Put six dots in a square and look at every triangle three of them make. The best possible arrangement keeps the smallest of those triangles at 1/8 of the square. Scatter the dots at random and, in this case’s simulation, the smallest triangle is typically about 36 times smaller. How large the smallest triangle can be as the dots multiply is Heilbronn’s triangle problem, asked before 1951 by a mathematician who had fled Nazi Germany and later taught in Toronto. One of the papers OpenAI released on October 6, 2026 claims that, for every large number of dots, the smallest triangle can be kept larger than the “almost 1/n²” formulation of the problem allows — by a margin whose explicit exponent has 235 zeros after the decimal point. Its machine check proves something narrower: the bound for infinitely many sizes, not every size. That is enough to disprove the formulation, and not enough to prove everything the paper says. A week later this case found no public response at all. The case is about reading that difference: what was claimed, what was checked, and why the two are not the same sentence.
Hans Heilbronn was born in Berlin in 1908, fled to England after losing his post at Göttingen in 1933, and was interned on the Isle of Man in 1940 as an enemy alien. He resigned from Bristol in 1964, moved to North America without a job, and took a chair in Toronto, where he died in 1975.[5] Before 1951 he asked a question that needs no training to state: place n dots in a square so that the smallest triangle any three of them form is as large as possible. How fast must that smallest triangle shrink as n grows?[4][6]
Heilbronn guessed it shrinks at least as fast as 1/n². Paul Erdős showed that careful dots on a curve achieve about that size, so the guess could not be improved if true.[4] In 1982 János Komlós, János Pintz and Endre Szemerédi showed it was not true: a smarter, randomised placement does better by a factor of log n.[7] From the other side, the best proof that triangles must shrink stood for four decades until 2023.[4][8] The gap between what can be built and what can be ruled out is still wide. One formulation, discussed in a recent survey, asked whether the truth sits within any small power of 1/n² — the “almost 1/n²” form of the problem.[1]
One of the results OpenAI released on October 6, 2026 claims to break it: for every sufficiently large n, dots can be placed so that every triangle has area at least a constant times n to the power −2 + η, for a fixed η greater than zero.[1] The paper calls η “fixed but extremely small.” Its explicit value is 2 divided by a number with 232 digits — a decimal point followed by 235 zeros.[1] No drawing could show the improvement. It was found by working with finite fields and integer lattices, not by placing dots. The paper also notes that two earlier preprints claimed stronger results and have gaps; in one, a construction contains three points on a line.[1][10][11]
Then the machine-check note, which says something precise. The formal proof “constructs an unbounded sequence of sizes n and point sets” with the improved bound, and it “consequently refutes the proposed upper bound.” But “the paper’s statement for every sufficiently large n is broader.”[2] Infinitely many sizes, possibly far apart, are enough to show the old expectation is false. They are not the same as every size from some point on. The disproof is checked. The full theorem is not.
η = 2/(45435k + 16), where k has 232 digits. The paper states it made no attempt to optimise the construction.[1]
Two sentences can look alike and say different things. “For infinitely many n” means there is always another size further on that works, with no promise about the sizes in between. “For every n beyond some point” means none are skipped. Which one a result needs depends on its purpose. To refute a claimed upper limit, it is enough to beat that limit again and again, at sizes as large as you like — and that is what the machine check proves.[2] To describe how the smallest triangle actually behaves at every size, the stronger statement is needed, and that rests on the paper’s own argument for passing from the special sizes to all of them.[1]
None of this is hidden; the scope note states it in one sentence. But summaries of the release, and of results like it, compress such sentences into “machine-checked” or “proved.” The skill is to ask, every time, which statement was checked. In UC-331 the machine check covered the weaker of two physical claims. Here it covers the disproof and not the full theorem. In UC-332 it covered the main statement itself. Three different answers to the same question, from the same release.
This case found no public response. The Wikipedia article on the problem was last edited on October 3, three days before the release, and the problem’s page on the Erdős problems site still lists the 1982 lower bound as the best known, with no comments.[9][12] Nothing here says the proof is wrong, and the refutation itself is the part a machine has already checked. The improvement is too small to change any picture or any application. What it would change is a sentence in every survey: the old expectation is false.
