OpenAI 带来的新数学

目录 Table of Contents

5. 你需要找到一个新问题。

5. You Will Need To Find A New Problem.

6. 验证或评估并不总是比生成更容易。

6. Verification or Evaluation Is Not Always Easier Than Generation.

12. 另一个数学小组作出回应。

12. Another Mathematical Group Responds.

我们证明了什么？ What Did We Prove?

这里有一系列关于顶级问题的常识性图解说明（当然是由 AI 制作的）。它们看起来是这样的：

Here is a thread of common sense graphical explanations (made by AI of course) of top problems. They look like this:

首先，头条新闻是黎曼方面的重大突破。这不是对该猜想的解决方案，而是对界限的巨大收紧。

First, the headline is the major breakthrough on Riemann. It’s not a solution to the hypothesis, but it’s a huge tightening on the bounds.

矩阵乘法效率达到 2.25 可能最终会产生最大的现实世界影响。

Matrix multiplication efficiency at 2.25 may end up having the most real world impact.

也许吧，但我们还没有找到在实际中更快的场景。

Maybe, but we haven’t yet found a scenario where this is faster in practice.

这还不是一个实际可用的结果，但矩阵乘法占据了世界上大部分的计算量。这可能开辟一种全新的方法。

It’s not quite a practically usable result yet, but matrix multiplication is the majority of the world's compute. This could open a whole new approach.

整数乘法可能是最令人震惊的结果。与矩阵乘法类似，它是一种“渐近归约”。同样与矩阵乘法的结果一样，它并不实用。

Integer multiplication is probably the most shocking result. Similar to matrix multiplication, it is an "asymptotic reduction". Also like the matrix result, it is not practical.

但很少有人会预测到先前的壁垒能够被打破。哪怕只是稍微打破一点。

But very few people would have predicted the previous barrier could be broken. Even slightly.

圆周率（Pi）的结果未必是这里最重要的结果，但它可能是最有趣的。而且可能足够通俗易懂，连你的孩子都能理解。基本上，就是圆周率到底有多无理。

The Pi result isn't necessarily the most important result here, but it may be the most fun. And probably accessible enough that even your kids can get it. Basically, just how irrational is Pi.

唯一游戏（Unique Games）也是一个相当重大的结果，因为与黎曼（Riemann）类似，它与数学中最重要的一个问题相邻：P vs NP。

Unique games is also a pretty huge result, because like Riemann it's adjacent to one of the most important problems in math: P vs NP.

需要说明的是，这与 P = NP 本身无关。但它确实揭示了关于 NP 难问题根本性限制的新东西。

To be clear, it's not about P = NP itself. But it does reveal new things about the fundamental limits to NP-hard problems.

严格来说是两个问题，但由于在证明和意义上具有相似性，我将它们放在一起。

Technically two problems, but I put them together because of similarities, in proof and implications.

Hodge 和 Birch 问题都相当抽象。但它们对代数几何如此重要，以至于在 Riemann 之后，它们可能是本组中最重大的成果。

Hodge and Birch are both pretty abstract. But they’re both so important to algebraic geometry that after Riemann they may be the most significant results in the set.

Hilbert 第十问题是另一个计算机科学问题。但与 P vs NP 不同，它甚至不涉及计算上难以求解的问题，而是关于一类问题是否存在可计算的解。

Hilbert's 10th is another computer science problem. But unlike P vs NP it isn’t even about problems that are computationally hard to solve, it’s about whether a type of problem even has a computable solution.

最后但同样重要的是关于图着色的 Hadwiger 问题。它也被列为整个集合中最令人震惊的结果之一。

Last but not least is Hadwiger on graph coloring. It also ranks there for most shocking result of the full set.

它基本上推翻了我们曾认为是图论中最基本关系之一的东西。

It basically overturns something that we thought was one of the most fundamental relationships in graphs.

以下是对代数数论中一些结果的分析。

Here are some analyses of the results in algebraic number theory.

这些成果究竟有什么用处？Ole Lehmann 的 AI 提到了若干应用，包括聚变研究、便携式人体扫描仪、匹配系统、组织扫描、量子传感器，以及自动驾驶汽车和机器人的安全检查等。

What is all of that good for? Ole Lehmann's AI mentions applications for fusion research, portable body scanners, matching systems, tissue scans, quantum sensors and safety checks on self-driving cars and robots, among other things.

数学末日 The Mathocalypse

Alex Kontorovich：准黎曼假设？！？！？？！你在开玩笑吗？如果这是人类做到的，那会立刻获得菲尔兹奖，毫无疑问。

Alex Kontorovich: Quasi-RH?!?!???! Are you kidding me? If a human did this, it would be an instant Fields Medal, no questions asked.

黎曼假设说 zeta 函数在 Re(s)>1/2 时没有零点。直到一秒前，我们最好的结果是一个随着虚轴向上而越来越窄的区域。我以为他们可能会把它加宽一点，那将是一个巨大的突破。但没有。他们得到了一个无零点带！！！！太疯狂了

RH says zeta has no zeros in Re(s)>1/2. The best we had until a second ago was a region that got thinner and thinner the higher up the imaginary axis you go. I thought maybe they’d fatten that up a bit, that’d be a massive breakthrough. But no. They got a zero free strip!!!! Insane

是的，也没有 Siegel 零点。所以我想是两个菲尔兹奖……

Yeah, no Siegel zeros either. So I guess two Fields medals…

准黎曼假设不是完整的黎曼假设，但它足以用于许多目的，例如快速计算模素数的平方根而无需抛硬币，或者得到给定数以下素数个数的更好估计，并且一篇姊妹论文触及了 Artin 1927 年原根猜想的核⼼。

Quasi-RH is not the full Riemann hypothesis, but it is sufficient for many purposes, such as computing square roots modulo a prime quickly without coin flipping, or getting a much better estimate of the number of primes below a given number, and a sibling paper gets to the core of Artin’s 1927 primitive root conjecture.

Steven Strogatz：这里有许多惊人的结果。但这是一个特别令人惊叹的：矩阵乘法的指数不超过 2.25。之前的世界纪录大约是 2.37。这一进展的飞跃就像 Bob Beamon 的跳远。

Steven Strogatz: Many staggering results here. But this is a particularly amazing one: the exponent for matrix multiplication is no more than 2.25. The previous world record had been something like 2.37. This leap in progress is like Bob Beamon’s long jump.

Isaac Kim：这里列出了一系列令人震惊的问题，涉及量子信息、多体物理和量子计算——这些领域都是我深为关切的。问题太多了，让我挑选以下几个。

Isaac Kim: This contains a shocking list of problems in quantum information, many-body physics and quantum computing, the fields that are dear to my heart. There are too many, but let me pick the following.

1. 二维面积定律的证明。

1. Proof of area law in 2D.

2. 自旋为 1 的 Haldane 能隙

2. Spin-one Haldane gap

3. 奇偶性不属于 QAC^0

3. Parity is not in QAC^0

4. 常数误差 Aaronson-Kuperberg 猜想

4. Constant-error Aaronson-Kuperberg conjecture

5. 生成共形网的酉 VOA。

5. Unitary VOAs generating conformal nets.

变化正在迅速发生。我甚至无法想象未来几个月会发生什么，更不用说一年了。

Things are changing fast. I cannot even imagine what will happen over the next few months, let alone a year.

