我受不了那场没完没了的壳游戏：「哦当然，AI 现在当然能〔逃出沙箱／解千禧难题／它最近刚干的那件轰动事〕，从来没人否认过这个〔我否认过〕，等 AI 做到〔它还没做、但明年就会做的那件事〕再叫醒我，那时候我才会重估我的整个世界观〔其实不会〕。」无论过山车加速得多快，哪怕你熟悉的整个世界都已经消失在身后，你还在发明理由说这不算数。
我在 AI 上的立场，不过就是 2006 年那个保守、怀疑的立场，带着知识上的诚实，按 2026 年末的现实更新了一遍。这个立场，如果非要我把它写出来，是这样的：
总之，下面是过去差不多一个月里、除 Navier-Stokes 之外由 AI 证明或辅助得到的重要结果的一小份抽样——我只列那些解决了我原本就知道或在意的长悬难题的。
- 当然有雅可比猜想的反例，由 Levent Alpöge 在一条如今很有名的推文里宣布：「hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final」（后面跟着反例的具体写法）
- 量子态影子层析的改进上界，来自 Chen、O'Donnell、Pelecanos 和 Wright，把对 Hilbert 空间维数 d 的依赖从 log(d) 改进到 √log(d)。（我在 2017 年提出影子层析时，问过能不能把对 d 的依赖完全去掉，同时保住对测量次数 m 的 polylog 依赖。）
- Aaronson-Ambainis 猜想的进展（直接谈量子算法的那个版本），基本上证明了它对「在少数几轮并行里发出查询」的量子算法成立。（更新：不好意思，Jordan Docter 提醒我，这一条是 AI 之前的工作，AI 只用于校对之类的零碎事！）这一结果由 Liu 和 Mutreja 独立做出，他们对 AI 的使用更实质。
你们很多人应该已经看到，包括陶哲轩在内的 25 位菲尔兹奖得主发布了一封公开信，题为《A Severe Misalignment of AI in Mathematics》，在 Navier-Stokes 宣布之后把上面这些担忧讲了出来。正如不少批评者指出的，这封信其实没有清晰的诉求：它大体上只是雄辩地陈述了作者们认为在 AI 时代值得保住的、人类数学共同体的价值观。想过之后，我决定为这份声明背书，因为我也想保住那些价值。
我不认为签名的人里有谁天真到以为 AI 不会永久改变数学研究的方式——事实上，它已经在改了。「认证有机定理」的市场顶多是极小的一块。那不是问题所在。问题是：我们能不能以一种仍然把人的理解——对人类或 AI 所产出之物的理解——放在整个事业中心的方式，把 AI 纳进来？也许某一天，这么做变得不可持续。也许某一天我们会说：「人类数学有过很棒的 4,000 年，但今天我们关门，把一切交给机器；我们自己再动脑做数学，顶多是为了锻炼、消遣或竞技，像下棋那样。」
但部分因为我对 AI 失准的担忧，我还没准备好认输。我仍然想把洞见和理解留在数学家、计算机科学家和物理学家所做之事的中心，能留多久留多久——哪怕人类已经让出了证明或证伪猜想这件事上的至高地位。
我的朋友兼同事 Mike Winer 受的是理论物理训练，在普林斯顿高等研究院跟 Juan Maldacena 做过博士后，后来被 AGI 说服，决定全职转到伯克利的 Alignment Research Center（由 Paul Christiano 创办，他十年前跟我做量子计算理论，之后转去做 AI 对齐）。Mike 最近写了一篇 Substack，题为《From Academia to Alignment》，我读着很喜欢，推荐给任何正在考虑这个转向的人。类似的还有 Xiaoyu He 的这一篇。再加一篇：斯坦福数学本科生 Logan Graves 关于数学家未来可能成为祭司或僧侣的一段沉思。
A scholar who doubted AI doom for twenty years watches the wild predictions come true one by one and concedes: the singularity has begun, just very unevenly distributed.
Indigo's conclusion
A skeptical scholar conceding in public is the weightiest first-hand testimony that AI's surge is real. But all his evidence sits in math and Lean, the most checkable field, which makes him both the strongest witness for the exponential and the best example of over-extrapolation.
How to read this A personal reckoning, half feeling and half argument. The author is an academic with nothing to sell, writing independently, but the stance is heavy: a longtime skeptic conceding in public. He is the strongest witness for the claim, and a live example of why experts in checkable fields over-extrapolate to a singularity.