How large can the smallest triangle among n dots in a square be? He guesses it shrinks like 1/n².[4]
The QuestionKlaus Roth publishes the first bound on the problem, with Erdős’s construction recorded.[6]
The StartHe leaves Bristol for North America and takes a chair in Toronto, where he dies in 1975.[5]
The ManCohen, Pohoata and Zakharov improve the upper bound for the first time in four decades.[8]
ProgressHow much smaller the smallest triangle typically is when six dots are scattered at random, in this case’s simulation.[13]
The linked construction is for an unbounded sequence of sizes; the paper’s statement for every sufficiently large n is broader. — OpenAI, Lean scope note for result 191 [2]
| Dimension | Evidence |
|---|---|
| Regulatory (D4) Origin · 84 | The machine check “constructs an unbounded sequence of sizes”; “the paper’s statement for every sufficiently large n is broader.”[2]Checked for Some n, Claimed for All |
| Quality (D5) L1 · 76 | The explicit exponent is 2/(45435k + 16) with k of 232 digits — 235 zeros after the decimal point. “Fixed but extremely small.”[1]Too Small to Draw |
| Operational (D6) L1 · 68 | Finite fields, field norms and lattice counts replace placing dots. Two earlier preprints claimed more and have documented gaps.[1][10][11]Built from Number Theory |
| Employee (D2) L2 · 62 | Wikipedia last edited October 3; the Erdős problems page still lists the 1982 lower bound, with no comments.[9][12]The Reference Pages Haven’t Moved |
| Customer (D1) L2 · 48 | For any number of dots anyone could place, the improvement is invisible. What changes is the sentence in the surveys.[1][4]No Picture Changes |
The cascade originates in D4 (Regulatory): the gap between the paper’s claim for every large n and a machine check that covers an unbounded sequence of sizes.[1][2] It propagates to D5 (Quality), where the improvement is real and fixed but its explicit exponent has 235 zeros,[1] and to D6 (Operational), where the method comes from finite fields and lattices rather than geometry, after two earlier claims with gaps.[1][10][11] It lands in D2 (Employee), the specialists who have not responded and the reference pages that still show the 1982 bound,[9][12] and in D1 (Customer), the readers and students of the problem, for whom no drawing or application changes. D3 (Revenue) is not scored.
This is the sixth result from the release in this library, and the one that most directly teaches a habit: read the scope of a check before repeating its conclusion. UC-331 and UC-332 showed machine checks covering less and more of their claims; this one covers exactly what a disproof needs and stops there. The honest limit: this case did not and cannot verify the proof; the simulation illustrates small cases only; silence measured in public may hide private reading.
-- UC-333: Infinitely Many Is Not Every: 6D Diagnostic Cascade
-- Heilbronn (1908-1975; Berlin, Cambridge, interned Isle of Man 1940, Toronto from 1964) conjectured before 1951 Delta(n)=O(1/n^2). Roth 1951 (JLMS 26:198-204). Erdos parabola gives ~1/n^2. Komlos-Pintz-Szemeredi 1982 (JLMS 25:13-24) log n/n^2. Upper: KPS n^(-8/7+e) (1981) until Cohen-Pohoata-Zakharov 2023 (n^(-8/7-1/2000)), then n^(-7/6+o(1)). OpenAI result 191 (paper Sep 25 2026, 24 pp): Delta(n) >= c n^(-2+eta) for every large n; eta = 2/(45435k+16), k with 232 digits (eta ~ 1e-235). Lean scope: unbounded sequence of n; refutes almost-n^-2 bound; every-large-n statement broader. Earlier preprints Ellmann v12 (heuristic), Agama v13 (collinear triple). Not withdrawn or revised. Wikipedia last edited Oct 3 2026; erdosproblems 507 unchanged, 0 comments. Simulation: 6 random dots median min area ~0.0035 vs optimum 1/8.
FORAGE infinitely_many_is_not_every
WHERE claim_scope_exceeds_check_scope = true
AND disproof_machine_checked = true
AND specialist_response_found = false
ACROSS D4, D5, D6, D2, D1
DEPTH 3
SURFACE infinitely_many_is_not_every
DIVE INTO quantifier_gap
WHEN checked_for = unbounded_sequence
AND claimed_for = every_large_n
TRACE scope_reading_cascade
EMIT infinitely_many_is_not_every_signal
DRIFT infinitely_many_is_not_every
METHODOLOGY 88
PERFORMANCE 36
FETCH infinitely_many_is_not_every
THRESHOLD 1000
ON MONITOR CHIRP high 'An AI-written paper claims that, for every large n, n dots in a square can avoid small triangles better than the almost-1/n^2 expectation, by an exponent with 235 zeros. Its machine check covers an unbounded sequence of sizes: enough to refute the expectation, narrower than the paper's claim. No public response found.'
SURFACE analysis AS json
Runtime: @stratiqx/cal-runtime · Spec: cal.semanticintent.dev · DOI: 10.5281/zenodo.18905193
Sizes that keep recurring, however far apart, are not the same as all sizes from some point on. The check proves the first; the paper claims the second.[2]
To refute an upper limit, beating it infinitely often is enough. To describe the true behaviour, every size matters. The same check can be complete for one purpose and partial for another.[1][2]
Random dots make tiny triangles; careful constructions avoid them. Every advance on this problem, from Erdős to 2026, has been a better construction.[4][7]
Across three cases from one release, the machine check covered less than the claim, all of it, and exactly what a disproof needs. “Machine-checked” is the start of a question.[2]
Primary sources are OpenAI’s paper, machine-check note and change history; Roth’s 1951 paper and Komlós, Pintz and Szemerédi’s 1982 paper; the 2023 upper-bound preprint; and the reference pages, all read directly. The random-placement figure comes from this case’s own simulation of 400 placements. No public response to this result was found to October 14, 2026. This case does not claim the proof is correct. Source 13 is an interactive companion visual made for this case.
This case separates a check that covers some instances from a claim about all of them — a gap that hides in test suites, audits and compliance reports as easily as in proofs. Book a discovery call to find where your verification statements say more than your checks do.