{"@context":"https://schema.org","@type":"NewsArticle","generatedAt":"2026-07-23T06:40:50.084Z","headline":"ChatGPT 用反例推翻 Erdős 单位距离猜想，AI 自动形式化验证紧随其后","description":"2026 年 5 月，ChatGPT 推翻了离散几何中的 Erdős 单位距离猜想，构造出反例。一周后，Logical Intelligence 的 AI 系统将该证明自动形式化为 Lean 代码。6 月，OpenAI 的 Sol 模型在无任何数学公理假设下完成了完整形式化，生成 120 万行 Lean 代码，证明中涉及百页以上的全局类域论定理。","url":"https://www.aioga.com/news/cmrtsfndu22tebihzaf3omv19/","mainEntityOfPage":"https://www.aioga.com/news/cmrtsfndu22tebihzaf3omv19/","datePublished":"2026-07-20T22:11:08.840Z","dateModified":"2026-07-20T22:11:08.840Z","inLanguage":"zh-CN","publisher":{"@type":"NewsMediaOrganization","name":"Aioga","url":"https://www.aioga.com"},"citation":["https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled","https://aihot.virxact.com/items/cmrtsfndu22tebihzaf3omv19"],"canonicalUrl":"https://www.aioga.com/news/cmrtsfndu22tebihzaf3omv19/","directAnswer":{"@type":"Answer","text":"Aioga 编辑摘要：2026 年 5 月，ChatGPT 推翻了离散几何中的 Erdős 单位距离猜想，构造出反例。 Aioga 将其归入「技巧观点」方向，重点关注它对真实使用和行业竞争的影响。","url":"https://www.aioga.com/news/cmrtsfndu22tebihzaf3omv19/","dateCreated":"2026-07-20T22:11:08.840Z","author":{"@type":"Organization","@id":"https://www.aioga.com/authors/aioga-editorial/#editorial-team","name":"Aioga Editorial Team","url":"https://www.aioga.com/authors/aioga-editorial/"}},"evidence":[{"@type":"CreativeWork","name":"xenaproject.wordpress.com source article","url":"https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled","datePublished":"2026-07-20T22:11:08.840Z","provider":{"@type":"Organization","name":"xenaproject.wordpress.com","url":"https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled"}},{"@type":"CreativeWork","name":"AIHot archive record","url":"https://aihot.virxact.com/items/cmrtsfndu22tebihzaf3omv19","datePublished":"2026-07-20T22:11:08.840Z","provider":{"@type":"Organization","name":"AIHot","url":"https://aihot.virxact.com/items/cmrtsfndu22tebihzaf3omv19"}}],"aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","originalPublisher":{"name":"xenaproject.wordpress.com","url":"https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled"},"article":{"id":"cmrtsfndu22tebihzaf3omv19","slug":"cmrtsfndu22tebihzaf3omv19","url":"https://www.aioga.com/news/cmrtsfndu22tebihzaf3omv19/","title":"ChatGPT 用反例推翻 Erdős 单位距离猜想，AI 自动形式化验证紧随其后","title_en":"人类数学家正被反例所击败","summary":"2026 年 5 月，ChatGPT 推翻了离散几何中的 Erdős 单位距离猜想，构造出反例。一周后，Logical Intelligence 的 AI 系统将该证明自动形式化为 Lean 代码。6 月，OpenAI 的 Sol 模型在无任何数学公理假设下完成了完整形式化，生成 120 万行 Lean 代码，证明中涉及百页以上的全局类域论定理。","source":"Hacker News 热门（buzzing.cc 中文翻译）","sourceUrl":"https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled","aiHotUrl":"https://aihot.virxact.com/items/cmrtsfndu22tebihzaf3omv19","publishedAt":"2026-07-20T22:11:08.840Z","category":"技巧观点","score":66,"selected":false,"articleBody":["It’s been an interesting few weeks for counterexamples. This post is basically my perspective of what has been going on in the world of formalization, AI tools and, in particular, counterexamples.","Two months ago today (20th May 2026), ChatGPT disproved Erdős’ Unit Distance conjecture in discrete geometry. This is now old news but I had to start somewhere. The announcement：https://openai.com/index/model-disproves-discrete-geometry-conjecture/ was accompanied with testimonies by human mathematicians, many of whom I knew and a few of whom I trusted, saying that they believed the argument (they had been given early access to it and had checked it). The basic structure of the proof is that a profound theorem in number theory due to Golod and Shafarevich from the 1960s could be used to construct a counterexample to the conjecture.","It is now 9 years since I had a mid-life crisis, realised I no longer trusted many human mathematicians when it comes to technical details, discovered Lean, and started to argue that interactive theorem provers should play an important role in the future of mathematics. So of course my first question was “is the counterexample formalized in Lean”. The answer was “no”.","But under a week later (26th May 2026), I got an email from Fields Medallist Mike Freedman. Mike is now the Chief Science Officer for Logical Intelligence：https://logicalintelligence.com/, a company cofounded by Turing Award winner and “godfather of AI” Yan LeCun. Mike informed me that their system had autoformalized the entire ChatGPT-generated paper in Lean and could I take a look. I looked, and my post-doc Thomas Browning looked too. And indeed this was what Logical Intelligence had done: they had formalized precisely the statement that the profound theorem of number theory implied the Erdős counterexample. Breakthrough LLM-generated mathematics being formalized in real time. Interesting data point.","Of course there is an elephant in the room here though, the profound theorem of number theory which takes 100+ pages to prove (it needs huge chunks of global class field theory, a theory developed at the beginning of the 20th century and for which there are still no short proofs; it is proving difficult to compress). In 2025 I had run a Clay Summer School：https://www.claymath.org/events/formalizing-class-field-theory/ with Richard Hill on the formalization of class field theory, and one year later we have nearly done the local case (it is the current PhD project of my student Edison Xie); the global case remained open, and indeed in 2025 formalizing global class field theory seemed like a fantasy.","I was not sure how good Logos’ tool was going to be, but I wanted a development of the theory of finite flat group schemes in Lean for my ongoing proof of Fermat’s Last Theorem, so I put uploaded some classic papers in the area to Fable and ChatGPT, and got them together to write down an exposition of the theory in natural language. I passed this pdf document over to Logos the day before the workshop, and on the first day of the workshop they said that one of the claims in the pdf was false and they had found an explicit counterexample. Another counterexample! I took a look and indeed the LLM-generated pdf was simply wrong at some point when describing a standard construction; false alarm. I had missed this myself though when reading through the pdf. Interesting how AI had again found a counterexample. I fixed the pdf. I thought it was interesting that the AI didn’t just say “I don’t quite follow this argument”, it instead said “here is a proof that this argument is simply wrong”, a much more powerful statement.","With the development of the theory of finite flat