本文为原文前 6,000 字符的节选翻译，完整内容请查看原文。

The Mathocalypse

… then they came for Navier–Stokes and I said nothing because I never worked on Navier–Stokes. But when they came for RL vs. L I realized that things are serious

–friend-of-the-blog Omer Reingold (shared with permission)

……然后他们冲着纳维-斯托克斯方程来了，我没说话，因为我从未研究过纳维-斯托克斯。但当他们冲着 RL vs. L（对数空间与对数空间归约）而来时，我意识到事情变得严重了。

——博客好友 Omer Reingold（经许可分享）

Last night my 9-year-old son was taunting my wife, complexity theorist Dana Moshkovitz, as follows: “mommy, I heard you got cooked! I heard that a robot solved the math problem you worked on for your whole career! OOF!”

昨晚，我 9 岁的儿子这样嘲讽我的妻子——复杂性理论家 Dana Moshkovitz：“妈妈，我听说你被‘烤’了！我听说一个机器人解决了你整个职业生涯都在研究的数学难题！噢！”

While my son was being a brat, he also wasn’t wrong. Whether you’re thrilled, depressed, angry, or whatever else about it, yesterday was surely one of the biggest days in mathematical history. And yes, among the 372 huge results released yesterday by OpenAI, on the recommendation of its advisory group of Timothy Gowers, Edward Witten, and other distinguished mathematicians, was a proof of Subhash Khot’s Unique Games Conjecture (UGC), a statement that my wife has worked toward proving for the entire time I’ve known her. (The UGC implies that a whole slew of optimization problems really are NP-hard, even if you just want an approximation that’s slightly better than what you get from semidefinite programming relaxation, which is one of our main tools.)

虽然我儿子当时很调皮，但他说的也没错。无论你对此感到兴奋、沮丧、愤怒还是其他什么情绪，昨天无疑是数学史上最重大的日子之一。没错，在 OpenAI 昨天发布的 372 项重大成果中（这些成果由 Timothy Gowers、Edward Witten 等杰出数学家组成的顾问小组推荐），包含了对 Subhash Khot 的唯一博弈猜想（UGC）的证明。这是我认识我妻子以来，她一直致力于证明的一个命题。（UGC 意味着一大批优化问题确实是 NP 难的，即使你只是想获得比半正定规划松弛——我们主要工具之一——所能提供的更好的近似解。）

Or at least, we’re pretty sure that it’s a proof! There’s a Lean certificate, as there are for some of the other 372 breakthrough results (not all of them). But it also appears that no human has understood just about any of these proofs yet; the race to do so has just started. If you want an on-the-ground sense of what that race is going to be like, here’s some of what Dana texted me last night:

或者至少，我们相当确定这是一个证明！它有一个 Lean 验证证书，就像其他 372 项突破性成果中的一部分（并非全部）一样。但目前看来，还没有人类真正理解这些证明中的任何一个；理解它们的竞赛才刚刚开始。如果你想了解这场竞赛的现场感，以下是 Dana 昨晚发给我的部分短信内容：

It feels like something written by someone who’s on psychedelics. So much unclear and doesn’t make sense. Lots of name dropping of previous work without discussing why it can be used despite impossibility results

感觉就像是某个嗑了迷幻药的人写的东西。太多地方不清楚，逻辑不通。大量堆砌过往研究的名称，却不讨论为什么在存在不可能性结果的情况下还能使用这些研究。

Basically the paper is so horribly written that it’s impossible to read it without AI help

基本上这篇论文写得太糟糕了，没有 AI 的帮助根本无法阅读。

I asked Astra for reasonable completeness and soundness claims of the noise gadget and it gave them by combining claims from all over the paper

我让 Astra 提供关于噪声小工具（noise gadget）的合理完备性和可靠性声明，它通过整合论文各处的声明给出了答案。

They also have direct optimal NP hardness of approximation proofs for the main applications of the UGC (Max Cut and all CSP) that bypass the UGC.

他们还针对 UGC 的主要应用（最大割问题和所有约束满足问题）给出了直接的最优 NP 难近似证明，这些证明绕过了 UGC 本身。

The UGC proof invents a completely new bizarre code with a noise test. It’s some crazy recursive construction.

UGC 的证明发明了一种带有噪声测试的、完全新颖且怪异的编码。这是一种疯狂的递归构造。

It’s not the long code, not the short code – some alien craziness

它不是长码，也不是短码——简直是外星人的疯狂产物。

I still think that there maybe is a proof that uses the half space code (which is natural)

我仍然认为，或许存在一种使用半空间编码（这是很自然的）的证明方法。

The citations are often irrelevant and confusing

引用文献往往不相关且令人困惑。

A possible future is a math world that’s heavenly if you have vision/creative ideas that AI could help check and implement.

未来的一种可能性是，如果你拥有愿景或创造性想法，而 AI 可以帮助你验证和实现，那么数学世界将变得无比美好。

And of course there’s a lot for us to learn from the aliens

当然，我们还有很多东西要向这些“外星人”学习。

If you’re wondering what emotions Dana is feeling—well, probably all of them! Even while a central career aspiration has fallen to a robot, there are at least two mitigating factors for her. First, she can feel vindicated that the UGC was true after all, something she never doubted even while many of her colleagues did! Second, all of us in math and theoretical computer science and mathematical physics, at least those who cared about solving crisply-stated problems, are now in the same boat.