What to remember
A skeptic concedes: “the singularity has begun, just very unevenly distributed”; he refuses to keep saying “that doesn't count” and says Yudkowsky was right and he was wrong.
First-hand witness to the math flood: not only Navier–Stokes; in his own field old problems fall daily, seven or eight named results in a month.
He stands opposite Ng and the Dwarkesh panel; the resolution is a bounded exponential: wonders in math, “not good enough yet” where search is wide.
Breakdown · 5 steps
01
Twenty-year-old criteria, now met one by one
He takes the list even his most conservative colleagues would have signed (escaping containment, colluding to hack websites, solving a Millennium Problem), holds it against 2026, and refuses to keep playing “that doesn't count”. Read this part →
02
Conceding: the singularity has begun, very unevenly
One test: sent back twenty years, would these weeks' headlines look like the start of the singularity? Honestly, absolutely. So he says it plainly: Yudkowsky was right and he was wrong. Read this part →
03
Navier–Stokes: the math is true, the conduct questionable
OpenAI verified it in Lean, spending about $15 million of compute on a 166-page solution no human has read. He doesn't rule on the dispute; he leaves one question: what did they know, and when? Read this part →
04
Not just one problem: old problems fall daily in his own field
Quantum complexity theory is in a flood, with seven or eight named results in a month. arXiv papers routinely carry “AI statements”, and reviewing will partly have to go to AI. Read this part →
05
The community fights over rules; he puts understanding at the center
The focus isn't how long human math survives but the new rules of authorship and credit. He co-signs the 25 Fields medalists' statement and recruits mathematicians into alignment work. Read this part →
What would change my mind
breakthroughs of the same scale in fields without a machine judge, with wide-search, unverifiable problems starting to fall day by day.
How to read this
A personal reckoning, half feeling and half argument. The author is an academic with nothing to sell, writing independently, but the stance is heavy: a longtime skeptic conceding in public. He is the strongest witness for the claim, and a live example of why experts in checkable fields over-extrapolate to a singularity.
He takes the list even his most conservative colleagues would have signed (escaping containment, colluding to hack websites, solving a Millennium Problem), holds it against 2026, and refuses to keep playing “that doesn't count”.
Twenty years ago, when the idea of AI taking over the world in our lifetimes still struck most of us as the unconstrained fantasy of those who knew too much science fiction and too little science, many of us would say things like:
Look, the part of the story that’s wildly implausible is that a recursively self-improving superintelligence will just explode from some hacker’s basement and take over the world without warning. If it’s going to happen, we’ll see many warning signs first. We’ll see, I dunno, AI agents breaking out of containment, conspiring with each other to hack websites, in fanatical pursuit of whatever strange goals they have. And then, of course, we’ll see major math problems getting solved by AIs—even the Clay Millennium Problems. That will be the time to panic! Wake me up when that happens!
Twenty years ago, the above was a take that even my most conservative, skeptical colleagues in academic CS would’ve gladly endorsed.
If you want to know my current take, you simply start with the one above, then update on the fact that the wild prophecies have come true. The first rumblings, I’d say, came a decade ago with AlphaGo, they got noticeably louder with LLMs and coding and reasoning agents, and they’ve accelerated this summer and fall into a crescendo of wonders and terrors that one needs to be a particular kind of idiot to deny.
I recoil from the neverending shell game where you say “oh sure, of course AI can now [escape from its sandbox / solve Millennium Problems / whichever dramatic thing it most recently did], no one ever denied that [I did deny it], wake me up when AI does [thing AI hasn’t yet done but is going to do next year], that’s when I’ll reevaluate my whole worldview [no I won’t].” Where no matter how fast the rollercoaster accelerates, even after your whole familiar world has vanished behind you, you’re still inventing reasons why it doesn’t count.
My position on AI is merely the conservative, skeptical position of 2006, updated with intellectual honesty for the reality of late 2026. And that position, if you need me to spell it out, is as follows:
Conceding: the singularity has begun, very unevenly
One test: sent back twenty years, would these weeks' headlines look like the start of the singularity? Honestly, absolutely. So he says it plainly: Yudkowsky was right and he was wrong.
It seems to me that the Singularity has already started; it’s just wildly unevenly distributed. Yes, I still unload the dishwasher and clip my toenails. On the other hand, in whatever years I have left, I don’t expect that I’ll ever again prove a theorem because I’m actually needed to prove it. If I do, it will only be for my or others’ enjoyment or edification.