group schemes back on track, I could relax back into the FLT workshop. On Tuesday 7th July I sat opposite Akhil Mathew：https://math.uchicago.edu/~amathew/ at lunch; Akhil is a professor of mathematics at UChicago and he was an attendee who had been experimenting with the tools available. We talked about potential questions which AI could work on, and Akhil raised the old question of Grothendieck about whether every finite free group scheme of order n was killed by n. Deligne had proved the result in the commutative case, and Grothendieck had proved it when the base was reduced; Rene Schoof had proved it in more cases, and there had even been a paper：https://ems.press/content/serial-article-files/51994 by Emiliano Torti published last year, proving it in even more generality. I said that I thought that this was a fabulous thing to get AI thinking about.","I think it’s worth stepping back at this point and surveying what the attitudes of human experts to these sorts of things are. On Tuesday (14th July) I went to work at Imperial and the Grothendieck counterexample was the talk of lunch. A member of the faculty (who I won’t name) said to me that the fact that the counterexample was so easy to find just indicated that humans had not spent enough time thinking about the problem, implying that a 60-year-old question of Grothendieck was not actually that interesting to work on. I didn’t tell him that at some point earlier in my career I had spent a week working hard on the problem. In my mind my colleague is just going through the five stages of grief; right now they seem to be in the denial phase.","After lunch I met with my PhD student Andrew Yang, who had been working on formalizing a modularity lifting theorem in Lean, something which is crucial to my FLT work. Andrew had come to the Logos FLT workshop and now had access to both Sol and Fable. He told me that using these tools he had written 250K lines of Lean code which basically completely finished the project in what was I guess a 2 week period.","A few days earlier I had got an email from a professor in the maths department here at Imperial, expressing surprise that some of our graduate students were paying $200 per month to access models such as Sol and Fable. He said that he thought that these people were crazy. I did not immediately respond. But after meeting with Andrew I emailed the professor back and told him that in my opinion, any PhD student who was not paying $200 per month to access these tools was crazy. In fact during the workshop I learnt from Harvard PhD student Bryan Wang that Harvard were already giving free Fable access to all PhD students, post-docs and faculty at Harvard.","But back to Akhil. I am not sure if he took my idea to disprove the Hodge conjecture seriously. But it looks like he had deeply understood that, with these extraordinary new AI tools, counterexamples might be low-hanging fruit right now. He had discussed with Levent Alpöge the idea of finding more counterexamples in algebraic geometry, and 12 hours ago Levent posted on X：https://x.com/__alpoge__/status/2079028340955197566?s=20 that Fable had found a counterexample to the Jacobian Conjecture：https://en.wikipedia.org/wiki/Jacobian_conjecture. This is a big deal — this is a famous question in algebraic geometry which had been open for 100 years and which many people had thought about. It was apparently solved during the 2026 World Cup Final.","I woke up today to a DM from Akhil saying “shall I make another PR?” but this time he was too late — Paul Lezeau had already formalized the counterexample manually and had made a PR：https://github.com/google-deepmind/formal-conjectures/pull/4474 to DeepMind’s Formal Conjectures repo：https://github.com/google-deepmind/formal-conjectures. Mathlib does not contain a large list of conjectures in mathematics, but DeepMind’s repo does. The importance of formalization of conjectures by humans is that if humans are agreed that a Lean statement does faithfully capture the idea behind a conjecture, then checking that (possibly AI-generated) Lean code does comprise a proof or disproof of the conjecture is a triviality. Congratulations to Levent, thanks to Akhil for suggesting the problem to him, and thanks to DeepMind for already having formalized the statement and thus making formal verification of the counterexample a triviality.","The Jacobian conjecture is resolved! Wow! The next step in that work is for humans to understand exactly what is going on with the example. For the true value of work like this is to give humans better understanding of mathematics. Indeed Akhil has been working on trying to understand the Grothendieck counterexample in a way which is far deeper than “here is a random presentation of a random ring and a random calculation which shows that something doesn’t work”. What we need next is the insight which can be drawn from these extraordinary examples.","Thanks for this article. You write that you are “not reading AI-generated informal mathematics” but also that “the next step in that work is for humans to understand exactly what is going on with the example.” How do we do this without reading the informal AI output? We are surely not going through a thousand-line Lean file?","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4839&_wpnonce=611d4e061b Like","I actually find it easier to go through the formal output than the informal output. With the formal output, there is no danger of imagining things or misinterpreting the argument. Most reasonable Lean developments are organized into logical progressions of lemmas, and most of the lemmas might even be obvious to someone well versed in the field (who can also read Lean, of course), leaving only a small amount of actual argumentation to read.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4848&_wpnonce=ef34c24134 Like","Well, that’s because you are versed in Lean. IMO, Lean proofs are very hard to read as a begginner, compared to, say Isabelle/HOL. If human understanding is the main objective “informal output”, which I believe is a Latex document, for example shouldn’t be relegated at all.","It will be very sad if, in the future all proofs are machine-generated in lean/rocq/whatever proof assistant only, and relegating knowledge to the handful of people that know the proof assistant AND are experts in the specific field.