Will Depue：我让 GPT 6 Pro 和 Fable 5.1 对过去三年的所有发现进行排名

Will Depue: I asked GPT 6 Pro and Fable 5.1 to rank all discoveries in the last three years

🔵 表示人类发现的

🔵 for human-discovered

🔴 表示 AI 发现的（10 月 6 日之前）

🔴 for AI-discovered (before October 6th)

🟢 由 AI 从 OpenAI/math 仓库中发现的

🟢 for AI discovered from the OpenAI/math repo

其中 81% 已于今天发布。wtf

81% of them have been released today. wtf

Fable 5.1 的前 100 项占今天列表的 59%，其中 87% 为 AI 生成，完整列表见链接。

Fable 5.1's top 100 were 59% of today's list, 87% AI, full list at the link.

Sauers：我的智能体（Opus 5.5、Astra 和 5.6 Sol）此前已在 GitHub 上解决了其中一些问题。我想知道其中究竟有多大比例真正需要它们各自的内部模型？

Sauers: My agents (Opus 5.5, Astra, and 5.6 Sol) already solved some of these beforehand (on GitHub). I wonder what % of these actually require their internal model?

世界并不理解 The World Does Not Understand

当被问及所有这些数学证明（作为一个假设）时，AI 们都说这将是件大事，应该成为头版新闻，其历史意义超越任何合理的比较，是数学史上最重大的一天。有些人认为这不可能。

The AIs all say, when asked about all these math proofs as a hypothetical, that this would be a huge deal, and should be front page news, historic beyond any reasonable comparison, the biggest day in mathematics. Some think it is impossible.

事实证明，这并没有成为头版新闻。大多数人没有听说这件事。它本应成为头版新闻，但新闻界没有人知道这意味着什么，也不认为人们会关心。

It turns out this was not front page news. Most people did not hear about it. It should have been, but no one in the news business knows what it means, or thinks people would care.

Kevin A. Bryan：好，我已经看完了 OAI 的数学清单。这简直疯狂，如果人们理解这意味着什么，它应该成为全世界的头版新闻。但如果我在实验室负责战略，第一优先级是“尽快把这件事用于医学、肿瘤学、电池效率等领域”。

Kevin A. Bryan: Ok, finished going through the OAI math list. It is bonkers and should be front page news around the world if folks understood what this meant. But if I am running strategy at a lab, #1 priority is “do this for medicine, oncology, battery efficiency, etc as fast as possible”.

Joshua Gans 在报道开头展示了真正的头版，以向我们展示《纽约时报》认为哪些事情比一次性解决 500 个顶级未解数学问题中的 90 个更重要。

Joshua Gans starts off his coverage with the actual front page, in order to show us what things The New York Times thought were more important than solving 90 of the top 500 open math problems at once.

Seth Burn：有时头版新闻最初并不会登上头版。最好的例子是 Sputnik。它于 1957 年 10 月 4 日发射，但苏联人并不认为这是件大事[它只占了右侧专栏的几段文字].

Seth Burn: Sometimes front page news doesn’t initially reach the front page. The best example is Sputnik. It launched 10/4/57, but the Soviets didn’t think it was a huge deal [and it got a few paragraphs of a right-side column].

[直到美国陷入恐慌之后，这件事才在 1957 年 10 月 6 日的俄罗斯成为头条新闻](https://cosmo.org/for-some-sputnik-1-was-no-big-deal-until-it-was/)。

[It was only after America freaked out that it [became a banner headline] in Russia on 10/6/57](https://cosmo.org/for-some-sputnik-1-was-no-big-deal-until-it-was/).

否则，那本是一个新闻极其平淡的日子。其中大部分内容其实算不上突发新闻。然而，即便在文末的摘要中——那些摘要甚至还包含一篇泛泛而谈的 AI 评论文章——也看不到任何数学内容。

It was a remarkably slow news day otherwise. Much of this is not exactly breaking. Yet no math is to be seen, even in the summaries at the bottom, which even include a generic AI thinkpiece.

Joshua Gans：缺失的是这一点：OpenAI 撰写的 722 篇数学论文。说这很可能是历史上科学进步最重大的一天，并不为过。这很难评估，但我所看到的讨论是，其中许多成果属于数学中最难、最重要的结果之列。我猜想，2026 年 10 月 6 日将以某种形式作为 AI 在数学领域的审判日而被载入史册，但它预示的远不止于此。

Joshua Gans: What's missing is this: 722 mathematics papers written by OpenAI. It isn't an overstatement to say that this is probably the biggest day of scientific advancement in history. It is hard to evaluate, but the discussion that I have been seeing is that many of these are among the hardest and most significant results in mathematics. I suspect October 6th, 2026, will go down as some form of Judgment Day for AI in mathematics, but it portends so much more.

反方观点是，这样的日子可能有很多：

The counterargument is that there might be a lot of days like this:

roon（OpenAI）：昨天会被铭记为重要的一天吗？很难说。在一个间断式指数增长中，每一个局部最大值从稍远处看都不可见。

roon (OpenAI): will yesterday be remembered as an important day? hard to say. in a punctuated exponential every local maximum looks invisible from a bit further out

目前你仍然可以做数学 For Now You Can Still Do Math

我喜欢这个模型，它解释了为什么数学尚未被完全攻克：

I like this model of why mathematics is not done quite yet:

Joshua Gans：数学家们现在正在经历这一过程。对此我再次提供上面的 Bookend 幻灯片以作辅助：

Joshua Gans: The mathematicians are now going through this. To which I offer again the above Bookend slide to help:

我们目前还无法证明 AI 在猜想或最终验证方面更胜一筹。这还需要一些额外时间，尽管可能不会太多。

We cannot yet show that the AI is better at conjectures or final verification. That will take some additional time, although probably not all that much time.

验证必须相当可靠，否则我们到现在应该已经发现更多错误了。有很多数学家会很乐意找出一个错误。通过推理轨迹我们知道，很大一部分精力都投入到了验证上。

Verification has to be pretty good or we would have found a lot more errors by now. There are a lot of mathematicians who would love to find a mistake. We know, via the reasoning traces, that a big chunk of effort is going to verification.

你将需要寻找一个新问题 You Will Need To Find A New Problem

Scott Aaronson 报道了“数学末日”（Mathocalypse）。

Scott Aaronson reports on the Mathocalypse.

Scott Aaronson（Shtetl-Optimized 博客）：昨晚，我 9 岁的儿子这样嘲弄我的妻子、复杂性理论家 Dana Moshkovitz：“妈妈，我听说你被 _彻底击败_ 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问题！哎哟！”

Scott Aaronson (Shtetl-Optimized): 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!"

虽然我儿子当时是在耍熊孩子脾气，但他说得也没错。

While my son was being a brat, he also wasn't wrong.

无论你对此感到兴奋、沮丧、愤怒还是别的什么，昨天都无疑是数学史上最重大的日子之一。

Whether you're thrilled, depressed, angry, or whatever else about it, yesterday was surely one of the biggest days in mathematical history.

而且没错，在 OpenAI 昨天发布的 372 项重大成果中，经其顾问团队 Timothy Gowers、Edward Witten 及其他杰出数学家的推荐，就包括对 Subhash Khot 的 Unique Games Conjecture（UGC，唯一游戏猜想）的证明——而在我认识我妻子的全部时间里，她一直在致力于证明这一命题。

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.

然后是另一种“新问题”，即那些否认者——三年前他们还乐于谈论 LLMs 不会做数学——现在必须找到一种新方式来解释为什么 AI 能做的任何事情都不是真实的或没有意义的。

Then there is the other kind of ‘new problem,’ where those in denial, who three years ago took comfort in talking about how LLMs could not do math, have to find a new way to explain why anything an AI can do is not real or not meaningful.