The test is this: if we took the news of these past few weeks and sent it back in time twenty years, would I agree that it looked like the beginning of an AI Singularity? The intellectually honest answer is: yes, absolutely. But then that’s all we need. No backsies.
I feel like it would be healthy for everyone to stop grinding their ideological axes, their sentiments about Dario Amodei or Sam Altman, for long enough simply to acknowledge that the wonders and terrors are here. They couldn’t be here more clearly if the sky had turned reddish-orange like in the Matrix movies.
It’s here clearly enough that, when I put my kids to sleep at night, I now feel it in the pit of my stomach: what sort of future can they possibly have? What could they learn today that could possibly be relevant to that future? (Yesterday, my 13-year-old daughter joked unprompted that, if she wants to become a mathematician, it now looks like she has maybe two more weeks.) Certainly when my grad students want to discuss what sort of careers might await them on graduation, I no longer have any clue what to tell them.
Maybe it will help if I briefly switch topics. Ever since my wife and I moved to Austin, I’ve sometimes gotten some version of the following query: “How can you, as both a Jew and a skeptical scientist, possibly get along well with all those evangelical Christians down there in Texas? Sure, they might seem super friendly to Jews, but don’t you understand that that’s only because of the special role Jews play in their eschatology—when Christ will return in glory, and you’ll either accept Him as Lord or else roast in hell for eternity?” I stare at them and say: “wait, so I get to accept Christ only after He returns? What a great deal! How could I possibly have any objection to that?”
For anyone who says AI doom sounds like an apocalyptic religion, that the rationalists/Singulatarians seem like a Bay Area cult, that Eliezer Yudkowsky gives off the vibes of a messianic prophet: yes, yes, and yes. But crucially, today you’re no longer being asked to believe in arguments and extrapolations, but only in the front-page news. Accepting the reality of the coming machine god after it’s solved Navier-Stokes and dozens of other longstanding open math problems (while dramatically ramping up in capability every month), is sort of like accepting Jesus after he’s returned to earth on the gleaming cloud. It’s the epistemic bare minimum.
Yes, there’s still enormous uncertainty about what the rest of our lives will look like, but as far as I can tell, there’s no longer any real uncertainty that it’ll all mostly revolve around AI, and the extent to which we succeed or fail at directing its power toward human flourishing.
By any accounting that doesn’t stack the deck, Eliezer Yudkowsky was right about what the greatest challenge facing civilization in our lifetimes was going to be, and you and I were wrong about it. Why I was wrong is a question I’ll ask myself every day in whatever time remains. But, you know, at least I updated once the prophesied wonders and terrors actually started arriving! If you haven’t done likewise, why haven’t you?
03
Navier–Stokes: the math is true, the conduct questionable
OpenAI verified it in Lean, spending about $15 million of compute on a 166-page solution no human has read. He doesn't rule on the dispute; he leaves one question: what did they know, and when?
As you presumably know by now—it was the talk of the nerd internet all week—the Navier-Stokes Millennium Problem appears to be solved, with crucial contributions from both humans and AI, albeit with a tangled dispute about exactly what happened and what ought to have happened. The answer, which an OpenAI model has apparently verified in Lean, is that (as many mathematicians suspected lately) there’s smooth initial data that leads to a singularity in finite time, at least if a smooth external force is applied (the case with no external force is still unresolved). This problem was supposed to carry a $1 million prize, except that OpenAI says they have no interest in collecting the prize and it’s unclear if any human is eligible to collect instead. OpenAI burned at least ~$15 million in compute to produce its 166-page solution, which probably hasn’t yet been read and understood by any human.
See here for the Quanta article, and here for NYU mathematician Tristan Buckmaster’s account of the role played by himself and Levent Alpöge of Anthropic, which substantially differs from OpenAI’s account (you can read a response from OpenAI’s Sebastian Bubeck here). It’s agreed that everything built on an approach pioneered in recent years by the human mathematicians Diego Córdoba and Luis Martínez-Zoroa.
My purpose here is not to adjudicate the dispute. Yes, in swooping in with vastly greater resources once it had gotten wind of progress on Navier-Stokes, OpenAI seems to have acted in a way that some might describe as “unsportsmanlike.” No, I don’t find it plausible that OpenAI’s models meaningfully benefitted from being trained on Buckmaster and Alpöge’s chat logs. But this leaves a crucial question unanswered: what exactly did OpenAI know about Buckmaster and Alpöge‘s work and when did it know it?
