Warm a flat metal plate unevenly, insulate its edges, and wait. The temperatures even out, but one spot stays a little hotter than the rest for longest. In 1974 Jeffrey Rauch asked whether that last hot spot always ends up on the edge. It does not always: plates with holes can trap it inside. The plain case — a plate with no holes — was stated as a conjecture by Krzysztof Burdzy in 2005 and stayed open. One of the results OpenAI released on October 6, 2026 claims it for every smooth plate with no holes, with the full statement checked by machine. Two days later Burdzy posted a paper removing the smoothness assumption in the simple-eigenvalue case, and said how it was made: “This paper was generated by ChatGPT in response to multiple prompts from the human author.” He added that Rodrigo Bañuelos had done the same, independently, at the same time; that the OpenAI model “missed” this theorem; and that the release should have cited a 2024 counterexample. A week after the release, Burdzy and Rami Atar posted a new counterexample in three dimensions. This case is the field answering a machine with the same tools — and what that does to credit, checking and citation.
The question is one a child could ask about a frying pan. Heat a flat plate unevenly, insulate its edges so no heat escapes, and wait. Where is the last hot spot? Jeffrey Rauch raised it in a 1974 lecture at Tulane University; by his own account he never published it, and it first appeared in print in 1985.[5][15] Mathematicians restate it through the plate’s slowest pattern of cooling: does that pattern reach its highest and lowest values on the edge?[1][5] The companion page lets a reader paint heat onto a plate and watch.[14]
The answer depends on the shape. Partial results for long, symmetric and other special plates came in the 1990s and 2000s.[1][5] Then the counterexamples: in 1999 Burdzy and Wendelin Werner built a plate with two holes where the hot spot stays inside, and in 2005 Burdzy did it with one hole.[6][7] In that 2005 paper he stated the case left over — a plate with no holes — as his own conjecture.[7] Triangles were settled in 2020.[8] In 2024 Jaume de Dios Pont showed the question fails for some convex shapes in high dimensions.[9] And in 2025 de Dios Pont, Alexander Hsu and Mitchell Taylor reported that one of their proofs “was found with significant assistance of a large language model.”[10] This field was already working with AI before the release.
One of the results OpenAI released on October 6, 2026 claims the no-hole case for every plate with a smooth edge, in its strongest form: no hot spot, cold spot or saddle anywhere inside, even when the cooling pattern is not unique.[1] Unlike several results from the same release, the machine check covers the full statement.[2] The paper has not been withdrawn or revised.[3] What it assumes is a smooth edge — and that is where the reply began.
On October 8, Burdzy posted a seven-page paper extending the result to plates whose edges need not be smooth, in the case where the cooling pattern is unique and in a slightly weaker form.[4] Its first section is headed “Human commentary.” It reads: “This paper was generated by ChatGPT in response to multiple prompts from the human author. KB has not checked every detail, because the theorem is, to a human, an obvious corollary of the results in [3].” And: “Independently and at the same time, Rodrigo Bañuelos used ChatGPT to generate an almost identical theorem and proof.”[4] Six days later, Burdzy and Rami Atar posted a counterexample in three dimensions.[11]
Burdzy’s note is unusually explicit about three things a field usually leaves implicit. On credit: “Neither KB nor Rodrigo Bañuelos intend to submit their papers to a journal. If the results or arguments are cited, equal credit should be given to KB, Rodrigo Bañuelos and to any other mathematicians who obtained the result at the same time.”[4] When a tool can produce a result for anyone who asks, the old rule — credit to whoever got there first — stops sorting much.
On checking, he states plainly that he did not check every detail, and that his paper takes the smooth theorem “as an input and do[es] not assess or reproduce its proof.”[4] On citation, he writes that the OpenAI model “missed the theorem presented in this note,” and that “it is highly surprising that [3] did not cite the spectacular paper of de Dios Pont,” adding that “even if there were a reason for the omission, such as a gap in the proof, the paper should nevertheless have been cited, with an appropriate caveat.”[4] The release does cite a related 2025 paper by de Dios Pont and co-authors, but not the 2024 counterexample.[1][9][10]
None of this is a verdict on the proof, and none of it is hostile. The mathematician who posed the open case read the claim within two days, built on it with the same kind of tool, disclosed exactly how, and kept working: his October 14 paper cites the release as the planar state of play.[11] That is a different response from the silence around UC-332 and the waiting around UC-330. It is the field absorbing a machine’s result by using machines — and writing down, in the margin, the rules it is improvising.
