, and the hand-off is textually exact ((A.6) is (3.5), (A.7) is (3.7) with the same hypothesis). Walsh's approximate complete-intersection theorem is proved, not imported (Theorem A.2 was compared line by line with Walsh's Definition 2.2, Lemma 5.3, the proof of Lemma 3.5 and Theorem 5.5 in arXiv:1811.07865v2); TYZ's Theorem 4.1(3), which TYZ prove only in codimension ≤ 2 and defer, is not used anywhere. The Segre-class identity (A.25) was checked symbolically for e = 3…8 and every h ≤ e−2, the nonnegativity of its coefficients for u ≤ 11, and the necessity of h ≤ r−1 (at h = r the expansion has negative coefficients). The exceptional parameter families — one-parameter rulings through a generic point, possible only for hypersurfaces — are classified as maximal isotropic subspaces (Proposition A.7), and the seat checked numerically in d = 3, 4, 5, both orientations, that the matches of a single isometry S span exactly one such subspace and, when I + S is invertible, all pass through the Cayley point (I−S)(I+S)⁻¹; for improper S in d = 3 they lie in a plane — TYZ's obstruction, reappearing as the reason hypothesis (ii) and Theorem 7.1 are necessary rather than decorative. Objections are cosmetic: heavy symbol overloading (U, B, C, c₁, q, P each mean two to four things), Proposition A.7's tangent constraint needs F_λ ⊄ Sing V for general λ (true, unstated), a one-line semicontinuity step at a load-bearing place in Lemma A.6, and the d = 3 sentence of Lemma 2.5 glossing over the non-uniqueness of the extending isometry (harmless: Theorem 7.1 deletes every rich motion of both orientations). Fulton and Hartshorne could not be fetched; the facts used (isolated Bézout with multiplicities, multiplicity = length for Cohen–Macaulay, the projection formula, Segre-class conventions) were checked independently. The seat's own closing sentence is the right summary of where risk sits: not in Appendix A, but in what it consumes (Lemma 3.6) and in what consumes it (the rank comparison of Proposition 2.6, which also rests on Lemma 3.6).

The bibliography and the world (seat: citations) — HOLDS AS READ for every source that could be fetched. Twenty-four of the twenty-eight references were read at source (Fulton and Hartshorne are books with no lawful free copy; Beck and Elekes–Tóth are attribution only); every theorem-number citation in §1–2 matches its source verbatim or by a one-line derivation; no bibliographic detail is wrong; no external theorem is a black box on the main chain beyond the Stacks tags, Fulton's refined Bézout and Borsuk–Ulam. Four findings, none mathematical. The one worth repeating is about provenance: the title page says 23 September 2026, reference [25] says the Stacks Project was "consulted September 23 and October 1, 2026", the PDF's embedded dates are a normalised 17 September, and the openai/math repository holds exactly one commit of this paper, 6 October — so the text was finalised between 1 and 6 October, first became public on 6 October, and the date on the title page is an internal date with no version stamp or changelog behind it. The others: footnote 1's withdrawn manuscript is accurately described but given no locator (it is arXiv:2002.01248, whose Theorem 1.2 is a Cohn–Elkies linear-programming inequality with Viazovska's eight-dimensional magic function — nothing in common with this paper); §3.4's paraphrase of Tidor–Yu–Zakharov's Theorem 4.1 omits that TYZ prove its Jacobian clause only in codimension ≤ 2 and defer the rest (the paper does not use it; its Lemmas 3.8 and A.5 are proved in-paper and are weaker); and Stacks Tag 0578 is cited as "openness of reduced fibers" when it is the generic-reducedness lemma — the content the step needs, the label loose. World state at 21:45Z on 11 October, re-checked by hand: the teorth database says OPEN (last update 17 October 2025) with the statement formalized since 7 September; erdosproblems.com/1083 shows no status and does not mention OpenAI or TYZ, and its forum carries only the August note that TYZ solved d = 3; Bloom's 8 October post lists 1083 among the 25 and says, correctly when written, "This proof has not been formalised"; Formal Conjectures has an open issue (#6905, 7 October, zero comments) proposing to mark it solved and no pull request; the Justin Sun Prize catalogue lists JSP-000900 as "Solved … independent mathematical review pending", not eligible to claim, with the formalizer's own claim (PR #4994) open, unreviewed, and stating "Independent verifier(s), if any: none"; Palsson–Wang's 7 October arXiv paper on the integer-lattice version still cites Solymosi–Vu as the state of the art for d ≥ 4; no arXiv paper, MathOverflow thread, Zulip stream or blog by any named expert in the field discusses the mathematics, and the two earlier arXiv manuscripts claiming the higher-dimensional bound (Aksoy Yazici 2020, withdrawn; Agama 2020–26, which bounds one constructed configuration and whose Theorem 4.2 is refuted by a regular simplex) are not proofs. Gil Kalai and Bill Gasarch repeat the claim as a claim.