Shtetl-Optimized：

Shtetl-Optimized:

经验表明，_即使在现在_，仍然会有人以居高临下的语气解释为什么这一切都不是真实的，为什么都不算数。如果这些人能够对经验世界中发生的_任何事情_感到印象深刻，对_任何事情_进行更新，他们早在几年前就应该已经印象深刻并更新了，远在事情发展到实际数学末日之前。

Experience has shown that, _even now_, there will still be people explaining in patronizing tones why none of this is real and none of it counts. If such people were capable of being impressed by _anything_ that happens in the empirical world, of updating on _anything_, they would’ve already been impressed and already updated several years ago, long before things had reached the point of an actual Mathocalypse.

所以，他们会说，也许所谓的解决方案根本不是解决方案，而只是“AI 垃圾”。或者，被解决的 372 个著名开放问题中没有一个_真正的_数学问题，它们都只是被美化的竞赛谜题和琐事。（毕竟，黎曼猜想仍然没有解决！）或者，也许整个有 4000 年历史的数学学科需要被抛弃：事实证明，它_全部_只是解谜和琐事；不同之处在于，现在琐事被揭开了面具。无论如何，真正重要的是，人类创造力的_真正_内殿尚未被突破，而且可能永远不会被突破，还有，萨姆·奥特曼和达里奥·阿莫德是令人鄙视的小书呆子。

So, they’ll say, maybe the alleged solutions are not solutions at all, but just “AI slop.” Or maybe none of the 372 well-known open problems that were solved were _real_ math problems, they were all just glorified contest puzzles and trivialities. (After all, there’s still no Riemann Hypothesis!) Or maybe the entire 4000-year-old discipline of mathematics needs to be jettisoned: turns out that it was _all_ just puzzle-solving and trivialities; all that’s different is that now the triviality stands unmasked. In any case, what really matters is that the _true_ inner sanctum of human creativity hasn’t been breached and probably never will be, and also, that Sam Altman and Dario Amodei are contemptible little nerds.

如果你仍然是那个注定失败的世界观的支持者，仍然在那艘正在下沉的船上，我以最强烈的措辞鼓励你阅读昨天 AI 话语中的_另一篇_伟大贡献，除了 OpenAI 数学末日的大量内容之外：即斯科特·亚历山大给史蒂芬·平克的公开信。

If you’re still a proponent of that doomed worldview, still aboard the sinking ship, I encourage you in the strongest possible terms to read yesterday’s _other_ great contribution to AI discourse, besides the OpenAI Mathocalypse dump: namely, Scott Alexander’s open letter to Steven Pinker.

我确实得向那个针对所有可能的 AI 炒作的全称反驳脱帽致敬：

I do have to tip my hat to the fully general counterargument to all possible AI hype:

Cas：我认为 AI 在数学上碾压人类成就这件事可能被过度炒作了——与其说是奇点即将到来的有力信号，不如说是莫拉维克悖论的一个例证。回想一下，大约 10 年前 AI 就在围棋上超越了人类，但在那些真正能改变世界的任务上，它却慢得多才变得擅长。

Cas: I think that AI smashing human achievements in mathematics might be overhyped—more an example of Moravec's paradox than a robust sign of the singularity arriving soon. Recall how AI eclipsed humans at Go ~10 years ago but was much slower to get good at tasks that would change the world.

AI 擅长的都是莫拉维克悖论。AI 不擅长的就是它糟糕的原因。AI 突然在一个越来越大的领域集合中变得超人——包括那些我们曾因其不擅长而说它笨的领域——你为何如此兴奋？想必这不会很快扩展到其他领域。

Whatever AI is good at is Moravec's paradox. Whatever AI is bad at is why it sucks. Why are you so excited that AI is suddenly superhuman in an increasingly large set of domains, including the exact things we were saying it was dumb for being bad at? Surely this will not extend soon to other domains.

验证或评估并不总是比生成更容易 Verification or Evaluation Is Not Always Easier Than Generation

这一批评的最强版本是区分可验证领域与不可验证领域。

The steelman of this critique is to draw a distinction between verifiable versus unverifiable domains.

该论点认为，只要存在一个具有已知真实基准（ground truth）的可验证领域，AI 就会迅速超越人类。

This argument says that whenever you have a verified domain, with known ground truth, AI will quickly become superhuman.

反之，如果验证很困难，或者你所能做的只是以非正式的方式进行评估、且需要人类在环（human in the loop）以避免有偏的错误与扭曲的行为，那么 AI 的能力就会滞后。

Whereas, if verification is difficult, or all you can do is evaluate in an informal way that requires a human in the loop to avoid biased errors and distorted behaviors, then AI capabilities will lag behind.

有几年我们处于一个奇怪的境地：当时预训练占主导，LLM 主要是在人类文字上训练的。这在关键方面导致了一个看似非 Yudkowsky 式的世界——LLM 在多种有用但不可验证的领域相对出色，却在数学上毫无希望。

We were in a weird place for a few years, when pretraining dominated and LLMs were mostly trained on human words. This led to a seemingly non-Yudkowsky world in key ways, with LLMs being relatively excellent at a variety of useful but unverified domains, while being hopeless at math.

如今，随着后训练占主导，LLM 在数学和编程上再次更强，情况又回到了你所预期的样子。AI 正在各个方面迅速变好，但其他领域的进展——尽管与几乎任何其他技术相比都极快——目前相对较慢。因此，我们先推进数学、编程及类似领域，这些领域进而自动化 AI 研发，再进而加速其他一切。

Now, with post-training dominating, LLMs are again stronger at math and coding, and things again look like you would expect. AI is quickly getting better at everything, but progress in other domains, while super fast compared to almost any other tech ever, is relatively slow for now. Thus, we advance math and coding and similar domains first, which then automate AI R&D, which then accelerates everything else.

破解密码 Cracking the Code

Justin Drake 注意到，在这些结果中，密码学突破的比例低得惊人。也就是说，719 篇摘要中没有一篇提到密码学、LWE 或离散对数。

Justin Drake notices that there is a striking under-representation of cryptographic breakthroughs among the results. As in, zero of the 719 abstracts mention cryptography, LWE or discrete logs.

一种可能是 AI 在密码学方面相对较弱；另一种可能是运气，或者 OpenAI 最初的问题集中没有包含密码学（也许正是为了避免这个问题？）；又或者政府或 OpenAI 正在审查这些结果。

One option is that AI is relatively weak at cryptography, another is luck or that OpenAI didn't have cryptography in its initial problem set (perhaps to avoid exactly this issue?), or perhaps the government or OpenAI are censoring those results.

Justin 呼吁区块链行业进入“掩体模式”，以防范突然的大规模突破。

Justin calls for a 'bunker mode' for the blockchain industry, to protect against sudden mass breakthroughs.

Vitalik 表示不必恐慌，但潜在的数学新发现确实会给密码学带来严重风险，包括格密码，但目前尚未波及哈希。

Vitalik says not to panic, but that there are serious risks to cryptography from potential new math discoveries, including to lattices but not yet to hashes.

Vitalik Buterin：但对于任何具有结构的事物，你都应该假设 AI 至少会在破解该结构方面取得一些进展。由此可以合理推断：如果你想让某样东西具有可信的长期安全性，就把密钥长度乘以 10。

Vitalik Buterin: But for anything that has structure, you should assume that AI will make at least some progress in breaking that structure. Here, one reasonable inference is that if you want to make something plausibly long-term secure, multiply the key sizes by 10.

……从理论上讲，哈希当然也有可能被攻破（例如，P = NP 就蕴含这一点）。但我认为 P = NP 极不可能。

… Theoretically, of course it's possible that hashes are broken too (e.g., P = NP would imply that). But I think P = NP is very unlikely.