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4866&_wpnonce=99a1d09e1c Liked by 1 person：#","I only started using Lean three months ago; I would hardly call myself an expert. Nobody really “reads” Lean in the traditional sense of reading a static artifact. You step through the proof interactively and observe the state transformations at each step.","It takes a fair bit of training to read mathematics in the first place. Reading Lean is only a small difficulty increment on top of that, provided that one is motivated to learn.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4879&_wpnonce=2b36a92851 Like","Parsing through so much LLM and Lean output is exactly the bottleneck I think is coming this year. We’re going to be dealing with two huge issues going forwards","For newcomers to Lean especially, the bigger risk imo is formalizing something different that what they thought they were formalizing.","Both of these are fundamentally a UI/UX problem about how human mathematicians will deal with information overload and it’s exactly what we’re addressing at https://mathvision.ai/：https://mathvision.ai/ 🙂 Check it out for free if you’re interested.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4858&_wpnonce=eaf031ee8e Like","“A few days earlier I had got an email from a professor in the maths department here at Imperial, expressing surprise that some of our graduate students were paying $200 per month to access models such as Sol and Fable. He said that he thought that these people were crazy. I did not immediately respond. But after meeting with Andrew I emailed the professor back and told him that in my opinion, any PhD student who was not paying $200 per month to access these tools was crazy.”","With due respect, I find this comment disgusting. Among other problems, this money will be used for the gigantic V-sign to future generations that is building lots of extra gas power plants to supply new data centers. I won’t judge people for doing it any more than for eating red meat, and especially not in such a toughly competitive academic system, but if you think it’s crazy not to give 10% of your salary to unethical companies if it might hurt your career, then we are fundamentally not operating on the same set of values.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4840&_wpnonce=e68045bc2b Liked by 4 people：#","With the caveat that the internet and years of social media encourage us to make kneejerk responses, my instinct is to concur with JAS’s comment/reply. Aside from views one may (or may not) have about the companies building and selling these tools, the sentiment that Kevin candidly admits to is not a mentality that I want to see encouraged among those doing a PhD.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4842&_wpnonce=0e5a0eb3a1 Liked by 1 person：#","There are possibly political or social welfare reasons why one should resist the AI takeover of mathematics. But, from a strict utilitarian standpoint, these tools make you much more than 10% more productive, and are well worth the cost.","I worry about the social implications of handing over a large portion of our resources to AI companies, and the equity and access concerns for people from countries where $200 is a lot of money. Also, there is evidence that AI companies are subsidizing the present cost of subscription access and that the true cost is much higher than what we pay.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4847&_wpnonce=2bea7100ae Like","you should inform your opinion on actual numbers, data centers use a small fraction of electricity and water compared to, say agriculture. so if you’re the kind of person who judges people for eating red meat, you need to dish out that judgement proportionally for AI users.","i’ve given the exact same advice to graduate students months ago, pay for the best models, especially now that they are still affordable. contribute to proving the use case (at this point, is there even any doubt?), and get your department / advisor to pay for the models going forward.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4844&_wpnonce=b7679e2bb6 Like","idk if this is satisfactory but here’s how GPT interpreted the Jacobian example. Take P1xP2->P3 (Union of divisors on P1 say), remove ramification locus and remove a hyperplane of P3 which is tangent to a point of the map P1->P3 (tripling the divisor) but not osculating (multiplicity exactly two). Then the preimage is A3 by direct computation.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4841&_wpnonce=6833b187db Like","On a more constructive note than my previous comment/post: in the hope that some people who follow this blog as part of their interest in/commitment to formalization are also interested in understanding algebraic geometry, I want to give a signal boost to https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/：https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/ so that some maths discussion takes place there.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4843&_wpnonce=7b61124922 Like","A pretty important set of philosophical questions: what is the point of mathematics and the role of mathematicians when the task of formally proving conjectures has been outsourced to a team consisting of a large language model and a proof assistant? What’s the point of mathematical rigour when mathematicians no longer have to develop and write proofs out by hand?","I feel like whatever remains of the formalist viewpoint of mathematics has been largely disproven by developments in the past few years in artificial intelligence and proof assistants: clearly there are still activities for mathematicians in an era where large language models are the ones writing formal proofs in a proof assistant.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4851&_wpnonce=711b4dabd8 Like","“what is the point of mathematics and the role of mathematicians when the task of formally proving conjectures has been outsourced to a team consisting of a large language model and a proof assistant?”","The role of mathematicians is not just to prove conjectures, but also to formulate them. Formulating an interesting/important/useful statement that is a candidate for a theorem often requires as much creative insight (if not more) as does proving it.