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Anthropic published two AI-derived cryptanalysis results in July and OpenAI's model disproved an 80-year-old conjecture in May, but both authors argue the models are still weak exactly where deep new theory is required.
The Watch · Security desk
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Schneier and Rafi single out the counterexample to the Jacobian conjecture as their clearest case of the first type: once it existed, checking it was quick and straightforward, and the hard part was finding it among a large number of candidates, which they read as machine-learned intuition wired to an extensive computational search [9]. Plenty of practical cryptanalysis has that same profile, since locating a weak parameter set in a large space costs far more than confirming one that somebody hands you.
The scope argument is the part with defensive content. The essay notes that the unit-distance counterexample pulled in algebraic number theory, and that a specialist in that field who deliberately set out to find a counterexample would probably have succeeded, except that nobody with exactly that background had a reason to work on the problem [10]. Models do not have careers, so they do not have that gap. The cryptographic version of an unfashionable open problem is a construction that shipped, got one round of scrutiny at publication, and has been quietly load-bearing since.
OpenAI published ten further results from its latest model this month [6], and Anthropic published Claude's attempt at the century-and-a-half-old Riemann hypothesis, which did not land it [7]. Add the May disproof and the July pair and the public tally comes to thirteen named results in roughly three months [16]. It arrives without a denominator. No lab has said how many problems were attempted to produce thirteen, so the figure describes output, not hit rate.
Two soft spots in the essay matter, because the ceiling claim rests on its taxonomy of what AI has actually done [8]. The Jacobian counterexample carries most of the weight for the first category, and the essay does not say which lab or model produced it, or when [17]. Separately, the authors credit current systems with larger working memory, broader knowledge and faster processing than any individual human [13]. That is a description of scale, and the accompanying claim that true novelty remains out of reach is a judgment about what scale has not yet produced rather than a bound on what it cannot [12].
This is planning input. It helps whoever has to answer, in writing, whether AI will break the algorithms in their estate before the migration finishes. The authors' answer is that the capability is coming, their guess is sooner rather than later, and they will not say whether that means months, years or decades [15]. They also stress that none of these mathematical capabilities were designed for or planned, which is why they arrive without a roadmap [14].
Ranked by verification strength, evidence, and original report placement.
Schneier and Rafi argue that AI models are nowhere near as capable as experienced academic mathematicians, at least in the short term.
The essay was written by Bruce Schneier with Kasra Rafi, originally appeared in The Guardian, and was posted on schneier.com in an August 2026 archive entry.
The authors say these AI-powered advances in mathematics fall into two categories: counterexamples to statements people had been trying to prove, and novel applications of known techniques to existing problems that human experts either did not know or did not think of using.
The unit-distance conjecture is the essay's example of the second kind: most mathematicians expected the original elegant construction to be essentially optimal and so tried to prove rather than disprove it, the counterexample brings in ideas from algebraic number theory, and an expert with that background who deliberately set out to find a counterexample would probably have succeeded, but had no reason to focus on the problem. The authors say AIs do not have that limitation because of their scope.
The authors describe these results as relatively low-hanging fruit for AI, none of which required developing an extensive new theory, while saying this does not make the discoveries trivial.
The authors say what has not yet been seen is an AI developing a substantial new conceptual framework to solve a mathematical problem; current AIs are very strong at searching and recombining existing ideas but weak at building deep and sustained new theory.
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1 article · August 28, 2026
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One essay, no citations
Every dated result here — the May unit-distance disproof, July's two cryptanalysis papers, ten more from OpenAI in August, Claude on Riemann — reaches readers through a single paragraph of a Guardian column reposted on Schneier's own blog, with no links, no titles and no lab statements behind it. The mathematics would be checkable; the reporting is not, because nothing in it points at the underlying work. The one result the argument leans on hardest, the Jacobian conjecture counterexample, is the one that arrives with no lab, no model and no month.
Publication, not uptake
What can actually be counted is announcements: four of them from two labs across about three months, all narrated second-hand. Nothing tells us whether any result has been refereed, reproduced, or used by a working mathematician, and the strand with operational consequence — cryptanalysis — amounts to two unnamed papers in a single clause. Meanwhile the mathematicians themselves appear only as forty people in a room at OpenAI whose mood was inferred from other people's articles.
Crypto framing outruns the text
The overreach is ours more than theirs. Schneier and Rafi wrote about mathematics and mentioned cryptanalysis once; reading a near-term ceiling on breaking cryptography out of that clause is an extrapolation the essay never makes. Inside the piece, the deflationary verdict is delivered with more confidence than an anonymous counterexample can carry, and the taxonomy it rests on folds in on itself when the unit-distance result serves as both the disproof and the example of clever reuse. The direction of the argument still reads as under-hyped rather than over-hyped; the specifics are the part that has been stretched.
Two insiders on their own patch
Both authors are writing about the ground they stand on: Schneier has built a career on cryptography's failure modes, and Rafi is a mathematician judging whether mathematicians are still needed. Reassurance is the natural output of that pairing, and the essay's own opening notes that the profession is frightened. On the other side of the ledger, every result cited originates in OpenAI's or Anthropic's own announcements — labs that gain from having their models credited with eighty-year-old problems — and the gathering described took place at OpenAI's offices, off the record.
Sure of the view, unsure of the facts
We can be confident about what these two authors think and very little else. One publisher, no corroboration, no primary papers in view, and an argument about search-and-recombine versus theory-building that stands or falls on results no reader can inspect. Treat the shape as durable — models are visibly good at finding examples and unproven at inventing frameworks — and treat every date, count and attribution as the essay's word alone.