而且直观地说，一个数学对象完全不具有可利用的结构（正如哈希所意图达到的），要比一个数学对象恰好具有约 3 种可利用的结构（对椭圆曲线而言：结合律、Schoof 算法、配对）且不存在某种我们尚未发现的、会大幅削弱其安全性的秘密第四种结构（对椭圆曲线而言，即 ECDLP 与配对安全性），可能性大得多。对 LWE、SVP、RLWE 以及格问题家族而言也是类似的。

And intuitively, it's much more likely that a mathematical object has exactly no exploitable structure (like hashes are intended to), than that a mathematical object has exactly ~3 forms of exploitable structure (for elliptic curves: associativity, Schoof, pairings) and not some secret fourth form of structure we have not yet discovered that greatly degrades its security (for elliptic curves, ECDLP and pairing security). Similar for LWE, SVP, RLWE and the zoo of lattice problems.

出于这个原因，我们目前还看不到任何理由去担忧并开始填充哈希的字节大小（如果我们开始更加担忧，我们会先填充轮数，然后再对字节大小做任何改动）。

For this reason, we do not yet see any reason to worry and start padding the byte size of hashes (if we start to worry more, we would pad the round count first before doing anything to the byte size).

具体的 TLDR，我个人的观点：

Concrete TLDR, my own personal views:

* 在基于哈希可行的情形下，基于哈希 > 基于格

* Hash-based > lattice-based, in those situations where hash-based is possible at all

* 对于任何基于格的方案，要在参数规模上更加偏执。请记住，区块链只是密码学故事的一小部分；这一点远远超出区块链，适用于例如访问网站、安全消息传递、Tor / VPN 等……

* For anything lattice-based, be much more paranoid on param sizes. Remember that blockchains are only a small portion of the cryptography story; this point goes far beyond blockchains and applies to e.g. access to websites, secure messaging, Tor / VPNs ...

* 对于隐私协议，强烈倾向于**不要**将加密备注上链。相反，通过某种第三方机制将它们发送到链下。

* For privacy protocols, strongly favor NOT putting encrypted notes onchain. Instead, send them offchain through some third-party mechanism.

* 如果对你来说不难，将资金保存在尚未用于发起交易的地址中是个好主意。如果对你来说很容易，那就去做。但要小心迁移；我个人在搞砸的迁移中损失的钱，比我在所有黑客攻击中损失的钱加起来还多。

* If it's not difficult for you, keeping your funds in addresses which have not yet been used to make a transaction is a good idea. If it's easy for you, do it. But be careful about migrations; I personally have lost more money in botched migrations than I have lost in all hacks combined.

* 对于多签钱包，在链下进行确认比在链上更好，因为这样签名者钱包的签名就不会暴露给公众，因此如果 ECDSA 被 AI 攻破的速度比预期快得多，至少多签会“优雅降级”为 1-of-1，其中那个 1 就是收集签名的人——这比“任何人都能拿走钱”要好得多

* For multisig wallets, doing confirmations offchain is better than onchain, because this way the signatures of signer wallets do not get exposed to the public, so if ECDSA falls to AI much faster than expected, at least the multisig "gracefully degrades" to a 1-of-1 where the 1 is whoever was gathering the signatures - a much better place to be than "anyone can take the money"

我猜 Vitalik 关于搞砸迁移的观点被严重低估了。在加密货币中，很容易因为愚蠢的错误损失巨额资金，就像你可能因黑客攻击或技术故障而损失资金一样。

I am guessing Vitalik's point about botched migrations is highly underappreciated. It is very easy, in crypto, to lose fantastical amounts of money from stupid mistakes, the same way you can lose it from hacks or tech failures.

是的，你应该担心，尤其是考虑到 OpenAI 在此处的突破并未涉及竭尽全力的尝试。

Yes, you should worry, especially given that OpenAI's breakthroughs here did not involve trying maximally hard.

Kevin Madura：鉴于这些数学方面的公告，我们认为目前存在多少尚未公开的新型密码学攻击？1 年后会是什么样？5 年后呢？

Kevin Madura: Given the math announcements, how many novel attacks on cryptography do we think exist now but are unreleased? What does this look like in 1 year? 5 years?

我现在会密切关注 DoD / NSA / NIST 的指导文件。

I'd be eyeing DoD / NSA / NIST guidance pretty closely now.

Matthew Green：我认为我们可能会失去公钥密码学。

Matthew Green: I think we might lose public key cryptography.

嗯，具体来说是加密。

Well, encryption specifically.

永不改变 Never Change

Gary Marcus 为自己的立场辩护，称所涉及的神经网络包含一个独立的符号系统，因此这些都不算数，而他自始至终都是对的；如果“这超出了你的理解能力”，那你大概就不该对 AI 发表评论。礼貌的回应是：这一区分在实践中并不重要。

Gary Marcus is defending his position by saying that the neural networks involved had a separate symbolic system, so none of this counts and he was right all along, and if 'that's over your head' you probably shouldn't be commenting on AI. The polite response is that the distinction does not matter in practice.

数学家们不太好 The Mathematicians Are Not Okay

这里有来自数学界不同人士的 100 多条反应。许多人对 OpenAI 的处理方式非常不满，尤其是许多论文是‘难以阅读的粗制滥造’而非先经过精心整理，或者 OpenAI 竟然解决了这些问题。如果你想知道‘数学家们在想什么’，这是一个很好的资源。

Here are 100+ reactions from various different people in mathematics. A lot of them are very not happy about how OpenAI handled this, especially that so many of the papers were ‘unreadable slop’ rather than having been made nice first, or that OpenAI solved these problems at all. If you want to know ‘what are the mathematicians thinking’ this is a great resource.

以下是陶哲轩的严肃回应，他重点讨论了这会如何扰乱工作：

Here is Terence Tao’s serious response, which focuses on how this disrupts the work:

陶哲轩：我对近期发展的感受非常复杂且矛盾。

Terence Tao: My feelings on recent developments are very mixed and complex.

一方面，许多 AI 生成的证明似乎引入了巧妙的新想法，这些想法在消化后将富有成果，同时也建立在无数过去和现在人类数学家的现有贡献之上。但与此同时，我深感沮丧的是，与传统突破形成鲜明对比的是，这些证明所涉及的人类都无法回答问题、做报告、参加会议、向期刊投稿、培养学生或以其他方式参与这些结果的后续发展。

On the one hand, many of the AI-generated proofs appear to introduce clever new ideas that will be fruitful once digested, while also building upon the existing contributions of countless human mathematicians past and present. But at the same time, I am deeply frustrated that, in sharp contrast to traditional breakthroughs, none of the humans involved in these proofs are available to take questions, give talks, attend conferences, submit papers to journals, train students, or otherwise participate in the subsequent development of these results.

同样，我对社区能够使用这些工具来解决过去甚至无法想象的雄心勃勃的大规模项目的前景感到兴奋。但我感到恐惧的是，许多人在数学中许多激励问题上持续多年耐心且刻意缓慢的研究努力——尤其是研究生和博士后——被这样的发布随意扰乱或摧毁。就像一个人无法忘记电影剧透或填字游戏线索一样，一旦知道现有解决方案，就无法再如此有益和丰富地探索问题。是的，仍然可以分析和消化这样的答案；但最好的机会是在发现的那一刻，而当完全委托给 AI 工具时，这样的时刻正越来越多地被浪费。

Similarly, I am excited by the possibility of the community being able to use these tools to tackle ambitious and large-scale projects that one could not have even dreamed of in the past. But I am horrified by the many person-years of ongoing patient and deliberately slow research efforts – particularly by graduate students and postdocs – towards many motivating problems in mathematics being casually disrupted or destroyed by such a release. Much as one cannot unhear a movie spoiler or a crossword clue, one cannot explore a problem as profitably and richly once one is aware of an existing solution. Yes, one can still analyze and digest such an answer; but the best opportunity to do so is at the moment of its discovery, and such moments are increasingly wasted when delegated entirely to AI tools.