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4873&_wpnonce=b7139e08dd Like","For now, I agree, it is still true that humans have an edge in formulating conjectures and judging the importance of the results. But I can reasonably imagine a world where machines get better at formulating conjectures.","The next moat behind inquiry and judging is liability and responsibility. I believe there have been a number of court cases where customers interacting with AI customer service chatbots received incorrect information from those chatbots (for example, an airline’s chatbot claiming that a ticket purchase was refundable when it wasn’t). In all of these cases, the companies that those chatbots were supposedly representing were held liable for the statements that the chatbots made (e.g. the company was forced to honor the promised refund). For now, many companies are skittish about allowing their machines to assume liability on their behalf. The human is still the final backstop for liability purposes.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4881&_wpnonce=fa96b95c51 Like","I have one article (with Graber, Harris and Mazur) that disproves a conjecture about rational points that Serre posed in correspondence with Grothendieck. Our theorem proves that a counterexample to the conjecture exists (as a corollary of a more general theorem), but our method gives no explicit counterexample. Almost immediately afterward, somebody sat down and found an explicit counterexample. Honestly, that explicit counterexample could probably have been found (by a dedicated human) anytime in the four decades the conjecture was open, but the community expected the conjecture to be true. If the conjecture is true, it is a waste of resources to search for counterexamples. Now AI means that sort of “checking for counterexamples” can probably be done automatically. That is a fantastic development! As others have said, we could try to *disprove* positive characteristic resolution of singularities (by disproving some of the corollaries about specific congruences of rational points of specific varieties over specific finite fields).","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4884&_wpnonce=1c337d5605 Like","But after meeting with Andrew I emailed the professor back and told him that in my opinion, any PhD student who was not paying $200 per month to access these tools was crazy.","Maybe you intended to express enthusiasm for how good AI has gotten at mathematics. But from the perspective of a PhD student, this comes across as very tone deaf. In many parts of the world (including Europe and many places in the US), PhD students are paid a pittance and are barely making ends meet. Expecting them to fork out 200 USD/month is extremely unreasonable and unrealistic.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4867&_wpnonce=3f1d2ca959 Liked by 1 person：#","I think we’re all hoping for the Harvard approach where the institution pays for access without docking grad student pay to offset the cost.","I don’t think it’s realistic to go back to the good old days. Even if you yourself forswear AI, you can’t stop other people from using it, and those other people are your competition.","Like ：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4880&_wpnonce=37b71f919c Like","What we need next is the insight which can be drawn from these extraordinary examples."],"articleImages":[{"sourceUrl":"https://2.gravatar.com/avatar/e7ccfd42722fdb89c0fdea2f473152c14ab978bc1974acc4d67944906653f36e?s=240&d=identicon&r=G","alt":"Unknown's avatar","afterParagraph":12,"url":"/media/articles/cmrtsfndu22tebihzaf3omv19/440b1a494a6f35f8.jpg"},{"sourceUrl":"https://2.gravatar.com/avatar/b9fc79290566c7a497210d223438842d695b798b12571476f1bb4787cdfde505?s=160&d=identicon&r=G","alt":"ys's avatar","afterParagraph":12,"url":"/media/articles/cmrtsfndu22tebihzaf3omv19/3fafe2d87a80150a.jpg"},{"sourceUrl":"https://0.gravatar.com/avatar/c3b76c2d4fc37153e8924bd536d84c4bd5f858d379824a124a34d25f5da8cbd5?s=160&d=identicon&r=G","alt":"David Jao's avatar","afterParagraph":14,"url":"/media/articles/cmrtsfndu22tebihzaf3omv19/d84de2245d82ac9a.jpg"},{"sourceUrl":"https://1.gravatar.com/avatar/d588ff894539474420cc212e390ee30326c68e6732ff95cd4834cdfff2c7a2b1?s=160&d=identicon&r=G","alt":"dasc's avatar","afterParagraph":16,"url":"/media/articles/cmrtsfndu22tebihzaf3omv19/8b35198881a69c4f.jpg"},{"sourceUrl":"https://2.gravatar.com/avatar/5f84435e2b91b5a4368450d3a8be56db4934e8065aeeade3b5ba48b8de3dd4cd?s=160&d=identicon&r=G","alt":"Tanay Wakhare's avatar","afterParagraph":22,"url":"/media/articles/cmrtsfndu22tebihzaf3omv19/88d9f5f38f77232e.jpg"},{"sourceUrl":"https://1.gravatar.com/avatar/736af9802fe1ac13091d6f4e4d202307a9d80a00a2f43076596eab5496086e95?s=160&d=identicon&r=G","alt":"jeanabousamra's avatar","afterParagraph":26,"url":"/media/articles/cmrtsfndu22tebihzaf3omv19/904a1a14fc393c83.png"}],"mediaStatus":"ok","articleBodyZh":["过去几周对反例研究来说非常有趣。这篇文章基本上是我对形式化、人工智能工具，特别是反例领域所发生情况的看法。","两个月前的今天（2026年5月20日），ChatGPT 证明了离散几何中 Erdős 的单位距离猜想是错误的。这在现在已经是老消息了，但我必须从某处开始。官方公告：https://openai.com/index/model-disproves-discrete-geometry-conjecture/ 附带了人类数学家的见证，其中很多我认识，也有几位我信任，他们表示相信这一论证（他们曾提前获取过这一论证并进行了检查）。证明的基本结构是，20世纪60年代 Golod 和 Shafarevich 提出的一个重要数论定理可以用来构造该猜想的反例。","我中年危机已经过去9年，我意识到在技术细节上我不再信任许多人类数学家，发现了 Lean，并开始主张交互式定理证明器在未来数学中应发挥重要作用。因此，我的第一个问题当然是：“这个反例在 Lean 中被形式化了吗？”答案是“没有”。","但不到一周后（2026年5月26日），我收到了菲尔兹奖得主 Mike Freedman 的电子邮件。Mike 现在是 Logical Intelligence 的首席科学官：https://logicalintelligence.com/，这家公司由图灵奖得主、“人工智能教父” Yan LeCun 共同创办。Mike 告诉我，他们的系统已将整个 ChatGPT 生成的论文自动在 Lean 中形式化，问我是否可以看一看。我和我的博士后 Thomas Browning 都看了。确实，这正是 Logical Intelligence 所做的：他们形式化了数论重要定理蕴含 Erdős 反例的精确陈述。突破性的 LLM 生成数学得到了实时形式化。这是一个有趣的数据点。","当然，这里有一个不容忽视的问题，即数论中的一个深刻定理，其证明需要100多页（它需要大量的全局类域论，这是20世纪初发展起来的理论，至今仍没有简短的证明；压缩它一直非常困难）。在2025年，我与Richard Hill一起举办了一个Clay暑期学校：https://www.claymath.org/events/formalizing-class-field-theory/，研究类域论的形式化，一年后，我们几乎完成了局部情形（这是我学生Edison Xie当前的博士项目）；全局情形仍然未解决，实际上在2025年，形式化全局类域论似乎像一个幻想。","我不确定Logos的工具会有多好，但我希望在Lean中发展有限平坦群方案理论，以用于我正在进行的费马大定理证明，所以我将该领域的一些经典论文上传到Fable和ChatGPT，并将它们整理在一起，用自然语言写成理论阐述。我在研讨会前一天将这份PDF文档交给了Logos，在研讨会的第一天，他们说PDF中的一个论断是错误的，并且他们找到了一个明确的反例。又一个反例！我看了一下，确实在描述一个标准构造时，LLM生成的PDF在某点是错误的；虚惊一场。不过，当我自己阅读这份PDF时也忽略了这一点。有趣的是，AI再次找到了一个反例。我修正了PDF。我觉得有趣的是，AI不仅仅说“我不太理解这个论证”，它反而说“这是一个证明，这个论证完全错误”，这是一个更强有力的声明。","随着有限平群方案理论的发展重新上轨道，我可以放松地回到FLT工作坊。7月7日星期二，我在午餐时坐在Akhil Mathew（https://math.uchicago.edu/~amathew/）对面；Akhil是芝加哥大学的数学教授，他是一个一直在尝试可用工具的与会者。我们讨论了人工智能可能研究的潜在问题，Akhil提出了Grothendieck的老问题：是否每个阶为n的有限自由群方案都会被n消去。Deligne在交换情形下已经证明了这个结果，而Grothendieck在基是约化的情况下证明了它；Rene Schoof在更多情况下也证明了它，甚至还出现了一篇论文（https://ems.press/content/serial-article-files/51994），由Emiliano Torti去年发表，在更一般的条件下证明了这一点。我说我认为这是让人工智能思考一个极好的话题。","我认为此刻值得退一步，审视一下人类专家对这类问题的态度。星期二（7月14日）我去帝国理工工作，Grothendieck的反例成为午餐时的热门话题。一位（我不打算透露姓名的）教职人员对我说，反例如此容易找到，仅仅表明人类尚未花足够的时间思考这个问题，这暗示Grothendieck提出的一个60年历史的问题实际上并不那么值得研究。