Burdzy and Werner build a plate where the hot spot stays inside.[6]
CounterexampleBurdzy does it with one hole and states the no-hole case as his own conjecture.[7]
The Open CaseRaised in a lecture, never published by its author, first in print in 1985.[5][15]
Burdzy states the no-hole case as his own conjecture, after a one-hole counterexample.[7]
October 6 to October 8, 2026, with ChatGPT — twice, independently.[4]
Atar and Burdzy’s 3D construction, October 14, citing the release as the planar state of play.[11]
This paper was generated by ChatGPT in response to multiple prompts from the human author. — Krzysztof Burdzy, October 8, 2026 [4]
| Dimension | Evidence |
|---|---|
| Employee (D2) Origin · 82 | Two days after the release, the mathematician who posed the open case extended it in a paper “generated by ChatGPT,” alongside an independent one by Bañuelos.[4][7]The Author Replied, With ChatGPT |
| Regulatory (D4) L1 · 78 | “Equal credit should be given to KB, Rodrigo Bañuelos and to any other mathematicians who obtained the result at the same time.” No journal submission intended.[4]Rules Written in Footnotes |
| Quality (D5) L1 · 74 | The smooth-plate statement is machine-checked in full; the smoothness assumption and an uncited 2024 counterexample are where the reply landed.[1][2][9]Checked in Full, Within Its Scope |
| Operational (D6) L2 · 70 | A 2025 proof in this field “was found with significant assistance of a large language model.” A 3D counterexample followed within a week of the release.[10][11]AI Was Already in the Room |
| Customer (D1) L2 · 52 | Readers now meet papers that disclose they were generated by a model and not fully checked by their author — stated openly, which is the point.[4]How Was This Paper Made? |
The cascade originates in D2 (Employee): how the specialists who own a problem respond to a machine’s answer — here, within two days and with the same kind of tool.[4] It propagates to D4 (Regulatory), where credit, checking and citation are being set by footnote rather than by rule,[4] and to D5 (Quality), where the claim is machine-checked in full but limited to smooth edges, and a known counterexample went uncited.[1][2][9] It lands in D6 (Operational), the way the field now works — AI-assisted proofs in 2025, ChatGPT-generated extensions in 2026,[4][10] — and in D1 (Customer), readers of the literature who must now ask how each paper was made. D3 (Revenue) is not scored.
This is the seventh result from the release in this library, and the first where the response used the same tools as the claim. UC-328 recorded specialists reading, UC-329 a crowd building, UC-330 waiting, UC-331 a prediction arriving, UC-332 silence and UC-333 a check narrower than its claim. The honest limit: no specialist has reviewed the smooth-plate proof; Burdzy states he did not check every detail of his own; and Bañuelos’s parallel result is known only from Burdzy’s note.
-- UC-334: Answered in Kind: 6D Amplifying Cascade
-- Rauch 1974 lecture (Tulane), never published by him; in print 1985 (Kawohl). Banuelos-Burdzy 1999 (JFA 164:1-33). Burdzy-Werner 1999 (Ann. Math. 149:309-317) two-hole counterexample. Burdzy 2005 (Duke 129:481-502) one hole; states simply connected planar case as Conjecture 1.2(ii). Judge-Mondal 2020 triangles. de Dios Pont 2024 (arXiv 2412.06344) convex counterexample, high d. de Dios Pont-Hsu-Taylor 2025 (arXiv 2508.16321): proof found with significant LLM assistance. OpenAI result 369 (Sep 24 2026, 21 pp): strict hot spots for smooth bounded simply connected planar domains, all first-eigenspace functions; Lean covers full statement. Burdzy arXiv 2610.12622 (Oct 8): non-strict, simple-eigenvalue, nonsmooth extension; generated by ChatGPT; not every detail checked; Banuelos independently same; equal credit; OpenAI model missed it; de Dios Pont 2024 uncited. Atar-Burdzy arXiv 2610.19908 (Oct 14): 3D starlike counterexample, cites OpenAI.
FORAGE answered_in_kind
WHERE conjecture_author_responded = true
AND response_generated_with_ai = true
AND claim_machine_checked_in_full = true
ACROSS D2, D4, D5, D6, D1
DEPTH 3
SURFACE answered_in_kind
DIVE INTO same_tool_response
WHEN response_days <= 2
AND credit_terms_stated = true
TRACE norms_by_footnote_cascade
EMIT answered_in_kind_signal
DRIFT answered_in_kind
METHODOLOGY 88
PERFORMANCE 40
FETCH answered_in_kind
THRESHOLD 1000
ON MONITOR CHIRP high 'An AI-written, machine-checked proof claims Rauch's 1974 hot spots conjecture for smooth plates with no holes. Two days later the mathematician who posed that case extended it to rough edges with ChatGPT, disclosed he had not checked every detail, asked for equal credit with an independent parallel result, and noted an uncited counterexample. A week later he co-posted a 3D counterexample.'
SURFACE analysis AS json
Runtime: @stratiqx/cal-runtime · Spec: cal.semanticintent.dev · DOI: 10.5281/zenodo.18905193
The specialist who posed the open case extended a machine’s proof with a machine, in two days, and said so on the first page.[4]
“Equal credit should be given to … any other mathematicians who obtained the result at the same time.” When anyone can generate a result, being first stops meaning much.[4]
“KB has not checked every detail.” Stating what was not checked is more useful than silence about it — and rarer.[4]
Primary sources are OpenAI’s paper, machine-check note and change history; Burdzy’s October 8 response and Atar and Burdzy’s October 14 paper; and the 1999, 2005, 2020, 2024 and 2025 papers on the problem, read at source or checked against their published records. Bañuelos’s parallel result is reported only in Burdzy’s note; no preprint was found. This case does not claim the proof is correct. Source 14 is an interactive companion visual made for this case.
This case follows a field that answered an AI result with AI — and wrote its rules for credit, checking and citation in a footnote. The same moment is arriving in audits, research and engineering reviews. Book a discovery call to set those rules on purpose, before they are set by default.