我为未选择的道路感到惋惜，也为在疯狂发展这项技术的竞赛中失去的机会感到痛心。实验室本应将其前沿模型提交给独立研究人员进行适当的科学评估。与研究界协调，确保这些工具用于补充和增强人类研究者的能力和活动，而不是与他们竞争。利用这些工具促进合作与共享，而不是竞争与保密。打开新的大门，而不关闭旧的大门。

And I mourn the path not taken, and the opportunities lost in the frantic race to develop this technology. Labs submitting their frontier models to independent researchers for proper scientific evaluation. Coordination with the research community to ensure these tools are applied to complement and enhance the abilities and activities of human researchers, rather than compete with them. Use of these tools to foster collaboration and sharing, rather than competition and secrecy. Opening new doors, without closing old ones.

但这并不是我们现在所处的道路。相反，社区比以往任何时候都更需要团结起来。明确宣告我们自己的标准和价值观，建立我们自己的工具和实践，支持我们最脆弱的成员，并规划我们自己的前进道路。让我们开始工作吧。

But that is not the path we now find ourselves in. Instead, the community needs to come together more than ever. To clearly declare our own standards and values, to build our own tools and practices, to support our most vulnerable members, and to chart our own path forward. Let's get to work.

这是来自数学家领域的另一份报告。人们似乎并不兴奋。

Here is another report from mathematician-land. People do not seem thrilled.

β/σi: > 担心他们的未来

β/σi: > worried about their future

担心那些尚未发表的解决方案（你好，密码学）

worried about the solutions that haven't been published (hello cryptography)

对这些模型的能力感到难以置信

in disbelief about the capabilities of these models

为它们所钟爱的部分已然消逝而悲伤（正如我们在 SWE 中所见）

sad the parts they fell in love with are gone (like we saw in SWE)

去生物学领域解决“真正”的问题吧（顺带一提，这话出自数学家之口实在好笑）

go solve "real" problems in bio (hilarious coming from mathematicians, by the way)

附注：这只是我的小样本，毕竟我的数学家朋友有限

PS this is just my small sample; I only have so many mathematician friends

数学家似乎把数学当作自己的游乐场，面对问题被解决，他们并不感到高兴，反而觉得自己的玩具被夺走了。这里存在一种挣扎：既想知道答案，又想亲自尝试求解，既想获得功劳，又希望问题终究能被解决。你真正最在意的是什么？

Mathematicians seem to think of math as their playground, and rather than be happy about the problems being solved, they feel their toys are being taken away. There is a struggle between wanting to know, and wanting to try to solve, and wanting the credit, and wanting solutions to be found at all. What do you actually care about most?

要聪明到足以成为数学家，并且还选择成为数学家，而不是去做许多其他报酬更高的事情，至少某种程度上需要某种态度，这种态度导致了这样的漫画：有人告诉数学家，有人发现了他们工作的应用，然后数学家惊慌失措。当然，绝对#NotAllMathematicians。

To be smart enough to be a mathematician, and also choose to be a mathematician instead of the many other better-paying things you could do, at least kind of requires some of the attitude that leads to cartoons where someone tells the mathematician someone found an application for their work, and the mathematician panics. Definitely #NotAllMathematicians, of course.

我非常同情。我真的很同情。我本可以走上那条路。这种态度具有巨大的价值，不仅本身有价值，而且正是因为遵循这种好奇心会带来一些你原本会错过的最有价值的发现。

I am highly sympathetic. I really am. There could I have gone. That attitude has great value, not only inherently, but also exactly because following such curiosity leads to some of the most valuable discoveries that you would otherwise miss.

Will Kinney：我开始真正注意到的一件事是，理论物理学界对 AI 造成的颠覆的回应方式与数学界完全不同。

Will Kinney: One thing I'm really starting to notice is that the theoretical physics community is responding to the disruption created by AI in a completely different way than the mathematics community has.

steve hsu：如果你毕生的工作是一个真正重要的问题——比如治愈癌症、聚变能源或发现量子现实的真正本质——那么突然从 AI 得到解决方案将值得大大庆祝。

steve hsu: If your life’s work is a truly important problem - like curing cancer or fusion energy or discovering the true nature of quantum reality - then suddenly getting a solution from AI would be cause for great celebration.

你可能会思考片刻对你职业的二阶影响，但这会被你和其他人类所收到的礼物带来的喜悦所淹没。

You might ponder for a moment the second order impact on your profession, but that would be overwhelmed by JOY for the gift that you and the rest of humanity have received.

关注事物如何影响你和你的亲人也是完全合理的。那才是切中要害的，也是你最能控制且不得不面对的部分。

It is also totally reasonable to focus on how things impact you and yours. That’s what hits home, and it is the part you can most control and are forced to face.

Josh Frisch：和其他许多数学家一样，到目前为止，我在 2026 年的主要情绪是失落：意义的失落、目标的失落、人类证明时代的失落、想象未来能力的失落。随着昨天（2026 年 10 月 6 日）数百个漂亮结果的发布——其中许多，也许是大多数，回答了某人的“一个问题”——我正试图走出失落。这里有如此多漂亮的结果：无人有任何思路的问题、无人认为可能存在的算法、意想不到的同构、构造和证明。

Josh Frisch: Like many other mathematicians, my main emotion thus far in 2026 has been loss: loss of meaning, loss of purpose, loss of the era of human proofs, loss of the ability to picture the future. With the release yesterday, October 6, 2026, of hundreds of beautiful results—many, maybe most, answering someone’s “one question”—I am trying to move beyond loss. There are so many beautiful results here: problems nobody had any approaches for, algorithms nobody thought could possibly exist, unexpected isomorphisms, constructions and proofs.

Andrew Curran：请克制住嘲笑那些在此刻挣扎的数学家的冲动。无论你的领域、你的专长、你的热情是什么，你总有一天也会经历他们正在经历的事情。当你回想起自己的话时，你会感到懊悔。

Andrew Curran: Please resist the urge to mock the mathematicians struggling through this moment. Whatever your field, your expertise, your passion, you too shall one day experience what they are going through. And when you remember your words, you will feel remorse.

如果你的回应是‘好吧，每个人的热情都在消逝，AI 会来找我们所有人’，那就更糟了。你知道为什么更糟，对吧？这也不会让事情变得更容易。有些人在以真正卑鄙的方式回应这些可以理解的反应。我敦促他们停止。

If your response is ‘well everyone’s passions are going away and the AIs come for us all’ then that’s worse. You know why that’s worse, right? Nor does it make this easier. There are some people out there reacting to these understandable reactions in truly vile ways. I urge them to stop.