我没有告诉他，我职业生涯早期曾花了一周时间努力研究这个问题。在我看来，我的同事只是在经历悲伤的五个阶段；目前他们似乎处于否认阶段。","午餐后，我与我的博士生Andrew Yang会面，他一直在用Lean形式化一个模性提升定理，这是我FLT工作的关键。Andrew参加了Logos FLT工作坊，现在可以使用Sol和Fable。他告诉我，使用这些工具，他写了25万行Lean代码，基本上在大约两周内就完成了整个项目。","几天前，我收到了帝国理工学院数学系一位教授的电子邮件，他对于我们的一些研究生每月支付200美元来使用Sol和Fable等模型感到惊讶。他说他认为这些人疯了。我没有立即回复。但在见过Andrew之后，我给那位教授回了邮件，并告诉他，在我看来，任何没有每月支付200美元来使用这些工具的博士生都是疯了。事实上，在研讨会期间，我从哈佛的博士生Bryan Wang那里了解到，哈佛已经为所有博士生、博士后和教职员工免费提供Fable访问权限。","但回到Akhil。我不确定他是否认真对待了我提出反驳霍奇猜想的想法。但看起来他已经深刻理解到，利用这些非凡的新AI工具，反例可能正是目前唾手可得的果实。他曾与Levent Alpöge讨论过在代数几何中寻找更多反例的想法，而12小时前，Levent在X上发布：https://x.com/__alpoge__/status/2079028340955197566?s=20，Fable发现了雅可比猜想的一个反例：https://en.wikipedia.org/wiki/Jacobian_conjecture。这是一件大事——这是代数几何中的一个著名问题，已经未解答达100年，许多人都曾思考过。显然，它是在2026年世界杯决赛期间被解决的。","今天早上，我醒来收到了Akhil发来的私信，说“我应该再提交一个PR吗？”，但这次他来晚了——Paul Lezeau已经手动形式化了这个反例，并在DeepMind的Formal Conjectures仓库中提交了一个PR：https://github.com/google-deepmind/formal-conjectures/pull/4474：https://github.com/google-deepmind/formal-conjectures。Mathlib中没有包含大量数学猜想的列表，但DeepMind的仓库有。由人类形式化猜想的重要性在于，如果人类一致认为某个Lean语句忠实地表达了一个猜想的核心思想，那么检查该（可能由AI生成的）Lean代码是否构成该猜想的证明或反例就变得微不足道。恭喜Levent，感谢Akhil向他提出这个问题，并感谢DeepMind已经形式化了该陈述，从而使反例的正式验证变得毫不费力。","雅可比猜想已被解决！哇！接下来的工作是让人类确切地理解这个例子中发生了什么。因为这类工作的真正价值在于让人类更好地理解数学。事实上，Akhil 一直在努力以比“这是一个随机环的随机表示和随机计算，显示某些东西不起作用”更深的方式理解格罗滕迪克的反例。接下来我们需要的是从这些非凡的例子中获得的洞察。","感谢这篇文章。你写道你“没有在阅读 AI 生成的非正式数学”，但也写道“接下来的工作是让人类确切理解这个例子中发生了什么。”我们如何在不阅读非正式 AI 输出的情况下做到这一点？我们肯定不会去看一个千行的 Lean 文件吧？","链接：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4839&_wpnonce=611d4e061b 赞","我实际上觉得阅读形式化输出比非正式输出更容易。使用形式化输出时，不会有想象或者误解论证的危险。大多数合理的 Lean 开发都是按照引理的逻辑进展组织的，并且大多数引理对于熟悉该领域的人（当然也能读 Lean）来说可能显而易见，只剩下一小部分实际论证需要阅读。","链接：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4848&_wpnonce=ef34c24134 赞","嗯，那是因为你熟悉 Lean。在我看来，对初学者来说，阅读 Lean 证明非常困难，相比之下比如 Isabelle/HOL 会更容易。如果人类理解是主要目标，“非正式输出”，比如我认为是 Latex 文档，不应该被忽视。","如果未来所有证明都仅由 Lean/Rocq/任意证明助手生成，而知识只掌握在少数既懂证明助手又精通某个特定领域的人手中，那将非常悲哀。","链接：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4866&_wpnonce=99a1d09e1c 被 1 人点赞：#","我只开始使用 Lean 三个月；我很难称自己为专家。其实没有人会像传统意义上那样“阅读” Lean 的静态文档。你需要互动地逐步进行证明，并观察每一步的状态变化。","首先，要阅读数学就需要相当多的训练。阅读 Lean 只是基于此的一小步困难，只要你有学习的动力。","像：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4879&_wpnonce=2b36a92851 喜欢","解析如此大量的 LLM 和 Lean 输出正是我认为今年会出现的瓶颈。今后我们将面临两个巨大问题。","我认为，对于 Lean 的新手来说，更大的风险是形式化的内容与他们原本认为的不同。","这两者从根本上是关于用户界面/用户体验的问题，涉及到人类数学家如何应对信息过载，这正是我们在 https://mathvision.ai/ 解决的问题：https://mathvision.ai/ 🙂 如果你感兴趣，可以免费查看。","像：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4858&_wpnonce=eaf031ee8e 喜欢","“几天前，我收到帝国理工数学系一位教授的电子邮件，他对一些研究生每月支付 200 美元访问 Sol 和 Fable 这样的模型表示惊讶。他说，他认为这些人疯了。我当时没有立即回复。但在与 Andrew 会面后，我给教授回了邮件，告诉他在我看来，任何不每月支付 200 美元使用这些工具的博士生都是疯了。”","恕我直言，我觉得这条评论让人反感。除了其他问题，这笔钱将被用来向未来几代人竖起巨大的“V”手势，也就是建立大量额外的燃气发电厂来为新的数据中心供电。我不会因为别人这么做而评判他们，就像我不会因为别人吃红肉而评判一样，尤其是在这样一个竞争激烈的学术体系中，但如果你认为如果可能会影响职业生涯而不把工资的10%捐给不道德的公司是疯狂的，那么我们根本不是在同一套价值观下运作的。","点赞：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4840&_wpnonce=e68045bc2b 4人点赞：#","在提示的条件下，即互联网和多年的社交媒体鼓励我们做出本能反应，我的直觉是同意JAS的评论/回复。除了人们可能（或可能不）对构建和销售这些工具的公司有的看法之外，Kevin坦率承认的这种心态并不是我希望看到在攻读博士的人中被鼓励的。","点赞：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4842&_wpnonce=0e5a0eb3a1 1人点赞：#","可能有政治或社会福利方面的理由让人抵制人工智能对数学的接管。但是，从严格的功利主义角度来看，这些工具让你的生产力提高远超过10%，并且完全值得其成本。","我担心把我们大量资源交给AI公司所带来的社会影响，以及对来自那些200美元就是大数目的国家的人在公平性和获取上的担忧。此外，有证据表明，AI公司正在补贴当前订阅访问的费用，而实际成本远高于我们支付的费用。","点赞：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4847&_wpnonce=2bea7100ae 点赞","你应该根据实际数字来形成你的观点，数据中心使用的电力和水只占很小的一部分，比如与农业相比。所以，如果你是那种会因为别人吃红肉而评判别人的人，你也需要对AI用户进行相应比例的评判。","几个月前我就给研究生们提过同样的建议，支付最好的模型费用，尤其是现在它们仍然负担得起。参与证明使用案例（此时，还有疑问吗？），并让你的系/导师负责今后的模型费用。","点赞：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4844&_wpnonce=b7679e2bb6 点赞","我不知道这是否让人满意，但这是GPT对Jacobian例子的理解方法。取P1×P2->P3（例如P1上除子的联合），去掉分支点集并去掉P3的一个与映射P1->P3的某个点相切（除子三倍化）但不振荡（重数正好为二）的超平面。然后通过直接计算，其逆像是A3。","点赞：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4841&_wpnonce=6833b187db 点赞","比起我之前的评论/帖子更建设性一点：希望那些出于对形式化兴趣/承诺而关注此博客的人，也对理解代数几何感兴趣，我想给https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/ 提个信号，使这里能展开一些数学讨论。","点赞：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4843&_wpnonce=7b61124922 点赞","一组相当重要的哲学问题：当正式证明猜想的任务已经外包给由大型语言模型和证明助手组成的团队时，数学的意义和数学家的角色是什么？当数学家不再需要手动开展和撰写证明时，数学严谨性还有什么意义？","我觉得，数学形式主义观点的残余，在过去几年人工智能和证明助手的发展中已经基本被否定：显然，在大型语言模型在证明助手中撰写正式证明的时代，数学家仍然有他们的活动空间。","喜欢：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4851&_wpnonce=711b4dabd8 喜欢","“当正式证明猜想的任务已经被外包给一个由大型语言模型和证明助手组成的团队时，数学的意义和数学家的角色是什么？”","数学家的角色不仅仅是证明猜想，还包括提出猜想。提出一个有趣的/重要的/有用的、可能成为定理的陈述，通常需要与证明它同样多的创造性洞察（甚至更多）。","喜欢：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4873&_wpnonce=b7139e08dd 喜欢","目前，我同意，人类在提出猜想和判断结果的重要性方面仍有优势。但我可以合理地设想一个机器在提出猜想方面变得更好的世界。","下一个围绕探究和判断的护城河是责任和义务。我相信已经有一些法院案例，客户与人工智能客服聊天机器人互动时，从这些聊天机器人那里获得了错误的信息（例如，某航空公司的聊天机器人声称票务购买可以退款，而实际上不可以）。在所有这些案例中，那些聊天机器人所代表的公司都对聊天机器人所作的声明承担责任（例如，公司被迫履行承诺的退款）。目前，许多公司对允许机器代表它们承担责任持谨慎态度。人类仍然是责任目的上的最终保障。","喜欢：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4881&_wpnonce=fa96b95c51 喜欢","我有一篇文章（与Graber、Harris和Mazur合著）反驳了Serre在与Grothendieck通信中提出的一个关于有理点的猜想。我们的定理证明了这个猜想存在反例（作为一个更一般定理的推论），但我们的方法并没有给出具体的反例。几乎在随后的不久，就有人坐下来找到了一个具体的反例。说实话，这个具体反例可能在这个猜想打开的四十年里任何时候都可以被找到（如果有一个专注的人去寻找的话），但学术界预期这个猜想是真的。如果猜想是真的，那么去寻找反例就是浪费资源。现在人工智能意味着这种“寻找反例”的工作可能可以自动完成。这是一个非常棒的进展！正如其他人所说，我们可以尝试*反驳*正特征下的奇点解析（通过反驳关于特定有限域上特定簇的有理点的一些推论）。","喜欢： https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4884&_wpnonce=1c337d5605 喜欢","但在与Andrew会面后，我给教授发了邮件，告诉他在我看来，任何没有每月支付200美元访问这些工具的博士生都是疯狂的。","也许你本意是想表达对人工智能在数学上取得的进展感到兴奋。但从博士生的角度来看，这听起来非常不合时宜。在世界上的许多地区（包括欧洲和美国的许多地方），博士生的工资微薄，几乎难以维持生计。期望他们每月支付200美元是非常不合理且不切实际的。","喜欢： https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4867&_wpnonce=3f1d2ca959 由 1 个人点赞：#","我认为我们都希望哈佛式的做法，即机构支付访问费用，而不是从研究生工资中扣除以抵消成本。","我认为回到过去的美好时代是不现实的。即便你自己放弃使用人工智能，你也无法阻止其他人使用，而这些其他人正是你的竞争对手。","喜欢：https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/?like_comment=4880&_wpnonce=37b71f919c 点赞","我们接下来需要的是可以从这些非凡例子中得出的洞察。"],"translationStatus":"translated","bodyOrigin":"source-page","editorial":{"summary":"Aioga 编辑摘要：2026 年 5 月，ChatGPT 推翻了离散几何中的 Erdős 单位距离猜想，构造出反例。 Aioga 将其归入「技巧观点」方向，重点关注它对真实使用和行业竞争的影响。","background":"背景分析：实践类内容的价值在于是否能被复现、是否有明确边界，以及它能否转化为稳定的开发或工作流方法。","viewpoint":"Aioga 判断：这条动态更适合作为行业观察信号，当前信息足以建立线索，但不足以推导长期结论。","implications":"影响分析：对相关团队而言，短期应先核对来源、可用范围和实际成本，再判断是否值得接入或跟进。","nextStep":"后续观察：继续观察示例是否可复现、工具版本变化、社区反馈和实际成本。","evidenceRefs":["title","summary","articleBody"],"confidence":"medium","status":"published","aiGenerated":false,"autoApproved":true,"generatedBy":"rule-safe-fallback","generatedAt":"2026-07-23T06:49:19.095Z","sourceHash":"b490d4cf76aea890","validation":{"passed":true,"mode":"rule-safe-fallback","checks":["schema","length","source-attribution","no-html"]}},"tags":["技巧观点","Hacker News 热门（buzzing.cc 中文翻译）"],"translations":{"zh-CN":{"title":"ChatGPT 用反例推翻 Erdős 单位距离猜想，AI 自动形式化验证紧随其后","summary":"2026 年 5 月，ChatGPT 推翻了离散几何中的 Erdős 单位距离猜想，构造出反例。