Natanael Wildner

Testing the outer bounds of what LLMs can do

[Natanael Wildner @NatanaelWF]

I’ve solved Dujella’s Problem 4.4 on Diophantine quadruples with the aid of LLMs.

The proof shows that, for any fixed Diophantine triple {b,c,d} with b<c<d, there is at most one positive integer a<b such that {a,b,c,d} is a Diophantine quadruple.

Full proof @dujella1 :
https://t.co/5nViZhAT1a

[Natanael Wildner @NatanaelWF]

Here is the webpage: https://t.co/HcK6bbB5O1

[Natanael Wildner @NatanaelWF]

I built a page to track how far behind open-source models are compared to proprietary ones.

Right now, the gap is about 7.5 months. At this pace, we could see open-source models reaching Fable 5.1/Astra-level performance around April 2027. 👇

[Natanael Wildner @NatanaelWF]

Robots are now committing another crime: suicide.

[宝玉 @dotey]

Figure AI 让旗下的 Figure 02 人形机器人自己跳进芬兰一家铸造厂的钢水里，以此让这一代机器人退役。熔出来的金属会被加工成限量纪念品。

Figure 02 是 Figure 的第二代人形机器人，公司很多第一次都由它完成：第一次进 BMW 工厂上岗，第一次运行 Helix（Figure 自研的机器人 AI 模型，负责看画面、理解指令并控制动作），第一次做家务，第一次做物流工作。现在第三代 F.03 的数量越来越多，Figure 说继续维护 F.02 已经不划算。

退役的麻烦在于保护技术。机器人里有大量自研的执行器（可以理解为驱动关节的电机）等核心硬件，Figure 不想让它们流出去；但一台台拆解又太占工程师的时间，会推迟第四代 F.04 的发布。

Figure 于是在网上征求处理办法，施瓦辛格回复说"熔了它们"。他主演的《终结者2》结尾，T-800 机器人就是沉进钢水里的。Figure 联系了施瓦辛格，他表示想参与，项目随后启动。

找场地并不顺利。美国和墨西哥的铸造厂都不愿意让装着锂电池的机器人跳进自家昂贵的设备，Figure 还请《流言终结者》的前主持人帮忙找过。最后只有芬兰伊马特拉的一家铸造厂同意。

训练在 Figure 位于加州圣���塞的园区进行。团队铺了大气垫，让机器人从二楼往下跳。他们参考特技演员的动作，在仿真环境里训练了一个新的 AI 模型，让机器人在一个从没去过的铸造厂里，准确跳进盛钢水的容器。

铸造厂用的是一座 75 吨的电弧炉，靠三根石墨电极通电熔化废钢。团队只有 24 小时、6 炉的机会，每炉钢水只有 20 分钟可用，之后表面就会冷却结壳。Figure 称，现场的高温和强电磁场让摄像器材等电子设备出了故障，机器人则一直正常运行自己的 AI 控制程序。

熔出的金属锭已运回美国，正在加工成一批纪念 F.02 的限量藏品，数量很少，面向个人出售，博文没有公布价格和购买方式。大部分 F.02 已被熔掉，只有几台留在总部仓库。

Figure 在文末特意说明，视频不是 AI 生成的：机器人确实是自主起跳，被运到芬兰，跳进了钢水。

[Natanael Wildner @NatanaelWF]

OpenAI did a pretty great job advertising Anthropic today. I’ve been an OpenAI user since 2022, and today I finally subscribed to Claude.

[Natanael Wildner @NatanaelWF]

David came up with a fun way to compete for math. I'm just playing around with it, but it’d be awesome if you @dwrensha built something similar for the 'Elliptic Curve Rank Leaderboard'.

[Natanael Wildner @NatanaelWF]

@thsottiaux Speculation: The “1 point” appears six times, so that’s six “.1”s. Could this mean Astra 6.1?

[Natanael Wildner @NatanaelWF]

Everyone’s talking about Opus 5.5.

So I told Astra: Opus is humiliating OpenAI. Save the company’s reputation.

Astra cooked for 6 hours and dropped this clip.

[donald @donaldjewkes]

I made this with one prompt using Opus 5.5

I spoke to my computer for 5mins, claude worked for 12 hours, and I woke up to this

full prompt:

[Natanael Wildner @NatanaelWF]

I remember that before Astra launched, OpenAI said it had solved that list of 10 really hard problems. This is definitely not the Astra we got.
We’re not getting the same Bel they have either. The gap between internal and released models keeps getting wider.

We’ll have to work some magic with our little toys

[Natanael Wildner @NatanaelWF]

I’ve published a solution to Dujella’s Problems 1.12 and 1.13 on D(1)-tuples @dujella1. A determinant bounds each height range; the Quantitative Subspace Theorem controls the high-height tail. Together they yield a bound depending only on the field degree.

https://t.co/XsnvnhrRpM

I also added a short note at https://t.co/UeU6sxCJBj, sharing a bit about my personal experience using LLMs. Over time, I’ll continue sharing more results and lessons learned.

[Natanael Wildner @NatanaelWF]

I’ve just published a solution to Dujella’s Problem 4.3 @dujella1. Odd powers of the Pell unit 485 + 66√54 generate an explicit infinite family of pairs of distinct Diophantine quadruples with the same largest element; a polynomial identity proves the match.
https://t.co/xeeadsRJVx

[Natanael Wildner @NatanaelWF]

Over the past few days, I’ve made significant progress on the conjecture in Green’s Open Problem 24.

Before formalizing it in Lean and going further, I decided to pause and work on a system for making AI better at collaborating and solving hard problems.

I think we’re still using LLMs in fairly primitive ways in science. We need to build mechanisms for this new kind of intelligence, just as paved roads came after cars.

[Natanael Wildner @NatanaelWF]

@ErnestoSOFTWARE Cool, but I think that fly is rated like 100 Elo

[Natanael Wildner @NatanaelWF]

@Shaun_Fosmark Most researchers already expect P ≠ NP. The breakthrough would be proving it, but the immediate practical effects might be modest. P = NP would be the real shock

[Natanael Wildner @NatanaelWF]

Honestly, the behavior seems a bit random. For now, the best experiences getting Astra to work for longer have come from giving it an objective task list to execute alongside a detailed prompt, and telling it at the end not to be lazy, that quality is more important than speed, and to take as much time as it needs. These phrases produced better results than the traditional 'work hard' or 'do your best,' but I'm still experimenting..

[Natanael Wildner @NatanaelWF]

@dhvanil Exactly. I remember asking GPT-4o when it expected a model to win a gold medal at the IMO, and it estimated 2029. LLMs place a lot of weight on the barriers that need to be overcome.

[Natanael Wildner @NatanaelWF]

10000 agents 🤯

[OpenAI @OpenAI]

We’re sharing a solution to the Navier-Stokes Millennium Prize Problem, one of the deepest problems at the frontier of mathematics.

The proof was produced by a group of agents, using an OpenAI next-generation model significantly more capable than GPT-6 Astra.

The problem concerns whether the description of smooth three-dimensional fluid motion modeled by the Navier-Stokes equations can break down. It has remained unresolved for roughly 90 years.

[Natanael Wildner @NatanaelWF]

Sometimes people forget that 'matrix multiplication' would not produce a single meaningful token without centuries of accumulated human knowledge.

[Nick @nickcammarata]

humans, a bipedal primate notable for early contributions to matrix multiplication

[Natanael Wildner @NatanaelWF]

@sama This message appears for pretty much everything in Astra. It's annoying and keeps us from knowing what's going on during thinking. But it's a good model btw