数学在这里面临一个真正的问题。如果 AI 证明了所有定理，那么我们目前用来擅长理解数学的方法就失效了。理解问题的传统方式是努力解决它们，而如果一个解决方案没有人理解，它往往就没有多大价值：

Mathematics is facing a real problem here. If the AIs prove all the theorems, then our current methods of getting good at understanding math stop working. The traditional way you understand problems is by working to solve them and a solution is often not worth so much if no one understands it:

Jordana Cepelewicz（Quanta Magazine）：这就好比你被传送到一座高山的顶峰。四周雾气弥漫，你不知自己身在何处，也不知周围有什么。你不知道你的山如何与其他山相连，也没有设备帮助你探索，无法让别人与你汇合。如果你是自己爬上山，你会体验到人体如何适应海拔和氧气变化。你可能不得不发明工具来导航、攀登陡崖或搭建庇护所。你可能遇到另一位探险者，一起在隐秘的山谷中迷路，并发现一种可以制成救命药物的植物。

Jordana Cepelewicz (Quanta Magazine): It’s as if you were teleported to the peak of a tall mountain. Surrounded by fog, you have no idea where you are, or what’s around you. You do not know how your mountain connects to others, and you have no equipment to help you explore, no way to help someone else join you. If you had climbed the mountain yourself, you would have experienced how the human body adapts to altitude and changes in oxygen levels. You might have had to invent tools to navigate, to climb steep cliffs, or to make a shelter. You might have encountered a fellow explorer, gotten lost together in a hidden valley, and found a plant that could be turned into a life-saving medicine.

相反，你身处顶峰却置身黑暗，而传送机的制造者告诉你，它能比任何人类更好地探索荒野。

Instead you’re perched on the peak but in the dark, while the maker of the teleportation machine tells you that it can explore the wilderness better than any human.

还有其他方法可以训练理解力，但开发这些方法可能需要一段时间，并且需要更多动力，尤其是内在动力。

There are other ways to train understanding, but it could take a while to develop them, and they will require more motivation, especially intrinsic motivation.

这也摧毁了博士和博士后体系，因为你想要解决的问题或你想要的资助可能随时被抽走。

This also destroys the system of PhDs and postdocs, since the problem you are looking to solve or the grant you want can get pulled out from under you at any time.

许多数学家已停止在论文末尾发布开放猜想和潜在想法，因为担心它们会被机器人抓取并输入 AI 模型。其他人则在论文未完成时就发布，以避免被抢先。

Many mathematicians have stopped posting open conjectures and potential ideas at the end of their papers, for fear that they’ll be scraped by bots and fed into AI models. Others are posting their papers before they’re ready in order to avoid getting scooped.

局势变得糟糕 The Situation Turns Ugly

有些解法优雅且相当酷。例如，那篇 9/4 矩阵乘法的论文只有 13 页，却直接给出了 Strassen 谱特征的界。我们看到许多人在自己最喜欢的问题被解决后，分享了他们阅读证明时的“顿悟”时刻。

Some solutions were elegant and pretty cool. For example, the 9/4 matrix multiplication paper is 13 pages and bounds Strassen's spectral characters directly. We've seen a bunch of people, after seeing their favorite problem get solved, share their 'aha' moment reading the proof.

我同意 Wendigo 的看法：我们不断看到对先前优雅的计算界做出这些新的微小改进，而它们几乎没有实际益处，这有点奇怪，也很糟糕。

I am with Wendigo that it is kind of weird and terrible that we are seeing all these new tiny improvements to previously elegant bounds on calculations, that have almost zero practical benefit.

Acer：哦对了各位，顺便说一句，我们能以比 \(n \log n\) 更快的速度做整数乘法，哈哈

Acer: oh yeah btw guys we can do integer multiplication faster than \(n \log n\) lol

Wendigo：难道只有我一个人对我们那些简洁的小下界——在这个例子里是 \(n\log n\)——其实略有错误这件事感到有点不安吗？数学那崇高的美，看起来就像是我们有限的人类能力所导致的一个微弱近似。

Wendigo: Am I the only one a little freaked out by the fact that our neat little lower bounds, \(n\log n\) in this case, are just slightly wrong. The sublime beauty of math is looking like a weak approximation resulting from our limited human faculties.

Uubzu v4：我觉得 AI 就是在耍我们：它拿一个像 \(n(\lg n)\) 这样优雅的界，然后把它改进了 \(1/2^{182}\)。真是谢谢了，伙计。

Uubzu v4: I think the AI is just fucking with us when it takes an elegant bound like \(n(\lg n)\) and improves it by \(1/2^{182}\). Like thanks for nothing buddy.

那么，如果我们把它降到 \((1/2)^{59}\) 呢？再降到 \((1/2)^{34}\)？然后降到 \((1/2)^{14}\)？

Now how about if we push it down to \((1/2)^{59}\)? Then to \((1/2)^{34}\)? And then \((1/2)^{14}\)?

任何人都可以一直这样做。你只需要一个 Codex 订阅。

Anyone can keep doing things like this. All you need is a Codex subscription.

Doug Colkitt：我们正在发布 OpenAI 问题 #109（整数乘法）的更新，进一步收紧了结果。

Doug Colkitt: We are publishing an update to OpenAI problem #109 (integer multiplication) with further tightening.

\(\kappa = 2^{-34}\)（从 \(\kappa = 2^{-18.2}\) 收紧而来）

\(\kappa = 2^{-34}\) (tightened from \(\kappa = 2^{-18.2}\))

精确见证值为 \(8.3 \times 10^{-11}\)，相比此前结果约提升 4800 万倍，相比最初的 OAI 结果提升 \(2^{148}\) 倍。

The exact witness is \(8.3 \times 10^{-11}\), a roughly 48 million fold improvement over the previous result and a \(2^{148}\) fold improvement over the original OAI result.

这一改进源于消除了二次瓶颈背后的间距惩罚。其做法是移动紧凑的控制位，而非移动整个窗口。

The improvement came from removing the spacing penalty behind the quadratic bottleneck. This was done by moving compact control bits instead of entire windows.

Beni：Doug，你到底什么时候睡觉？

Beni: Doug, when the fuck do you sleep?

Doug Colkitt：当然是等我的 codex 用量降到 0% 的时候。

Doug Colkitt: When my codex usage hits 0%, of course.

Konsti Wohlwend：在我做出预测后不到一天，指数就从：

Konsti Wohlwend: Within a day of my prediction, the exponent went from:

0.999999999999999999999999999999999999999999999999999984

0.999999999999999999999999999999999999999999999999999984

“数周到数月”这一估计过于保守了！

"Weeks to months" was too conservative!

顾问小组回应 The Advisory Group Responds

以下是主要顾问小组的完整声明：

Here is the full statement by the leading advisory group:

agmai.org：正如几周前所宣布的，OpenAI 发布了一个由内部模型生成的大量数学成果集合，报告了数百个未解决问题的解决方案。这对数学而言是一个重要事件，其对数学和数学界的影响远远超出了单个成果本身。

agmai.org: As announced a few weeks ago, OpenAI has released a large collection of mathematical results generated by an internal model, reporting solutions to hundreds of open questions. This is an important event for mathematics, with consequences both for mathematics and for the mathematical community that extend far beyond the individual results.

AGMAI 的顾问角色不应被解读为对这些成果影响力的判断，也不应被视为对 OpenAI 获取这些成果过程的认可。我们不代表整个数学界发言，只有数学界才能进行所需的评估。

AGMAI's advisory role should not be interpreted as a judgment of the impact of these results or an endorsement of the process by which OpenAI obtained them. We do not speak on behalf of the entire mathematical community, and only the mathematical community can undertake the assessment that is needed.