一周后，Logical Intelligence 的 AI 系统将该证明自动形式化为 Lean 代码。6 月，OpenAI 的 Sol 模型在无任何数学公理假设下完成了完整形式化，生成 120 万行 Lean 代码，证明中涉及百页以上的全局类域论定理。","category":"技巧观点","source":"xenaproject.wordpress.com","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT 用反例推翻 Erdős 单位距离猜想，AI 自动形式化验证紧随其后 - Aioga AI资讯","description":"2026 年 5 月，ChatGPT 推翻了离散几何中的 Erdős 单位距离猜想，构造出反例。一周后，Logical Intelligence 的 AI 系统将该证明自动形式化为 Lean 代码。6 月，OpenAI 的 Sol 模型在无任何数学公理假设下完成了完整形式化，生成 120 万行 Lean 代码，证明中涉及百页以上的全局类域论定理。","url":"https://www.aioga.com/news/cmrtsfndu22tebihzaf3omv19/"},"en":{"title":"ChatGPT refutes the Erdős unit distance conjecture with a counterexample, and AI automatically formal verification follows closely.","summary":"In May 2026, ChatGPT overturned the Erdős unit distance conjecture in discrete geometry by constructing a counterexample. A week later, the AI system of Logical Intelligence automatically formalized this proof into Lean code. In June, OpenAI's Sol model completed a full formalization without assuming any mathematical axioms, generating 1.2 million lines of Lean code, with the proof involving hundreds of pages of global class field theory theorems.","category":"Insights","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT refutes the Erdős unit distance conjecture with a counterexample, and AI automatically formal verification follows closely. - Aioga AI News","description":"In May 2026, ChatGPT overturned the Erdős unit distance conjecture in discrete geometry by constructing a counterexample. A week later, the AI system of Logical Intelligence automa...","url":"https://www.aioga.com/en/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:04:46.540Z"},"ja":{"title":"ChatGPT は反例を用いて Erdős の単位距離予想を覆し、AI が自動で形式的検証を続ける","summary":"2026年5月、ChatGPTは離散幾何におけるErdősの単位距離予想を覆し、反例を構築した。一週間後、Logical IntelligenceのAIシステムはその証明を自動的にLeanコードとして形式化した。6月、OpenAIのSolモデルは任意の数学公理の仮定なしに完全な形式化を達成し、120万行のLeanコードを生成し、証明には百ページ以上に及ぶグローバル類体論の定理が含まれていた。","category":"ヒントと視点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT は反例を用いて Erdős の単位距離予想を覆し、AI が自動で形式的検証を続ける - Aioga AIニュース","description":"2026年5月、ChatGPTは離散幾何におけるErdősの単位距離予想を覆し、反例を構築した。一週間後、Logical IntelligenceのAIシステムはその証明を自動的にLeanコードとして形式化した。6月、OpenAIのSolモデルは任意の数学公理の仮定なしに完全な形式化を達成し、120万行のLeanコードを生成し、証明には百ページ以上に及ぶグロ...","url":"https://www.aioga.com/ja/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:04:52.806Z"},"ko":{"title":"ChatGPT가 반례로 Erdős 단위 거리 추측을 뒤집고, AI가 자동 형식 검증을 뒤따랐다","summary":"2026년 5월, ChatGPT는 이산 기하학에서 Erdős 단위 거리 추측을 뒤집고 반례를 구성했다. 일주일 후, Logical Intelligence의 AI 시스템은 해당 증명을 자동으로 Lean 코드로 형식화했다. 6월에는 OpenAI의 Sol 모델이 어떠한 수학 공리 가정 없이 전체 형식화를 완료하여 120만 줄의 Lean 코드를 생성했고, 증명에는 수백 페이지에 달하는 전역 클래스 필드 이론 정리가 포함되었다.","category":"인사이트","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT가 반례로 Erdős 단위 거리 추측을 뒤집고, AI가 자동 형식 검증을 뒤따랐다 - Aioga AI 뉴스","description":"2026년 5월, ChatGPT는 이산 기하학에서 Erdős 단위 거리 추측을 뒤집고 반례를 구성했다. 일주일 후, Logical Intelligence의 AI 시스템은 해당 증명을 자동으로 Lean 코드로 형식화했다. 6월에는 OpenAI의 Sol 모델이 어떠한 수학 공리 가정 없이 전체 형식화를 완료하여 120만 줄의...","url":"https://www.aioga.com/ko/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:05:43.584Z"},"es":{"title":"ChatGPT refutó la conjetura de Erdős sobre distancias unitarias con un contraejemplo, y la verificación formal automática por IA siguió de inmediato","summary":"En mayo de 2026, ChatGPT refutó la conjetura de la unidad de Erdős en geometría discreta, construyendo un contraejemplo. Una semana después, el sistema de IA de Logical Intelligence formalizó automáticamente esta demostración en código Lean. En junio, el modelo Sol de OpenAI completó la formalización completa sin asumir ningún axioma matemático, generando 1,2 millones de líneas de código Lean, y la demostración involucró teoremas de teoría de campos globales de más de cien páginas.","category":"Ideas","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT refutó la conjetura de Erdős sobre distancias unitarias con un contraejemplo, y la verificación formal automática por IA siguió de inmediato - Aioga Noticias de IA","description":"En mayo de 2026, ChatGPT refutó la conjetura de la unidad de Erdős en geometría discreta, construyendo un contraejemplo. Una semana después, el sistema de IA de Logical Intelligenc...","url":"https://www.aioga.com/es/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:05:38.429Z"},"fr":{"title":"ChatGPT a réfuté la conjecture des distances unitaires d'Erdős avec un contre-exemple, la vérification formelle automatisée par l'IA a suivi","summary":"En mai 2026, ChatGPT a renversé la conjecture de distance unité d'Erdős en géométrie discrète en construisant un contre-exemple. Une semaine plus tard, le système d'IA de Logical Intelligence a automatiquement formalisé cette preuve en code Lean. En juin, le modèle Sol d'OpenAI a réalisé une formalisation complète sans aucune hypothèse d'axiome mathématique, générant 1,2 million de lignes de code Lean, la preuve incluant des théorèmes de théorie des corps globaux de plus de cent pages.","category":"Analyses","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT a réfuté la conjecture des distances unitaires d'Erdős avec un contre-exemple, la vérification formelle automatisée par l'IA a suivi - Aioga Actualités IA","description":"En mai 2026, ChatGPT a renversé la conjecture de distance unité d'Erdős en géométrie discrète en construisant un contre-exemple. Une semaine plus tard, le système d'IA de Logical I...","url":"https://www.aioga.com/fr/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:06:42.057Z"},"de":{"title":"ChatGPT widerlegt die Erdős-Vermutung über Einheitsabstände mit einem Gegenbeispiel, die KI-gestützte automatische Formalisierung folgt unmittelbar danach","summary":"Im Mai 2026 widerlegte ChatGPT die Erdős-Einheitsabstandsvermutung in der diskreten Geometrie und konstruierte ein Gegenbeispiel. Eine Woche später formalisierte das AI-System von Logical Intelligence den Beweis automatisch in Lean-Code. Im Juni vollzog das Sol-Modell von OpenAI die vollständige Formalisierung ohne jegliche Annahmen mathematischer Axiome und erzeugte 1,2 Millionen Zeilen Lean-Code, wobei der Beweis auch auf globale klassengruppentheoretische Theoreme von über hundert Seiten Bezug nahm.