#推理

Grok 4.5 just constructed an explicit counterexample to hypercontractivity for the Poisson semigroup (the square root of the Laplace–Beltrami operator) on the 4-sphere. Back in 2021, with Rupert Frank https://arxiv.org/abs/2101.06209 we proved that hypercontractivity holds in dimensions ≤3 and fails in sufficiently large dimensions (for example, in dimension 13). Grok's example shows that it already fails in dimension 4, making our earlier result sharp. I also tested this problem on several other frontier AI models. One of them also managed to find a counterexample, but I particularly like Grok 4.5's solution: it is explicit, simple, and elegant. The attached files were generated entirely by Grok 4.5 build (with zero intervention on my side).

@devberkayy 5.6 on Cerebras is insane. 10x speed.

My view of: Fable 5 vs GPT-5.6-Sol. They are not easy models to compare, these are my vibes - take them as you will. My overall feel is that Fable is a 'wise owl' who is very thoughtful and very well spoken, GPT-5.6-Sol is like a rottweiler who will grab the problem by the throat and not let go until it is done. In other words, Fable, is a fundamentally smarter model - even at low reasoning it can be very insightful and writes in a clear compelling way. GPT-5.6-Sol on the other hand is extremely diligent, I can give it a list of 8 things to do and you will be sure that they will be done. Fable feels more arrogant to me, I was both to get it to build a new benchmark for me - 5.6 worked between 6 hours and 2 days (I tried several times) and it came up with very thoroughly tested, working benchmark. Fable came back within 40 minutes (twice) and the benchmark sounded smart, but was ultimately was 'vibe' based slop and since it was Fable's vibes that was doing the judging, it decided that it was good to go (it kept giving Fable 100% score btw). Some thoughts by category: UI & App building: Fable will still craft a better UI from scratch, the flow of the app would probably be a bit nicer. But I find that Fable often misses quite key things, which GPT-5.6-Sol doesn't. GPT's Frontend skills are big jump vs previous GPT models, but still not as great overall. Writing: Fable is better hands down, Sol feels quite difficult to align to what I want to say or explain things to me simply. Though I think the 'Pro' model writes clearer. Robustness & Reliability: This is where I think GPT-5.6-Sol wins for me hands down. Fable seems to do things of high quality, but I can never relax with it, it always misses something. With 5.6 this just almost never happens. Other things where I liked GPT-5.6-Sol, but can't compare to Fable directly. - Video editing is actually working now, it is not completely perfect, but with the right skill/guidance you can just give it 1h footage and it can give you a 5 min highlight clip no problem - Computer use - getting really rather good, very usable - Sub agents - it is very fluent at managing sub-agents and speaking to different threads, can help with some new workflows - Adhering to existing code patterns - I love this, even without asking it would implement something in a way that aligns with you app - major problem for slop generation - Research - I think it is getting quite a bit better, it still has some bad patterns (e.g being too tactical), but it feels like it is more steerable to be a good researcher - Multi-day runs - the /goal feature is pretty insane with 5.6-Sol, you can run it for days if you wanted to and it does work. Useful to have another thread or /side to check up on it, but I have some great results with it - Token efficiency - it is so much more token efficient and faster than 5.5, in reality it is now much faster than Fable too On the downside, you can feel that Fable is naturally smarter, and I did have some baffling moments with 5.6 when I was getting it to make a fairly simple change in 8 turns - it seemed to get stuck in a dumb stream that was hard to get out of. So it is not AGI, don't get too carried away by the hype. I have some phenomenal examples that I'm honestly blown away by that I'll share, but as a side anecdote, I have a kind of 'swear meter' which counts how often I'm rude to Codex. In GPT-5.5 era, the % was at around 4-5%, it dropped to 1-2% when I was testing GPT-5.6-Sol and it shot up to 7% when I went back to 5.5 - it was so shocking to go back to 5.5 and experience how much worse it was. So is GPT-5.6-Sol better than Fable? On pure intelligence - no. But man, I missed it when I just wanted to get sh*t done. It is insanely capable workhorse that you can give any task to and just expect it to be done. No lectures or 'you are absolutely rightisms', nothing is beneath it, if it takes 2 days to do some dirty work, it will do it. It feels like the first time in a while when we have quite different types of frontier intelligences that benchmark sort of similarly, but feel very different. If you can, you would be probably better off using both and iteratively finding what you'd use Fable or GPT-5.6-Sol for. Perhaps, something like - an architectural discussion with Fable, implementation with 5.6 and docs & comms with Fable.

I think AI has just hit a gigantic threshold, and Grok 4.5 is the PERFECT example as to why that is. One of the hardest parts of working with AI is iterating on a project or task that you're working on. As the models have gotten smarter (and more expensive), it's taking longer and longer to get an answer or action back. This creates a ton of stall time per query or action, which is actually quite bad for creativity and staying in a state of flow. You have SO many extended starts and stops. Which inevitably leads to your brain going somewhere else. And then when the AI comes back, you have to redirect your brain to that original task, spool your brain back up to what you were working on at that moment, and then adjust as needed. There's a ton of mental friction involved. This ESPECIALLY sucks when the AI takes a REALLY long time to get something back for you, but it's not quite what you were looking for or asked for. And what sucks EVEN MORE is that these "mistakes" are getting MORE expensive!!! So wait time is going up. AND it costs more per run. HOWEVER - even after using Grok 4.5 for about an hour - what's become obvious is that it's SO MUCH MORE ENJOYABLE AND BETTER to use a model that is FAST... and capable ENOUGH. Capable ENOUGH is the real unlock here. Imagine having Fable 5 performance but at the speed of Gemini 3.5 flash. Or Haiku. That's where we're inevitably going. I think Grok 4.5 (and models like it) have really solved for one of the biggest unlocks in AI - a model that will get you a GOOD ENOUGH answer VERY FAST, at which point iteration can happen VERY QUICKLY. This - counter intuitively - keeps the user in a state of flow and creativity for MUCH longer because you are constantly ENGAGED with your project... instead of letting the AI loose for a long time. And as long as humans are involved, I think 'not quite right' will be a FOREVER problem with AI - because AIs, by default, CANNOT have human taste. Because they are NOT human. But they can be UNBELIEVABLE tools. And unbelievable tools are the ones that are VERY GOOD and VERY FAST. I think that's the true unlock with Grok 4.5 and models like it. Difficult to describe until you experience it. I think this is a VERY big deal for @SpaceXAI and @elonmusk.