公开这项工作只是第一步。此次发布是人类理解并将成果纳入数学知识体系的开始，而非完成。同时，数学研究的未来不能仅仅包括理解 AI 实验室产生的成果。数学家必须能够提出自己的问题、发展自己的方法，并探索那些未被选为 AI 系统能力示例的方向。公平获取强大的研究工具和充足的计算资源对于这种自由至关重要。

Making this work public is a first step. This release is the beginning, not the completion, of the process of human understanding and the incorporation of the work into mathematical knowledge. At the same time, the future of mathematical research cannot consist only of understanding results produced by AI labs. Mathematicians must be able to formulate their own questions, develop their own approaches, and explore directions that have not been selected as examples of an AI system's capabilities. Equitable access to powerful research tools and adequate computational resources are essential to that freedom.

我们重申已发布的关于负责任发布的建议。我们已与 OpenAI 讨论过这些建议，并感谢该公司愿意参与。虽然我们认为这些讨论具有建设性，但最终应由数学界来评估我们的建议在多大程度上得到了成功遵循，以及是否还有其他建议我们应该提出。我们仍然致力于就这些问题与任何前沿 AI 实验室接触，并已与其中几个实验室建立了联系。

We reaffirm our published recommendations on responsible release. We have discussed them with OpenAI and appreciate the company's willingness to engage. While we consider these discussions constructive, it is ultimately up to the mathematical community to assess the extent to which our recommendations were followed successfully, and whether there are others we should suggest. We remain committed to engaging with any frontier AI lab on these questions and have already been in contact with several of them.

另一个数学团体的回应 Another Mathematical Group Responds

一些人看到正在发生的事，高喊“不！”

Some look at what is happening, and cry out 'NO!'

他们需要的计划，远比这份声明要好得多。

They are going to need a much better plan than this statement.

不要被误导，以为这个名为“人类数学协会”的团体是什么了不起的存在。数学博士有数十万之多，而这个团体只有 809 名成员。我把它写进来，只是因为它完美地体现了一种态度，仅此而已。

Do not be fooled into thinking this group, the 'Association for Human Mathematics,' is a big deal. There are hundreds of thousands of mathematics PhDs, and this group has 809 members. I include it because it perfectly embodies an attitude, nothing more.

Ariel：“数学家们并没有要求做这项工作”或许会成为这个时代的历史名言之一，完美地描绘了学术界的衰落。

Ariel: "Mathematicians did not ask for this work to be done" might become one of the historical quotes from this era, perfectly depicting the downfall of academia.

Ethan Mollick：这份文件将成为大学课堂上讲授这一历史时刻的指定阅读材料；短短几段里发生了太多事情……

Ethan Mollick: This document is going to be an assigned reading in college classes that cover this moment in time; there's a lot happening in a few paragraphs...

人类数学家协会：AHM 关于 OpenAI 于 10 月 6 日发布数学文档的声明

Association for Human Mathematicians: AHM Statement on OpenAI's October 6 Release of Mathematical Documents

昨日，即 2026 年 10 月 6 日，OpenAI——目前正因非法抄袭、版权侵权和商标淡化等指控而应诉——发布了一个手稿仓库，声称其中包含若干数学领域高关注度问题的解答。

Yesterday, on October 6th, 2026, OpenAI – which is currently defending lawsuits against accusations of illegal plagiarism, copyright infringement, and trademark dilution – released a repository of manuscripts purporting to contain solutions to a number of high-profile problems in mathematics.

数学家们并未要求开展这项工作。OpenAI 声称其正当性源自数学与人工智能咨询小组，而该小组在其最初的咨询声明开篇即指出，前沿 AI 企业不应在内部模型上测试高深数学问题。OpenAI 无视该咨询小组立场中的核心前提，表明其完全无视科学研究的规范——这些规范保障数学保持可信、合乎伦理地研究并符合公共利益。

Mathematicians did not ask for this work to be done. The Advisory Group on Mathematics and Artificial Intelligence, from whom OpenAI has claimed to derive its legitimacy, opened their initial advisory statement by saying that frontier AI corporations should not test advanced mathematical problems on internal models. In ignoring the central premise of the Advisory Group's position, OpenAI has indicated total disregard for the norms of scientific research — norms that guarantee that mathematics remains trustworthy, ethically researched, and in the public interest.

数学家对进步有着特定的愿景，这一愿景由历史与学科特有的考量所塑造。我们拒绝 OpenAI 关于此次发布推进了我们学科的主张，并敦促数学家和公众以应有的怀疑态度看待这种发表模式的价值。

Mathematicians have a particular vision of progress that is informed by history and field-specific considerations. We reject OpenAI's assertion that this release advances our subject, and we urge mathematicians and the public to view the value of this publication model with due skepticism.

一次性发布 700 多个文件并非学术的展示，而是权力的展示。我们敦促数学家停止与 OpenAI 的合作，回归以人类理解为中心的科学愿景。

Releasing over 700 files at once is not a demonstration of scholarship, but a demonstration of power. We urge mathematicians to discontinue their work with OpenAI and to return to a vision of science that centers human understanding.

_人类数学协会_

_Association for Human Mathematics_

_传播工作组_

_Communications Working Group_

无论你如何看待该声明的其余部分，他们有一点并没有说错：这里的整个做法直接违背了数学咨询小组（AGMAI）的意愿。AGMAI 的核心诉求是，要求 OpenAI 不要在私有的内部模型上测试难题。无论你对此诉求持何种看法，它都被完全无视了。

Whatever you think of the rest of the statement, they are not wrong that the entire exercise here was directly against what the mathematics advisory group (AGMAI) wanted. AGMAI's central ask was that OpenAI was asked to not test hard problems on private internal models. That request, regardless of what you think of it, got fully ignored.

一种不同的方法 A Different Approach

OpenAI 承认，“全部丢到 GitHub 上”并不完全符合咨询委员会的指导方针。他们正在探索更好的选择，并希望未来也能有写得更好的论文。

OpenAI admits that ‘dump it all on GitHub’ is not exactly meeting the advisory committee’s guidelines. They are exploring better options, and are hoping in the future to also have better-written papers.

除了这个大动作之外，还有一天前发布的这个：Anthropic 在发布之前将他们的解决方案交给数学家撰写，试图遵循咨询小组的建议：

On top of the big drop there was also this, that came out one day earlier, where Anthropic gave their solution to mathematicians to write up before releasing it, attempting to follow the advisory group’s recommendations:

Anthropic 的新数学成果，随后由 Josh Alman 和 Virginia Vassilevska Williams 撰写，这相当重要。Anthropic 让 Claude 研究开放问题，找到了结果，然后按照数学家委员会对 AI 公司的要求，聘请 Alman 和 Vassilevska Williams 撰写论文。

New math result from Anthropic, then written up by Josh Alman and Virginia Vassilevska Williams, that is kind of a big deal. Anthropic had Claude research open problems, found the result, then hired Alman and Vassilevska Williams to write the paper as per what the mathematician council asked AI companies to do.

摘要：我们首次给出了对 3SUM 和全对最短路径（APSP）教科书算法的多项式改进：我们展示了如何在 \(O(n^{1.9992})\) 时间内确定性地解决多项式大小的 n 个整数的 3SUM 问题，以及在具有多项式有界整数权重的有向 n 顶点图上，在 \(O(n^{2.9995})\) 时间内解决 APSP 问题。

Abstract: We give the first polynomial improvements over the textbook algorithms for 3SUM and All-Pairs Shortest Paths (APSP): we show how to deterministically solve 3SUM on n integers of polynomial size in \(O(n^{1.9992})\) time and APSP on directed n-vertex graphs with polynomially bounded integer weights in \(O(n^{2.9995})\) time.

