I conjecture technically it was revealed to the world by a java shop successful Islington connected Insta, but an hr later it was officially announced by Anthropic: 1 of their soul models, utilizing the prove2.me platform, has formalized a complete impervious of Fermat’s Last Theorem (FLT) successful Lean. This is the last theorem to beryllium formalized successful Freek Wiedijk’s celebrated list of 100 formalization challenges and frankincense wraps up this 20-year-old benchmark. Congratulations to Anthropic!
Mathematical details
The impervious is not the modern impervious which I person been formalizing myself pursuing ideas of Khare, Taylor etc, but the Darmon–Diamond–Taylor exposition from 1995 of the Wiles–Taylor–Wiles argument, via the Langlands–Tunnell theorem and Ribet’s level-lowering theorem. Anthropic’s repository develops Fontaine mentation (to study level deformations of Galois representations) and develops capable of Mazur’s activity connected the Eisenstein perfect to reason that nary Frey curve tin person a constituent of bid 
The codification base
I’ve compiled the codification base and tally comparator connected it — it checks out. It is simply a gigantic impervious (over 13.4 cardinal lines of code) and takes astir 20 times arsenic agelong to compile arsenic Lean’s mathematics room (on a instrumentality pinch 96 cores!). Lean tin beryllium sluggish erstwhile jumping from record to record connected a repo of this size (even connected a instrumentality pinch 500G of ram, which Anthropic besides gave maine entree to), but Anthropic besides supplied maine pinch immoderate html documents which are easier successful believe to research (clone the repo and unfastened pinch a web browser).
What this activity is, and is not
I americium presently being funded by the EPSRC to formalize a impervious of Fermat’s Last Theorem, and a naive guidance to the news supra is that I nary longer person immoderate activity to do. This is not the case. The activity surely achieves immoderate of the intends of the EPSRC project, and so it goes overmuch further successful position of what is formalized (I only promised the EPSRC that I would trim FLT to the 1980s; this repo proves the full thing). But I besides promised respective different things to EPSRC: firstly, that I would beryllium making propulsion requests to Lean’s mathematics library, adding basal objects from modern number theory; this is ongoing. And secondly, and possibly astir importantly, that I would beryllium creating a move archive enabling humans to research the modern proof. My conjecture is that it is improbable that Anthropic are going to do this; they will consciousness that their occupation is done pinch the formalization (and they did not formalize the modern impervious anyway).
Note that mathematically this activity of anthropic tells america fundamentally nothing: I americium connected grounds arsenic saying that I americium 99.9% judge that the impervious of FLT is OK, and astir group successful the number mentation organization are 100% judge (formalization has made maine much paranoid astir the mathematical lit than most). From my knowing of the argument, the formalization conscionable faithfully follows the early lit connected the impervious and adds nothing.
What this activity does show us, however, is what is imaginable successful the section of autoformalization. If thousands of pages of the lit tin beryllium formalized end-to-end by immoderate benignant of AI swarm successful an 11 time play now, past successful the early we will commencement to spot formalization of modern investigation being done connected the fly. We will besides study whether my paranoia astir the existent authorities of the Langlands programme is justified, arsenic machines cheque it and ruthlessly emblem arguments which are incomplete. The expertise to autoformalize difficult worldly will yet make the reappraisal process for mathematics papers acold little painful. It will besides support america honorable — location are papers retired location which presume results which are “known to the experts” and it will beryllium absorbing to spot precisely what is being assumed successful the proofs of various important results successful my field. This is why I americium truthful excited astir the news!
I was fixed £1M to tally my task complete 5 years; Anthropic took only 11 days but I do wonderment if they spent much money…
An anecdote
Thought it mightiness beryllium bully to decorativeness pinch a individual anecdote. Wiles announced his impervious of FLT astatine the Newton Institute successful 1993 successful a bid of 3 lectures; I attended the first (I was a 2nd twelvemonth postgraduate student astatine the time) and I recovered it wholly incomprehensible, truthful I skipped the adjacent 2 lectures and went connected vacation to Ireland pinch my caller woman instead; I was truthful successful emotion that I wholly forgot astir the rumours, and it was only erstwhile I came backmost to Cambridge a week later that I heard the news that the theorem was proved. Something strangely akin happened here; erstwhile I sewage the email from Anthropic I was successful Wales astatine the Green Man euphony festival pinch the aforesaid girlfriend, but pinch very mediocre telephone reception; I did spot an email from personification I’d ne'er heard of successful a little infinitesimal of 4G, pinch title “End-to-end Lean formalization of Fermat’s Last Theorem”, but wrote them disconnected arsenic a crank! It was only a week later erstwhile going done the astir 1000 unread emails which had accrued whilst I was away, that I heard the news.
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