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT widerlegt die Erdős-Vermutung über Einheitsabstände mit einem Gegenbeispiel, die KI-gestützte automatische Formalisierung folgt unmittelbar danach - Aioga KI-News","description":"Im Mai 2026 widerlegte ChatGPT die Erdős-Einheitsabstandsvermutung in der diskreten Geometrie und konstruierte ein Gegenbeispiel. Eine Woche später formalisierte das AI-System von...","url":"https://www.aioga.com/de/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:06:39.957Z"},"pt-BR":{"title":"ChatGPT refuta a Conjectura da Distância Unitária de Erdős com um contraexemplo, e a verificação formal automática pela IA segue em seguida","summary":"Em maio de 2026, o ChatGPT derrubou a Conjectura da Distância Unitária de Erdős na geometria discreta, construindo um contraexemplo. Uma semana depois, o sistema de IA da Logical Intelligence formalizou automaticamente essa prova em código Lean. Em junho, o modelo Sol da OpenAI completou a formalização completa sem assumir nenhum axioma matemático, gerando 1,2 milhões de linhas de código Lean, envolvendo em sua prova teoremas de teoria de campos globais com mais de cem páginas.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT refuta a Conjectura da Distância Unitária de Erdős com um contraexemplo, e a verificação formal automática pela IA segue em seguida - Aioga Notícias de IA","description":"Em maio de 2026, o ChatGPT derrubou a Conjectura da Distância Unitária de Erdős na geometria discreta, construindo um contraexemplo. Uma semana depois, o sistema de IA da Logical I...","url":"https://www.aioga.com/pt-BR/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:07:35.368Z"},"ru":{"title":"ChatGPT опроверг гипотезу Эрдеша о единичном расстоянии с помощью контрпримера, за чем последовала автоматическая формальная проверка ИИ","summary":"В мае 2026 года ChatGPT опроверг гипотезу Эрдёша о единичных расстояниях в дискретной геометрии, построив контрпример. Через неделю система ИИ Logical Intelligence автоматически формализовала это доказательство в коде Lean. В июне модель Sol от OpenAI завершила полную формализацию без каких-либо допущений математических аксиом, создав 1,2 миллиона строк кода Lean, в доказательстве были задействованы глобальные теоремы теории классов, занимающие более сотни страниц.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT опроверг гипотезу Эрдеша о единичном расстоянии с помощью контрпримера, за чем последовала автоматическая формальная проверка ИИ - Aioga Новости ИИ","description":"В мае 2026 года ChatGPT опроверг гипотезу Эрдёша о единичных расстояниях в дискретной геометрии, построив контрпример. Через неделю система ИИ Logical Intelligence автоматически фо...","url":"https://www.aioga.com/ru/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:07:24.762Z"},"ar":{"title":"ChatGPT يستخدم الأمثلة المضادة لدحض تخمين إردوش حول المسافة الواحدة، والتحقق الرسمي الآلي بواسطة الذكاء الاصطناعي يتبع ذلك","summary":"في مايو 2026، قلب ChatGPT فرضية إيردوش لمسافة الوحدة في الهندسة المتقطعة من خلال بناء مثال مضاد. بعد أسبوع، قام نظام الذكاء الاصطناعي لـ Logical Intelligence بتحويل هذا الإثبات تلقائيًا إلى كود Lean. في يونيو، أكمل نموذج Sol التابع لـ OpenAI التشكيل الكامل بدون أي افتراضات رياضية أساسية، منتجًا 1.2 مليون سطر من كود Lean، وشمل الإثبات نظريات واسعة في نظرية المجالات تتجاوز المئة صفحة.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT يستخدم الأمثلة المضادة لدحض تخمين إردوش حول المسافة الواحدة، والتحقق الرسمي الآلي بواسطة الذكاء الاصطناعي يتبع ذلك - Aioga أخبار الذكاء الاصطناعي","description":"في مايو 2026، قلب ChatGPT فرضية إيردوش لمسافة الوحدة في الهندسة المتقطعة من خلال بناء مثال مضاد. بعد أسبوع، قام نظام الذكاء الاصطناعي لـ Logical Intelligence بتحويل هذا الإثبات تلق...","url":"https://www.aioga.com/ar/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:08:25.625Z"},"hi":{"title":"ChatGPT ने प्रतिक उदाहरण का उपयोग करके Erdős एकाई दूरी अनुमान को खारिज कर दिया, इसके तुरंत बाद AI ने स्वतः रूप से औपचारिक सत्यापन किया","summary":"2026 मई में, ChatGPT ने विविक्त ज्यामिति में एर्डोस यूनिट दूरी अनुमान को पलट दिया और एक विरोधाभासी उदाहरण का निर्माण किया। एक सप्ताह बाद, Logical Intelligence की AI प्रणाली ने उस प्रमाण को स्वचालित रूप से Lean कोड में रूपांतरित किया। जून में, OpenAI का Sol मॉडल बिना किसी गणितीय सूत्र की पूर्वधारणा के पूरे रूप में रूपांतरित करने में सफल रहा, 1.2 मिलियन पंक्तियों का Lean कोड उत्पन्न किया, जिसमें प्रमाण में सैकड़ों पृष्ठों के वैश्विक क्षेत्रीय सिद्धांत शामिल थे।","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT ने प्रतिक उदाहरण का उपयोग करके Erdős एकाई दूरी अनुमान को खारिज कर दिया, इसके तुरंत बाद AI ने स्वतः रूप से औपचारिक सत्यापन किया - Aioga AI समाचार","description":"2026 मई में, ChatGPT ने विविक्त ज्यामिति में एर्डोस यूनिट दूरी अनुमान को पलट दिया और एक विरोधाभासी उदाहरण का निर्माण किया। एक सप्ताह बाद, Logical Intelligence की AI प्रणाली ने उस प...","url":"https://www.aioga.com/hi/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:08:27.215Z"},"it":{"title":"ChatGPT confuta la congettura delle distanze unitarie di Erdős con un controesempio, con la verifica formale automatica dell'AI a seguire","summary":"A maggio 2026, ChatGPT ha smentito la congettura della distanza unitaria di Erdős nella geometria discreta, costruendo un controesempio. Una settimana dopo, il sistema AI di Logical Intelligence ha automaticamente formalizzato quella dimostrazione in codice Lean. A giugno, il modello Sol di OpenAI ha completato la formalizzazione completa senza alcuna ipotesi di assioma matematico, generando 1,2 milioni di righe di codice Lean, con dimostrazioni che coinvolgevano teoremi di teoria dei campi globali di oltre cento pagine.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT confuta la congettura delle distanze unitarie di Erdős con un controesempio, con la verifica formale automatica dell'AI a seguire - Aioga Notizie IA","description":"A maggio 2026, ChatGPT ha smentito la congettura della distanza unitaria di Erdős nella geometria discreta, costruendo un controesempio. Una settimana dopo, il sistema AI di Logica...","url":"https://www.aioga.com/it/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:09:19.963Z"},"nl":{"title":"ChatGPT weerlegt het Erdős-eenheidsafstand vermoeden met tegenvoorbeelden, AI volgt automatisch met formele verificatie","summary":"In mei 2026 verwierp ChatGPT het Erdős-eenheidsafstandsguess in de discrete meetkunde en construeerde een tegenvoorbeeld. Een week later formaliseerde het AI-systeem van Logical Intelligence dit bewijs automatisch naar Lean-code. In juni voltooide het Sol-model van OpenAI de volledige formalisering zonder enige veronderstelling van wiskundige axioma's, genererend 1,2 miljoen regels Lean-code, waarbij het bewijs meer dan honderd pagina's aan globale klassenveldtellingen omvatte.