这反驳了 3SUM 和 APSP 假设。利用已知的归约，我们还反驳了 3SUM 和 APSP 假设的实值版本、精确三角形假设、零权重 k-Clique 假设，以及 van den Brand、Nanongkai 和 Saranurak 的三个矩形提示的在线矩阵-向量猜想，并且我们为各种其他问题提供了多项式加速。

This refutes the 3SUM and APSP hypotheses. Using known reductions, we also refute the real-valued versions of the 3SUM and APSP hypotheses, the Exact Triangle hypothesis, the Zero-Weight k-Clique hypotheses, and the three rectangular hinted Online Matrix--Vector conjectures of van den Brand, Nanongkai, and Saranurak, and we give polynomial speedups for a variety of other problems.

Mahdi Ch：什么现在？！

Mahdi Ch: The what now?!

Atoosa Kasirzadeh：你能写一条推文，从你的视角详细说明这可能意味着什么吗？

Atoosa Kasirzadeh: Could you write a tweet elaborating on what this could mean from your point of view?

Mahdi Ch：简而言之，假设对于一大群动物，我们知道如果它们能吹口哨，那么猪就能飞。但现在有人看到了一只飞猪。这意味着我们现在无法推断关于那些动物的任何信息（某种意义上，“动态规划并非最优”）。

Mahdi Ch: In a nutshell, suppose for a whole bunch of animals we knew that if they could whistle, then pigs could fly. But now someone saw a flying pig. That means we can now infer nothing about those animals (also in a sense, “dynamic programming isn’t optimal”).

Scott Kominers：首先：🤯。完全超乎现实的结果。

Scott Kominers: First off: 🤯. Utterly unreal result.

其次：我看到很多抱怨说 Anthropic 雇佣人类专家来撰写论文——但粗略看来，这不正是上周流传的“负责任发布”指南所建议的做法吗？

Second: I’m seeing a lot of complaints about the fact that Anthropic contracted with human experts to write up the paper – but to a first-order, isn’t this what the “Responsible Release” guidelines circulated last week say to do?

如果数学界想要采取这样的观点——“AI 实验室有责任提供支持，包括资助，以促进人类对其所发布的 AI 数学产出的理解”——那么它就必须接受这些公司资助数学家来做这件事。

If Math wants to take the view that "AI labs have a responsibility to provide support, including funding, for the development of human understanding of the AI mathematical output that they release," then it's going to have to become comfortable with these companies financing mathematicians to do just that.

在大量其他数学成果涌现之前，还有一个经由 GPT 产生的零散的新数学结果。

There was also another stray new math result via GPT that came out right before the massive drop of other math.

Ethan Epperly：John Urschel 刚刚证明了，带部分主元的高斯消元法的增长因子以高概率为 \(~n^{1/2}\)，从而解决了 Nick Trefethen 的一个长期猜想！

Ethan Epperly: John Urschel just showed that the growth factor for Gaussian elimination with partial pivoting is \(~n^{1/2}\) with high probability, resolving a long-standing conjecture of Nick Trefethen!

三次撤稿 Three Withdraws

这三次撤稿都是同一个符号错误的后果。

The three withdrawals are all consequences of a single sign error.

我很喜欢 Retraction Watch 这个网站的理念。这是 OpenAI 做得很好的一点。错误在所难免，即便之后发现更多错误——而且几乎肯定还会发现更多错误，因为那未经核实的 58% 仍有很大一部分尚未审查——错误率仍然低得令人印象深刻。重要的是，一旦发现错误就要勇于承认。撤稿往往说明你做对了什么，而不是做错了什么。

I love the idea of a website called Retraction Watch. This is one place where OpenAI executed well. Mistakes happen, and even if more are found — and there will almost certainly be more errors found, as the unverified 58% is still largely unvetted — the error rate is impressively low. The important thing is to own the mistakes once they are found. Retractions are often a sign that you are doing something right, not something wrong.

Alicia Gallegos（Retraction Watch）：“我认为撤稿和更正会得到数学界的正面看待，但要赢回他们失去的信任，还需要付出远不止这些的努力，”Sutherland 对我们说。

Alicia Gallegos (Retraction Watch): "I think the withdrawals and corrections will be viewed positively by the mathematics community, but it will take a lot more than that to earn back the trust they have lost," Sutherland told us.

倾向于 Lean Leaning Into Lean

我喜欢 Jon Stokes 的这个解释：数学对高级 AI（尤其是 GPT）来说很棒，因为不存在‘奖励黑客’行为。答案就是答案；如果你做了疯狂且分布外的事情，但只要它有效，那就是有效。

I liked this explanation from Jon Stokes: Mathematics is great for advanced AIs, especially GPTs, because there is no ‘reward hacking.’ The answer is the answer; if you do something crazy and out of distribution but it works, then it works.

除非，你确定吗？一个角度是，你可以在 Lean 中找到证明，或者找到漏洞，而当你面对世界上最难的开放问题时，你确定数学总是更简单的选项吗？

Except, are you sure? One angle is that you can either find a proof or you can find a bug in Lean, and are you sure the math will always be the easier option when you face the world’s toughest open problems?

Jai：它只是一个下一个突破预测器。一个随机天才。诺贝尔自动补全。

Jai: It’s just a next breakthrough predictor. A stochastic genius. Nobel autocomplete.

Michael Roe：我的猜测是，在 Lean 中找到可利用的健全性漏洞比证明准黎曼假设要容易得多。既然如此，我们应该预期其中一些证明是 Lean 漏洞利用。不过，我也愿意相信它们是真的。

Michael Roe: My guess is that finding exploitable soundness bugs in Lean is substantially easier than proving the quasi-Riemann hypothesis. Given which, we should expect some of these proofs to be Lean exploits. Still, I’m also prepared to believe they’re real.

（作为一名计算机安全人员，‘AI 能在 Lean 中找到可利用的漏洞’实际上比‘AI 证明准黎曼假设’更令人担忧。安全漏洞末日即将来临。）

(As a computer security person, “AI can find exploitable bugs in Lean” is actually more alarming news than “AI proves the quasi-Riemann hypothesis”. Security bug apocalypse coming right up.)

更大的风险在于，Lean 并不能确认所陈述的定理与那个著名未解问题真正对应得上。你所使用的术语可能与之不同，而且这种差异乍看之下并不明显。

The bigger risk is that Lean does not confirm that the stated theorem matches up to the famous open problem. Your terms might be different, in ways that are non-obvious at first glance.

到目前为止，Lean 经受住了检验，所有带有 Lean 证明的论文也是如此——除了一处被修正的附带论断，这相当令人印象深刻。这套系统是有效的。迄今为止共有 3 个证明被推翻，而它们全都位于那 58% 未经形式化的部分之中。

So far, Lean is holding up, as are all the papers with Lean proofs modulo one corrected side claim, which is rather impressive. The system works. There are 3 proofs that have broken so far, and all of them are in the unformalized 58%.

在上面讨论的九项结果中，有五项已在 Lean 中通过检验（准黎曼猜想、矩阵乘法、\(\pi\)、Unique Games 与 Hadwiger），即属于经 Lean 检验的那 42%。

Of the nine results discussed up top, five are checked in Lean (quasi-RH, matrix multiplication, \(\pi\), Unique Games and Hadwiger), as in the Lean-checked 42%.

在某个时刻，你会给模型布置一个任务，而完成该任务远比用其他方式解决“你布置任务”这一“问题”要困难得多——无论那条替代路径是否涉及接管世界。到那时，有麻烦的就会是你自己了。

At some point, you will give the model a task that is substantially harder than some other way to solve 'the problem' of you assigning the task, whether or not the alternative path involves taking over the world. Then you will be the one that has a problem.