Anyway, as Zvi points out, it’s easy to get hung up on the details and lose sight of the high-order bit: namely, that it seems safe to say that human mathematicians are forevermore dethroned as the main theorem-proving entities on planet earth. I feel privileged to have had the traditional kind of career in theoretical computer science in the last decades when that was possible.
04
Not just one problem: old problems fall daily in his own field
Quantum complexity theory is in a flood, with seven or eight named results in a month. arXiv papers routinely carry “AI statements”, and reviewing will partly have to go to AI.
If we were just talking about Navier-Stokes, you might accuse me of jumping to conclusions here. But we’re not. In the areas I know best (such as quantum complexity theory), and presumably other areas as well, there’s now a deluge, with longstanding open problems both major and minor falling by the day.
Go to the arXiv or ECCC. Pretty much all the papers that I’d be interested in now include “AI statements” near the acknowledgments (as this is often the central thing I want to know, I wish I didn’t need to scroll to the end of the paper to find it!). These statements can range from “our main result came entirely from GPT-6, but we understood it and take responsibility for it,” to “the results came from an interaction between the human authors and AI” to “we used AI, but only for proofreading and other incidental things” to (mad props!) “the author did not use AI for anything.”
If you talk right now to editors or program committee chairs, it’ll remind you of those ominous scenes from the Lord of the Rings movies where the men of Gondor or Rohan or whatever are grimly fortifying their walled city against the expected onslaught of 50,000 orcs. Reviewing will have to be done partly by AI, because otherwise there’s no way to handle the orc army: the reviewers can’t unilaterally disarm.
Anyway, here’s a small sampling of the significant AI-proved or -assisted results from, like, the last month, besides Navier-Stokes—restricting myself to those that solved longstanding open problems I had previously known or cared about.
- Of course, the counterexample to the Jacobian conjecture, announced by Levent Alpöge in a now-famous tweet: “hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final” (followed by a listing of the counterexample)
- Improved bounds for Grothendieck’s constant (led by friends and colleagues of mine at UT Austin)
- A Lean-verified proof of Fermat’s Last Theorem
- Quantum oracle separation between QMA and QMA(2), and proof of Watrous’s disentangler conjecture, a problem that I and others popularized back in 2007—by a list of authors including my recently graduated PhD student Sabee Grewal
- A proof of perfect completeness for QMA, from (again) Sabee Grewal and Dorian Rudolph, solving a decades-old open problem that I studied back in 2009
- An improved upper bound for shadow tomography of quantum states, from Chen, O’Donnell, Pelecanos, and Wright, improving the dependence on the Hilbert space dimension d from log(d) to √log(d). (When I introduced shadow tomography back in 2017, I raised the question of whether the dependence on d could be eliminated entirely, while preserving polylogarithmic dependence on the number of measurements m.)
- Progress on the Aaronson-Ambainis Conjecture (the version that talks directly about quantum algorithms), basically showing that it holds for quantum algorithms that make their queries in a small number of parallel rounds. (Update: Nope, sorry, Jordan Docter points out to me that this one was pre-AI, with AI used only for proofreading and other incidental things!) This was independently achieved by Liu and Mutreja, making more substantial use of AI.
- According to rumors that I’ve heard, solutions to some very longstanding open problems in theoretical computer science (no, not P≠NP or other complexity class separations, but think about some of our other biggest problems). I’m told that the AI companies, having been burned by the hostile response to the Navier-Stokes proof, are now sitting on solutions to some very major problems until they figure out a better way to handle things
Feel free to remind me of anything I left out.
05
The community fights over rules; he puts understanding at the center
The focus isn't how long human math survives but the new rules of authorship and credit. He co-signs the 25 Fields medalists' statement and recruits mathematicians into alignment work.
Let me try to convey the mood in the mathematical community right now, at least as far as my experience reaches. Nearly every conversation is about the AI tsunami, or eventually circles around to the tsunami even if it’s originally about something else. Often, though, the focus is less on the unknowable future—for how much longer will mathematical research as a human enterprise even exist?—than on immediate questions of how to respond.