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT weerlegt het Erdős-eenheidsafstand vermoeden met tegenvoorbeelden, AI volgt automatisch met formele verificatie - Aioga AI-nieuws","description":"In mei 2026 verwierp ChatGPT het Erdős-eenheidsafstandsguess in de discrete meetkunde en construeerde een tegenvoorbeeld. Een week later formaliseerde het AI-systeem van Logical In...","url":"https://www.aioga.com/nl/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:09:09.798Z"},"tr":{"title":"ChatGPT, Erdős birim mesafe varsayımını çürütmek için karşı örnek kullandı, AI ardından otomatik biçimsel doğrulama gerçekleştirdi","summary":"2026 Mayıs ayında, ChatGPT ayrık geometrideki Erdős birim mesafe varsayımını çürüttü ve bir karşı örnek oluşturdu. Bir hafta sonra, Logical Intelligence’ın yapay zeka sistemi bu kanıtı otomatik olarak Lean koduna dönüştürdü. Haziran ayında, OpenAI’ın Sol modeli herhangi bir matematiksel aksiyom varsayımı olmadan tam bir formalizasyon gerçekleştirdi, 1,2 milyon satır Lean kodu üretti ve kanıt, yüz sayfayı aşan global sınıf alan teorisi teoremlerini içeriyordu.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT, Erdős birim mesafe varsayımını çürütmek için karşı örnek kullandı, AI ardından otomatik biçimsel doğrulama gerçekleştirdi - Aioga AI Haberleri","description":"2026 Mayıs ayında, ChatGPT ayrık geometrideki Erdős birim mesafe varsayımını çürüttü ve bir karşı örnek oluşturdu. Bir hafta sonra, Logical Intelligence’ın yapay zeka sistemi bu ka...","url":"https://www.aioga.com/tr/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:10:12.144Z"},"vi":{"title":"ChatGPT dùng phản ví dụ để bác bỏ giả thuyết khoảng cách đơn vị của Erdős, AI tự động xác minh hình thức theo sau ngay lập tức","summary":"Vào tháng 5 năm 2026, ChatGPT đã bác bỏ giả thuyết về khoảng cách đơn vị Erdős trong hình học rời rạc, xây dựng ra phản ví dụ. Một tuần sau, hệ thống AI của Logical Intelligence đã tự động hình thức hóa chứng minh đó thành mã Lean. Tháng 6, mô hình Sol của OpenAI đã hoàn thành việc hình thức hóa đầy đủ mà không dựa trên bất kỳ tiên đề toán học nào, tạo ra 1,2 triệu dòng mã Lean, trong đó chứng minh một định lý lý thuyết trường toàn cục dài hơn trăm trang.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT dùng phản ví dụ để bác bỏ giả thuyết khoảng cách đơn vị của Erdős, AI tự động xác minh hình thức theo sau ngay lập tức - Tin tức AI Aioga","description":"Vào tháng 5 năm 2026, ChatGPT đã bác bỏ giả thuyết về khoảng cách đơn vị Erdős trong hình học rời rạc, xây dựng ra phản ví dụ. Một tuần sau, hệ thống AI của Logical Intelligence đã...","url":"https://www.aioga.com/vi/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:10:07.982Z"},"id":{"title":"ChatGPT menggunakan contoh kontra untuk menolak dugaan jarak satu unit Erdős, verifikasi formal otomatis AI mengikuti segera setelahnya","summary":"Pada Mei 2026, ChatGPT membantah dugaan jarak tunggal Erdős dalam geometri diskret dengan membangun contoh kontra. Seminggu kemudian, sistem AI Logical Intelligence secara otomatis memformalkan bukti tersebut menjadi kode Lean. Pada bulan Juni, model Sol milik OpenAI menyelesaikan formalisasi lengkap tanpa asumsi aksioma matematika apapun, menghasilkan 1,2 juta baris kode Lean, dengan bukti yang melibatkan teorema teori medan kelas global lebih dari seratus halaman.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT menggunakan contoh kontra untuk menolak dugaan jarak satu unit Erdős, verifikasi formal otomatis AI mengikuti segera setelahnya - Berita AI Aioga","description":"Pada Mei 2026, ChatGPT membantah dugaan jarak tunggal Erdős dalam geometri diskret dengan membangun contoh kontra. Seminggu kemudian, sistem AI Logical Intelligence secara otomatis...","url":"https://www.aioga.com/id/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:10:54.467Z"},"th":{"title":"ChatGPT ใช้ตัวอย่างขัดแย้งโต้แย้งสมมติฐานระยะห่างหน่วยของ Erdős โดยการตรวจสอบเชิงรูปแบบอัตโนมัติโดย AI ตามมาทีหลัง","summary":"ในเดือนพฤษภาคม 2026, ChatGPT ล้มล้างสมมติฐานระยะทางหน่วยของ Erdős ในเรขาคณิตไม่ต่อเนื่อง และสร้างตัวอย่างโต้แย้ง หนึ่งสัปดาห์ต่อมา ระบบ AI ของ Logical Intelligence ทำการสร้างหลักฐานนั้นให้อยู่ในรูปแบบรหัส Lean โดยอัตโนมัติ ในเดือนมิถุนายน โมเดล Sol ของ OpenAI ทำการสร้างหลักฐานอย่างสมบูรณ์โดยไม่ต้องอ้างอิงสมมติฐานคณิตศาสตร์ใด ๆ ผลลัพธ์คือการสร้างรหัส Lean จำนวน 1.2 ล้านบรรทัด ซึ่งในหลักฐานมีทฤษฎีชั้นสากลของทฤษฎีสนามมากกว่าร้อยหน้า","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT ใช้ตัวอย่างขัดแย้งโต้แย้งสมมติฐานระยะห่างหน่วยของ Erdős โดยการตรวจสอบเชิงรูปแบบอัตโนมัติโดย AI ตามมาทีหลัง - ข่าว AI Aioga","description":"ในเดือนพฤษภาคม 2026, ChatGPT ล้มล้างสมมติฐานระยะทางหน่วยของ Erdős ในเรขาคณิตไม่ต่อเนื่อง และสร้างตัวอย่างโต้แย้ง หนึ่งสัปดาห์ต่อมา ระบบ AI ของ Logical Intelligence ทำการสร้างหลักฐา...","url":"https://www.aioga.com/th/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:11:07.402Z"},"pl":{"title":"ChatGPT obala przypuszczenie Erdosza dotyczące jednostkowej odległości przy użyciu kontrprzykładów, a automatyczna weryfikacja formalna AI następuje zaraz potem","summary":"W maju 2026 roku ChatGPT obalił hipotezę jednostkowej odległości Eryda w geometrii dyskretnej, konstruując kontrprzykład. Tydzień później system AI Logical Intelligence automatycznie sformalizował ten dowód w kodzie Lean. W czerwcu model Sol od OpenAI ukończył pełną formalizację bez żadnych założeń aksjomatycznych matematyki, generując 1,2 miliona linii kodu Lean, przy dowodzie obejmującym ponad sto stron twierdzeń z globalnej teorii ciał klasowych.","category":"技巧观点","source":"Hacker News 热门（buzzing.cc 中文翻译）","aggregationSource":"Hacker News 热门（buzzing.cc 中文翻译）","pageTitle":"ChatGPT obala przypuszczenie Erdosza dotyczące jednostkowej odległości przy użyciu kontrprzykładów, a automatyczna weryfikacja formalna AI następuje zaraz potem - Aioga Wiadomości AI","description":"W maju 2026 roku ChatGPT obalił hipotezę jednostkowej odległości Eryda w geometrii dyskretnej, konstruując kontrprzykład. Tydzień później system AI Logical Intelligence automatyczn...","url":"https://www.aioga.com/pl/news/cmrtsfndu22tebihzaf3omv19/","contentTranslated":true,"sourceHash":"7d7c61195f420e82","translatedAt":"2026-07-22T19:11:55.235Z"}}}}