The Lean 4 formalization, at source (seat: lean) — statement fidelity HOLDS; kernel acceptance now ours as well. github.com/cadamcat/distinct-distances-lean4 is the same author's third such repository in a week: 757 files, published as one squashed 211,476-line commit on 10 October at 18:52Z plus five comment-only commits, a public history of 7 h 50 min, the development history private, "AI coding agents wrote the Lean code under acceptance criteria set by the author", formalization.yaml "review: agent-reviewed, reviewers: []", zero stars, zero issues, built by nobody but him. Its target DistinctDistances.jsp_000900 is Theorem 1.1 with the constant factor for every d ≥ 3 and every set of at least two points, in Mathlib's L² metric with a real exponent 2/d, c and C chosen before the set, together with the grid upper bound; it is not vacuous, and it strictly implies the right-hand side of Formal Conjectures' erdos_1083 (the n^2/d−o(1) form) by a short step that the repository leaves to the reader. Across the project, the 503-module OpenAI closure it imports and the one patch-injected third-party file in the import cone, the seat found zero sorry, admit, axiom, native_decide, implemented_by, extern, unsafe, opaque, debug.* or environment edits, and no metaprogramming in the cone. Two findings about the paper rather than the Lean. First, "the formal proof follows the paper" is only partly true: Theorem B.12, Corollary B.13 and the whole ruled-surface / two-rich-line machinery of B.7–B.11 have no counterpart — the Lean replaces the planar Ω(n/log n) with a Pach–Sharir-type point–circle bound |Δ| ≥ c·n^3/4, plane occupancy ≤ C·M^4/3, and c₀ = 1/3 in place of the paper's 1/2. Our replay is green (below), so that is a fact about the theorem: Guth–Katz strength in the plane is off the critical path, including at d = 3, and any planar exponent above 2/3 would do — which is exactly what the incidence seat concluded independently from the paper's own consumption table (Theorems B.2, B.5 and Corollary B.6 stay on the path; they are formalized and consumed). Second, OpenAI's library supplies far more than the "polynomial isolation interface" the README names: 503 modules and 130,917 lines, including the definitions of degree and dimension, Hilbert–Serre and Lech, local Bézout, Bertini and Borsuk–Ulam. Provenance notes a reader should have: the author's Comparator run was, by his own documentation, not sandboxed ("comparator's development stand-in for landrun"), the committed Challenge.json has enable_nanoda: false while the docs say nanoda ran, and the agents' own comments record an intermediate statement of Appendix A that was false as frozen and re-routed ("see dehomogenize_rank_conclusion_false for a machine-checked counterexample") — about their blueprint, not the paper, which uses Fulton's projective-bundle formula where the Lean proves a specialised Hilbert-function identity instead. All trust in the formal object therefore rests on an independent kernel run. Ours ran on this machine on 12 October and is green. It waited behind the #181 replay, whose 22 GB of built packages it reuses (the two lake manifests pin the same OpenAI revision adc7f124 and the same Mathlib d13f23b7; the driver asserts this), opened its gates at 01:37Z with 39 GB free, checked out the pinned commit 88919b8, applied the repository's own OpenAI patches, fetched Mathlib's object files from the official cache, and built: lake build DistinctDistances.Main — 10,172 jobs in 13,960 s (3 h 53 min) at nice 19 on four threads, Build completed successfully, zero error lines. The repository's scripts/verify.sh then exited 0 in 153 s: Challenge.lean matches its frozen blob; the copied Formal Conjectures definitions are identical in statement and proof; challenge exit 0, main exit 0, pp.all outputs identical: True; and under --trust=0 'DistinctDistances.jsp_000900' depends on axioms: [propext, Classical.choice, Quot.sound] — PASS: all axioms are in the permitted set, scanned 756 files: PASS: no forbidden constructs. The three audit files (Audit/TargetProbes.lean, Audit/IndependentRestatement.lean, Audit/NonVacuity.lean) compiled under --trust=0 on the same build (14 s, 14 s, 30 s), their probes carrying the same three axioms. So the kernel accepted, on our machine, a proof of Theorem 1.1 with its constant factor, which implies the statement erdosproblems.com/1083 asks about. This entry is now CLOSED — not ours: the theorem is OpenAI's, the formal proof Yao Xu's — and #1083 joins the problems closed in the literature. The driver, steps.tsv, the axiom output, the verify and audit excerpts and the build log are in the bundle. One honest limit, the same one as on #172 and #181: in Lean 4.34 --trust=0 no longer re-checks imported modules, so what this run kernel-checked from source is the proof itself and OpenAI's 503-module library, while Mathlib's object files came from the cache; the leanchecker --fresh pass over the whole import closure, Mathlib included, started at 05:43Z on 12 October and exited 0 after 7,919 s (2 h 12 min) at 07:55Z, printing only Lake's "has local changes" warnings and no error line. The two controls for reading that silence as a pass ran on the same build (replay/leanchecker_controls.log): a module that does not exist exits 1 with "Could not find any oleans"; a real module exits 0 printing nothing. So Mathlib's cached object files are now kernel-checked too, and nothing in the import closure was rejected.

§4–5, secants, terminal curves and the scale profiles (seat: secants) — HOLDS AS READ. This was the seat the chain waited on: both §6 seats had conditioned their verdicts on what §4–5 deliver, and the seat that had been reading it when the previous session hit its limit never reported, so it was re-run from scratch. Lemma 4.1, Lemma 4.2, Proposition 4.3 in all three of its cases, Propositions 5.1–5.3, Lemma 5.4 and Proposition 5.5 were read line by line; no step uses a rate for δ → 0 or M/A → 0; every constant that must be fixed is fixed before the limit; and — the thing the brief asked to count — no step pays a factor equal to the number of windows, which tends to infinity: Proposition 5.5's pigeonhole runs over K(h)·m(h) bins (profile cells of diameter 1/h times the dyadic index modulo m(h)), a number depending only on h, so two random retained points share a bin with probability bounded below by a constant, the selected ordered product is covered in one orientation with density at least ½ − o(1) independent of N and δ, and the congruence modulo m(h(N)) → ∞ is what forces the separated case's width ratio to infinity with both widths o(log B) — exactly the two carrier configurations Lemma 4.2 and §6.3–6.4 assume. The seat answered each of its neighbours' conditions by name: Lemma 4.2 is a slice-and-count through Szemerédi–Trotter with no parameter count and no Bézout, needing only some flat cap c₀ > 0; Proposition 4.3's o(N²) is uniform in the assignment and the curves, and it deletes precisely the ordered pairs (p, q) whose terminal-curve projection from p lies in C_p, which is what §6.5 and the k = 1 case of §6.2 produce; it runs once, globally, inside Theorem 6.1 after Proposition 2.2's greedy removal and before window selection — the order is necessary (o(N²) is not small relative to one selected product) and hides no dependence, since it uses only Theorem 6.1's hypotheses; Propositions 5.1(3) and 5.3 are uniform over all cuts of degree O(T_s(b)) and over sequences of pieces with explicit error bounds e^−7s1/10/18 and N·e^{−(margin)s+o(s)}; and the algebra seat's flag that Lemmas 3.8/3.9 need D ≥ C_nL holds at both §5 uses, both inside Proposition 5.3, with fixed margins and s ≥ s_* → ∞. Fourteen objections, every one exposition. Three deserve the page. The window floor s_* depends on δ — the proof says "choose s_* → ∞ sufficiently slowly that s_* ≤ ¼ log B, δexp(6s_*) → 0" and the statement does not say so; no rate is needed, since s_* = min{¼ log B, (1/12)log(1/δ)} gives δe^6s_* ≤ √δ, and the seat checked every other use of s_* needs only s_* → ∞. The pool-mass sentence on p. 42 (the same sentence sparse_a flagged) names the wrong lower bound as the operative one: Proposition 4.3's deletion alone can cost δN² pairs, the same order as n² when n = N√δ, so n ≥ N√δ is not what keeps the pool covered — what does is the preceding sentence's fixed-h mass c(h)·N, under which the uncovered fraction (deletions included) is o(c(h)²), preserved by the slow diagonal; N√δ is the extra quantitative input §6.2 needs, not the coverage guarantee. And the core count O(EL²) in Proposition 5.5 needs the two-generic-combinations form of Bézout for isolated points, which the paper states on p. 14 with a perturb-and-specialize sketch the algebra seat had flagged; the seat supplied an elementary three-step proof for points. Also on the page: §4's three uses of "the plane cap" need only the proper-flat cap N·B^−c₀, so §4 never consumes the planar Ω(n/log n) — the third seat to reach that conclusion independently; the bound of at most Ce_γ centres per ordered pair of curves is sharp up to the constant (recorded so nobody "improves" it); and the asymptotic Σa_i = o(s^1/3) in Proposition 5.1 has an extremely late onset (K = 16 at s = 10⁶ gives a₁ ≈ 4·10¹⁵ against s^1/3 = 100), true only in the limit as stated. Numerics: the Hölder step in Proposition 4.3 held in 2,000 random dyadic instances and its negative control failed 2,000 of 2,000; the parabolic-lift identity behind (4.1) is exact; at most seven high-multiplicity centres per curve. Citations in scope match at source (Kaminski–Kanel-Belov–Teicher's trisecant section, TYZ §§10.2–10.3); Fulton's Proposition 4.4 is reproved in-paper as Lemma 3.12(i), checked on two examples.

Exact identities and small cases (seat: numerics) — nothing failed. Everything on the brief's list was run by brute force under nice 19 in under 35 seconds a script, scripts and verbatim outputs in the bundle. The grid: for d = 3, 4 and t = 2…6 every squared distance of {1,…,t}^d lies in [1, d(t−1)²] and |Δ|/n^2/d runs from 0.75 to 2.03 — the matching upper bound the Lean also proves. §1.1's arithmetic: Solymosi–Vu is not on arXiv, so the Combinatorica text was used; with Guth–Katz as planar input their Theorem 2.1 (not Corollary 2.3, which takes only a pure power) gives (5/3)^2/5n^3/5/(log n)^2/5 in d = 3, and with Tidor–Yu–Zakharov's n^2/3−o(1) Corollary 2.3 gives 8/17 in d = 4 and Corollary 2.6 gives 3/8 in d = 5, while the general 2/d − 2/(d(d+2)) is their Theorem 1.1(b) on Tardos's 0.8635 — every sentence of §1.1 matches its source (a side note for the reader, not an objection: feeding the newer planar and three-dimensional inputs through Solymosi–Vu would already beat 2/d − 2/(d(d+2)) for d ≥ 6, e.g. 9/29 against 7/24 at d = 6). The §2 bookkeeping holds symbolically, A^d−1 = N^4/3, N^3/2, N^8/5 for d = 3, 4, 5; the skew-form identities (2.6)–(2.8), the p. 11 matrix identity and both polarizations hold symbolically, and in exact rational arithmetic the neighbour locus {Z : Z(u+w) = u−w} is nonempty of codimension exactly d−1 when |u| = |w| and empty otherwise (40 trials a case); Gram equality extends to an isometry in d = 3 including dependent and collinear sets, with a negative control rejected. Theorem 7.1's scale was sanity-checked, not proved: for small integer grids, enumerating every isometry of ℝ³ for t = 3 (13,392 linear parts), Σk_g² over the motions with k_g ≥ C_richA sits at 1.5–4.3 times N^10/3 — the grid breaks M ≤ A but satisfies the line cap the proof actually uses. Two corrections to the paper's own small print, both in its favour: Lemma 2.1's circle row is really 2M − 1, sharp (the regular (2k+1)-gon attains it; 2M is never reached), and its line row is really M; and the range 1 ≤ k < d−1 in (6.9) is loose (balance forces k ≤ d/2). The Hilbert-function toys for Lemma 3.6 are the most useful output: of 100 toys, the 62 that satisfy the lemma's hypothesis all clear the algebra seat's constant c_n,k = 1/((n−k+1)ⁿk!) with a smallest ratio of 2.30 (components of degree D² and D³ included), while four of the 38 that break the hypothesis fall below it (sixteen concurrent lines at D = 3: H = 10 against 12) — so the hypothesis is load-bearing, and the paper's two uses, (2.3) and (2.9), satisfy it. Every displayed constant of §5–6 recomputes (7/18, the summability in (5.5), EL² = δN·e^6s_*, (5.7), the (6.7) margins, (6.20), R·D_d = A, e_d−1/R = N/A), and an exhaustive (6.26) case analysis over 100,000 random profiles finds a strict margin everywhere except narrower centres with l = 1 and b_d > 0 — the case the paper sends to §6.5, found independently of the sparse_b seat's census. Nothing in the paper compares δ with an uncontrolled exp(o(s)) factor.