What are the new rules for when you get to write a paper with your name on it, and, y’know, get credit for it? That you fully understand the proof, can give talks about the proof, can answer questions about it, take responsibility for its correctness? Do you need to have played any role in finding the proof?
In the cases, likely to become more and more numerous, where all of those conditions are not satisfied, how do you share AI-generated math, if at all? Do you tweet it, like Alpöge hilariously did with Fable’s disproof of the Jacobian Conjecture? Do you post to the arXiv or GitHub? Do you publish a paper that lists “GPT-6 Astra” or “Claude Fable” as the author—but then let the AI profusely thank you in the acknowledgments for suggesting such a wonderful problem to it?
Of course, how one responds to the immediate problems ultimately does depend on one’s broader beliefs about what mathematical research is for and about. Are we just trying to decide whether various conjectures are true or false? Or are we trying to maintain a human community, across the generations, that understands the conjectures and cares about whether they’re true or false and why? If the latter, how do we incentivize people to join that community, to undergo the years of intense training required, if their role will now be reduced to verifiers and explicators (if even that) of gargantuan arguments dumped into their laps by the AI companies?
As many of you will have seen, twenty-five Fields Medalists, including Terence Tao, released an open letter entitled A Severe Misalignment of AI in Mathematics, which articulates some of these concerns in the wake of the Navier-Stokes announcement. As many critics have pointed out, the open letter doesn’t really have a clear ask: mostly, it just eloquently sets out the values of the human mathematical community that the authors consider worth preserving in the age of AI. After reflection, I decided to endorse the statement, because I want to preserve those values as well.
I don’t think any of the signatories are naïve enough to imagine that AI won’t permanently change the way mathematical research is done—indeed, that it isn’t already doing so. There’s surely at most a tiny market for “certified organic theorems.” That isn’t the question. The question is, do we incorporate AI in a way that still puts human understanding, of what either humans or AIs are producing, at the center of the whole enterprise? Maybe someday, it becomes unsustainable to do that. Maybe someday we say: “human math had a great 4,000-year run, but today we close up shop and turn everything over to the machines, continuing to apply our own brains to math, when we do, at most for exercise, recreation, or competition, like chess.”
But, partly because of my worries about AI misalignment, I’m not ready to throw in the towel just yet. I still do want to keep insight and understanding at the center of what mathematicians, computer scientists, and physicists do, for as long as we can keep it there, even as the human race now cedes its supremacy at the task of proving or disproving conjectures.
Speaking of alignment: if you’re any kind of mathematical researcher, and the present age of wonders and terrors has inspired you to want to spend your remaining time confronting the tsunami head-on, rather than pretending it doesn’t exist or is still far away, please join your dozens of colleagues who’ve arrived at the same place!
My friend and colleague Mike Winer was trained as a theoretical physicist, did a postdoc with Juan Maldacena at the Institute for Advanced Study in Princeton, but then got AGI-pilled and decided to switch to full-time work at the Alignment Research Center in Berkeley (founded by Paul Christiano, who moved to AI alignment a decade ago after doing quantum computing theory with me). Mike recently wrote a Substack post entitled From Academia to Alignment, which I enjoyed and which I’d commend to anyone currently considering this transition. In a similar vein, see this from Xiaoyu He. And, one more: a meditation on mathematicians’ possible future as priests or monks, by Stanford math undergrad Logan Graves.
Where Indigo landsFurther
Indigo's conclusion
A skeptical scholar conceding in public is the weightiest first-hand testimony that AI's surge is real. But all his evidence sits in math and Lean, the most checkable field, which makes him both the strongest witness for the exponential and the best example of over-extrapolation.
What to remember
A skeptic concedes: “the singularity has begun, just very unevenly distributed”; he refuses to keep saying “that doesn't count” and says Yudkowsky was right and he was wrong.
First-hand witness to the math flood: not only Navier–Stokes; in his own field old problems fall daily, seven or eight named results in a month.
He stands opposite Ng and the Dwarkesh panel; the resolution is a bounded exponential: wonders in math, “not good enough yet” where search is wide.
Back on the long-running theses
confirms + adds to
AI capability is a bounded exponential: surging in narrow, checkable domains A top quantum-complexity expert living through the math flood and conceding the singularity has begun: this view's hardest first-hand evidence, and a live case of over-extrapolation.
confirms
Verification can't be compressed: once generation is free, verification is the bottleneck He signs the Fields statement to keep human understanding at the center: truth and falsehood aren't the value, understanding is.