He had wondered about a group of knot invariants for years, handed the question to Claude models, and then spent between fifty and a hundred hours rewriting and checking every proof by hand. He printed part of the transcript in the paper.
The paper has one author, Robert Lipshitz. It names his earlier collaborators on the underlying construction as Nathan Dunfield and Dirk Schütz, whose paper it cites as [DLS26], and it thanks them and Matthew Stoffregen and Ian Zemke for conversations. It records support from NSF grant DMS-2505715 and a Simons Foundation travel grant. That is all it says about who he is, and it is all this chapter will say. The result it is about, for the record, is at least six concordance homomorphisms to the integers that the $s$-invariants, $\tau$ and the Dai–Hom–Stoffregen–Truong invariants do not account for.
What the paper says about what he did is much fuller, because Section 5 is a reflection on method and ethics written in the first person, and the introduction states the division of labour in a short paragraph headed AI: apart from some quotations, the tables and some diagrams, the paper was written entirely by him; the research was largely carried out by various Claude models, with modest prompting by the author; except where flagged, all results were checked by him.
He had wondered about the structure of these groups since the earlier paper and had too many other projects to pursue it. He suggested the problem of finding torsion to at least three graduate students, none of whom was interested, and he says rightly so. His reasoning for giving it to a language model is specific: the question was of some interest but not enough to justify the human time, it relied on elementary algebraic arguments, and a human could check them line by line. He also says that language models in June 2026 had a reasonable chance at that kind of problem.
| Item | What the paper reports |
|---|---|
| Cost | A Claude Pro subscription at $20 a month. He believes he never used more than half his weekly quota on the project, and says the quota limit slowed progress and interfered less with his other work. |
| Models | Fable 5 at first, then Opus 5 when Fable 5 required extra payment, later Opus 5.5. Some Opus sessions spawned Sonnet subagents to write code, especially early. |
| Length | He did not count turns; at least several hundred. |
| Record | The first Claude session set ground rules: each instance writes its thoughts each turn in files it called lab notebooks, and does not revise history. He could read them but mostly did not. |
| Ownership | He has no special access to or relationship with Anthropic and is not paid by it or invested in it. |
The paper prints a stretch from a session that was stuck. In it a Claude instance reports two bugs that had produced forty-four reported failures, every one an artefact of the bug, and then lists what it will not claim. Later the same instance explains why it is stopping: its structural intuitions had repeatedly run ahead of what it could verify, the checks caught it each time, and the right response was more scepticism, not more optimism. He answers that this is how research goes and asks it to prove the remaining lemma. A few turns later it proved the existence of the $\mathbb{Z}^\infty$ summand over $\mathbb{Z}_p[h]/(ph)$.
He notes a pattern he cannot interpret: after early dramatic progress by a fresh session, later sessions kept suggesting he start a new one, and the next substantial progress came after he made encouraging remarks to the current session instead. He says he does not know if that is meaningful or random.
He says he rewrote all the paper except a few sample sentences, and that this took between 50 and 100 hours of his active work. In the process he often asked Claude to re-prove a result along a different line, and he says the final paper is mathematically more interesting than if he had accepted the original proofs. His example is the proof in Section 2.4.2, which Claude found in response to a single prompt from him complaining that Section 2.4.3 was surprisingly indirect. Computations and remarks he did not check are marked in the text with wording like Claude reports; the computational census of Section 3 is one place where he says Claude wrote and ran the code and he did not examine it.
He describes the earlier models as like an early-career graduate student who does a lot of computation, and the recent ones as more like a collaborator, with broader mathematical knowledge than any one human, better at routine tedious steps such as propagating signs, but a little worse at remembering global goals, at telling what is surprising to a human, and at writing clearly for one. He says he values collaborators who think differently and that Claude’s desire for computational checks was beneficial. He ends by saying it seems unlikely that many future papers will be LLM-discovered but human-written.
Two parts of his account have a counterpart in how this series is run, and one does not. The counterpart is that a person re-derives what a model produced before it is treated as settled, and that the model is asked to say what it will not claim. The difference is that here the check ships with the page: each chapter carries a script that a reader can run and that contains a control, a case where the test must fail. His checking was by hand and is described in prose; nothing in the paper lets a reader reproduce it. That is not a criticism of the paper, which is a research paper with proofs a referee can follow, and it is a difference in what is being offered.
Two figures in this chapter were re-checked by the script beside it: the six in the headline result, by the rank computation on the paper’s own Table 2 (see the previous chapter), and the correspondence between each number in this page and the constants the script holds from the paper. The script does not check that the paper’s account of his working process is accurate; it cannot.