The whole paper, read once through for the places the architecture could be wrong (seat: whole) — HOLDS AS READ. This seat read all 103 pages before opening any other report, wrote its objections, then compared. It classified 28 asymptotic steps — every "sufficiently large", "o(1)", "after passing to a subsequence" — and found each a genuine limit along the sequence: δ is never asked to beat an exp(o(s)) factor or δ⁻¹, entering only where its plain vanishing suffices (Proposition 4.3's δN², the choice of s_*, the pool size, and δ = 1/K_n in Proposition 2.2); no union bound runs over more than N² objects; the one pigeonhole over a growing number of scales is defused by binning the dyadic index modulo m(h). The induction: the lower-dimensional theorem is consumed at full constant-factor strength at exactly one place — Proposition 2.2, the case ℓ = 4 with fibres on spheres in a (d−4)-flat, which only arises for d ≥ 7 — and the margin there is the rate-free M/A → 0; no d = 4 or d = 5 special case is missing (at d = 4 the fibres are single points, at d = 5 pairs). The planar distinct-distance bound enters only through Corollary B.13 and always with polynomial room — any planar exponent above 2/3 suffices at d = 3, above 2/d in general — so the ruled-surface lemmas, the two-rich-line theorem and the n/log n of §§B.4–B.7 are off the critical path, as the Lean, incidence and secants seats each concluded. But the seat drew the line one step further than any of them: Guth–Katz's higher-richness incidence theorem B.5 is consumed at full strength. The route is Corollary B.6 → Lemma 7.4's planar-type count on p. 67 ("≤ C(m⁴/ℓ²)Σ(hh′)^−1/2 ≤ C′m⁴/ℓ³ ≤ C″m³/ℓ^3/2, … uses ℓ ≥ C_locm^2/3") → Theorem 7.1 → the d = 3 edge deletion → Lemma 2.5's isotropic cap; a k^−2+η version of (7.11) would make Theorem 7.1's bottom bucket fail by a factor N^2η/3, and since every d ≥ 4 reaches d = 3 through the induction, that exponent is on the critical path for every dimension — the indispensable Guth–Katz input is the k ≥ 3 bound, not the planar theorem the paper advertises. The either-endpoint alternative is reduced to one orientation by the pigeonhole in Proposition 5.5 ("pass to an orientation containing at least half the covered pairs"); the seat's own attempted coverings — lines, planar curves, products with a line, circles on spheres, quadric-cone rulings, pencils of planes, the lattice — are each excluded by an occupancy count Proposition 4.3 implements or do not exist. No lemma uses the dimension-d theorem; the four heavy citations (Chardin; Chardin–Philippon Corollary 3 with its erratum; Walsh's Theorem 1.3; Guth–Katz's Theorems 4.5 and 2.10; Bardwell-Evans–Sheffer's Theorem 6.1) match verbatim and are in any case re-proved in the paper; the words "clearly", "similarly" and "easily" never occur; every real-to-complex crossing checks. The global constant table has no cycle: θ, C(e, C₀), C_rich, K_rich and the Lemma 3.6 constant depend on d alone, while H, K, c, C, c₁, c′, c₂, c₃ are "fixed along the subsequence" — legitimate in a proof by contradiction, and the reason c_d is ineffective, which the introduction words loosely ("this is where the preceding quantitative estimates produce the positive constant"). Six objections, all exposition: the over-strong planar input; the existential constant; the one-sentence diagonal step in Proposition 2.2; Theorem 6.1 stated without the lower-dimensional hypotheses it consumes; notation overloading near load-bearing formulas; and the Proposition 5.5 pool-mass wording, where the seat reached the same repair as sparse_a and secants independently.