如果你想知道 Dana 现在是什么心情——嗯，大概是五味杂陈！尽管她职业生涯的核心抱负被一个机器人实现了，但对她来说至少有两个缓解因素。首先，她可以感到释然，因为 UGC 最终被证明是正确的，这一点即使在许多同事怀疑时她也从未动摇过！其次，我们所有从事数学、理论计算机科学和数学物理学的人，至少是那些关心解决明确定义问题的人，现在都坐在同一条船上了。

Besides the Unique Games Conjecture, here’s a small sampling of the treasures from Aladdin’s cave that I’ll probably be paying the most attention to over the coming weeks:

除了唯一博弈猜想，以下是我在未来几周可能会重点关注的“阿拉丁洞穴”中的一小部分宝藏：

Any of the above, alone, could easily have been “result of the year” in some area (and in some cases, like Unique Games and L=BPL, in all of CS theory). And there’s a lot that I’ve left out—feel free to share in the comments whatever is making your eyes bug out! There are equally astounding wonders in number theory, combinatorics, algebraic geometry, analysis, and pretty much every other area of math, most of which I’ll never understand, although I’ll note that it includes partial progress toward the Riemann hypothesis and the Hodge Conjecture and the Birch-Swinnerton-Dyer Conjecture (i.e., the majority of the remaining Millennium Problems).

上述任何一项成果，单独拿出来都可以轻松成为某个领域的“年度成果”（在某些情况下，如唯一博弈和 L=BPL，甚至可以成为整个计算机科学理论界的年度成果）。我遗漏了很多内容——欢迎在评论区分享那些让你目瞪口呆的发现！在数论、组合数学、代数几何、分析学以及几乎所有其他数学领域中，都有同样令人震惊的奇迹。其中大部分我永远无法理解，但我注意到这些成果包括了对黎曼猜想、霍奇猜想和贝赫-斯维讷通-戴尔猜想的部分进展（即剩余千禧年大奖难题中的大部分）。

We can take solace in what’s missing from the list. P≠NP isn’t there, nor even P=BPP or NEXP⊄P/poly, and surely not for lack of trying. Apparently the greatest open problems of theoretical computer science are indeed pretty hard!

我们可以从名单中缺失的内容里寻求慰藉。P≠NP 不在其中，P=BPP 或 NEXP⊄P/poly 也不在，这肯定不是因为没有尝试。显然，理论计算机科学中最伟大的未解难题确实非常困难！

Oh, lest I forget: one day before the OpenAI dump, meaning Monday evening, Virginia Williams and Josh Alman posted an arXiv preprint that solves the 3SUM problem in O(n1.9992) time, and the All-Pairs Shortest Paths problem in O(n2.9995) time, refuting half-century-old conjectures that the correct answers were n2-o(1) and n3-o(1) respectively. In this case, it wasn’t an OpenAI model that supplied the crucial idea; it was an Anthropic one! But Anthropic then took a different approach from OpenAI: rather than post the undigested solutions to the world, it gave Virginia and Josh the opportunity to write and announce a digested version in exchange for compensation.

哦，差点忘了：在 OpenAI 发布成果的前一天，也就是周一晚上，Virginia Williams 和 Josh Alman 在 arXiv 上发布了一篇预印本，以 O(n1.9992) 的时间复杂度解决了 3SUM 问题，并以 O(n2.9995) 的时间复杂度解决了全源最短路径问题，推翻了半个世纪以来认为正确答案分别为 n2-o(1) 和 n3-o(1) 的猜想。在这种情况下，提供关键思路的不是 OpenAI 的模型，而是 Anthropic 的模型！但 Anthropic 采取了与 OpenAI 不同的做法：它没有将未经消化的解决方案直接公之于众，而是让 Virginia 和 Josh 有机会撰写并发布一个经过整理的版本，作为交换，他们获得了报酬。

These have emerged as the two main models for communicating AI math breakthroughs, and they both have strengths and weaknesses. The “OpenAI model” sets up a crazy race among humans to digest and explain a messy AI proof (work that could easily be some combination of thankless, barely-credited, competitive, and unfun), while the “Anthropic model” puts a private company in the position of picking and choosing which human mathematicians get to be the emissaries of the AI. Dunno, what do you guys think?

这两种模式已成为传播 AI 数学突破的主要方式，它们各有利弊。“OpenAI 模式”在人类之间引发了一场疯狂的竞赛，去消化和解释混乱的 AI 证明（这项工作很容易变得吃力不讨好、几乎没有署名权、竞争激烈且毫无乐趣），而“Anthropic 模式”则让一家私营公司处于主导地位，去挑选哪些人类数学家可以成为 AI 的代言人。不知道你们怎么看？

For those who are wondering: apparently, the AI model that produced all these wonders was not bespoke contraption of 10,000 agents burning millions of dollars worth of compute, as was used for example to construct a finite-time blowup for the Navier-Stokes equations. Instead, it was simply the latest internal OpenAI model—one that might be released to paying ChatGPT customers within the next couple of months, depending on the recommendations of OpenAI’s safety board! (My 9-year-old son: “Oh they definitely shouldn’t release that. If it could solv

对于那些感到好奇的人：显然，产生所有这些奇迹的 AI 模型并不是像用于构建纳维-斯托克斯方程有限时间爆破的那种由 1 万个智能体、消耗数百万美元算力的定制装置。相反，它只是 OpenAI 最新的内部模型——根据 OpenAI 安全委员会的建议，该模型可能会在未来几个月内向付费的 ChatGPT 用户发布！（我 9 岁的儿子：“噢，他们绝对不应该发布那个。如果它能解……”）