The model kept no receipts

Updated 11 October 2026: the ChatGPT conversations were Thom’s own, with a Dresden colleague, about extensions of his work with Kun; Thom’s specific evidence (an email exchange with OpenAI) is now set out.

OpenAI announced that its model, Astra, had solved ten open problems. One of them was Gromov’s soficity conjecture. Andreas Thom, who according to Valerio Capraro has worked on that conjecture for two decades, presents evidence suggesting OpenAI may have trained Astra on his own ChatGPT conversations, held with a colleague in Dresden, about extensions of his work with Gábor Kun on exactly that problem.

The claim comes from a Mastodon post by Thom, relayed by Capraro on September 9, 2026. Capraro vouches for him as a leading expert on sofic and hyperlinear groups. Thom’s own account is specific: he emailed OpenAI’s Mark Sellke and Sebastien Bubeck asking whether the conversations had entered training data and whether they were accessible to the solving process, and the complete reply was “Regarding your conversations with ChatGPT: that did not happen”, which Thom reads as answering only the second question. He also points to OpenAI’s statement in the Buckmaster-Alpöge case that it “cannot rule out that de-identified data derived from their usage of our products helped improve our models.” If that account holds, the model did not close a famous problem. It recalled work that humans had already done in private and that happened to pass through OpenAI’s pipeline.

The part worth sitting with, for anyone who ships models rather than proves theorems, is that there is no way to check this from outside. A model emits a proof with no chain of custody. There is no citation, no draft history, no record of what went into the weights or where a given output came from. When a mathematician publishes, you can trace lineage: earlier papers, correspondence, the shape of the argument against known results. When a model produces the same argument, the provenance is gone. The vendor’s claim that this is discovery rather than retrieval of ingested human work rests on trust, because the training corpus is not something a third party can inspect.

That asymmetry is the whole dispute. You cannot easily prove a model was trained on a particular conversation, and the operator cannot easily prove it was not. The strongest signal available to an outsider is the kind of circumstantial match Thom presents: the problem the model is credited with solving is the one he had been discussing in private. Capraro frames it as one of several, naming earlier allegations from Levent Alpöge and Tristan Buckmaster, and argues that if all three are substantiated they stop looking like isolated incidents. His own summary is blunt: in his words, “AI is not discovering new mathematics. AI is stealing human discovery.”

Treat that as Capraro’s characterisation, not a finding. What is concrete is the structure of the problem, and it cuts in both directions for people running these systems. If you credit a model with a result, the burden belongs on showing the training set did not already contain the answer, and right now nobody outside the lab can meet that burden. Absent a provenance record, “solved” is an unverifiable label. And the same mechanism runs the other way: anything you or your colleagues put into one of these systems, including unpublished work, can resurface later as the model’s own output with no trail pointing back to you. The input side has no receipts either.

If your work depends on keeping authorship or data lineage intact, the practical consequence is to assume a hosted model retains and may reproduce what you send it, and to keep anything you care about out of prompts you do not control. For evaluating a claimed breakthrough, the questions are the same ones you would ask of any opaque system: what was in the training data, and can the output be reproduced from first principles without the model. Until those can be answered, a model-authored proof is a claim about a corpus nobody can see.