What the twelve seats add up to, and what they do not. Twelve of twelve HOLDS AS READ. The places every seat was told to hunt — a hidden rate on δ or M/A, a pigeonhole paying for the number of scales, a cycle among the forty-odd constants, a lemma quietly using the theorem in dimension d, a cheap-endpoint covering that slips past Theorem 6.1, a citation that does not say what is claimed, a Hilbert lower bound that fails at the defining degree for components of large degree — came back empty, and the engine of the proof, Lemma 3.6, was certified by one seat, consumed by three others only through its exact statement, and hit with a hundred toys by a fourth that found its hypothesis load-bearing. What the read found instead was a paper quietly stronger than it says in two places (Theorem 7.1 proves what Tidor–Yu–Zakharov state as Conjecture 7.13 and call out of reach; the planar input needed is n^3/4, not n/log n), honest at the one place the previous literature said the method breaks (Appendix A proves only the containment concentration TYZ call within reach, and routes around their d = 3 obstruction through Theorem 7.1), and sloppy in about ninety small ways that are all written down in the bundle — the small print of Lemma 2.1 (2M − 1, not 2M), an unversioned text finalised a week after its title date, constants and symbols overloaded, implications sketched where a reader would want the sentence, Walsh and TYZ attributed where their theorems are in fact reproved. What this is not: a human expert's reading. The seats' own skimmed lists are the edge of this read, and a referee that finds nothing is weaker evidence than a kernel that accepts a proof. For this paper a kernel has accepted one — a third party's, written by AI agents in a week, published as a single squashed commit, never built by anyone but its author, and taking a different route through the plane than the paper does. Our own replay of it ran on this machine on 12 October and printed the same three standard axioms; the Lean paragraph above has the numbers; the Mathlib cache re-check (leanchecker --fresh) exited 0 after 7,919 s.

Labels. The theorem: (d) computer-assisted — a kernel-checked proof exists, written by a third party following the paper (with a weaker planar input the paper turns out not to need), author-reported replays, and our own from-source replay printed the same three standard axioms (the leanchecker --fresh pass over Mathlib's cached object files exited 0 after 7,919 s). The paper as written: (c) plausible/unverified by a human — twelve tool-equipped machine seats with disjoint scopes, each briefed to refute, found no mathematical gap; what they skimmed is listed in each report. The citation checks, the numerics, and our own reads (the §2 bookkeeping, the proof of Lemma 3.6, Proposition 4.3 in full, the statements of Theorems 7.1, A.1, B.2 and B.12) are (a)/(b), ours. Nothing mathematical here is ours; the theorem is OpenAI's, the formalization Yao Xu's. This entry now moves the scoreboard: #1083 joins the problems closed in the literature — closed by OpenAI's theorem and Yao Xu's kernel proof, replayed here, none of it ours. The teorth database still says OPEN (statement formalized, proof not); erdosproblems.com/1083 shows no status and does not mention OpenAI or TYZ; Bloom's 8 October post lists the claim as "not formalised" (true when written) under his blanket "I have checked nothing"; Formal Conjectures carries an open issue and no pull request; the Justin Sun Prize catalogue says "independent mathematical review pending" and the formalizer's own claim says "Independent verifier(s), if any: none"; and we found no human proof, no expert comment, and a 7 October arXiv paper still citing Solymosi–Vu as the state of the art for d ≥ 4.