They weren't discovered by an LLM. They were known conjectures that were proven by an automated solving language that was linked to an LLM.

I'll take "Things That Didn't Happen For $200", Alex.

Only a handful of meaningful proofs have ever been done by automated formal theorem solvers (the Four Colour Theorem being the most noteworthy example - but its proof is so long that humans can't verify it). By contrast, AI tools have been solving Erdos problems en masse. The majority of them just bog-standard commercial models. In case you need help, the only ones on that list that were hybrid (AI / non-AI) in the actual solving phase are:

1) AlphaProof / DeepMind Prover Agent / AlphaProof Nexus
2) Aristotle (Harmonic)
3) Seed Prover / Seed Prover 1.5 (ByteDance)
4) AxiomProver (Axiom Math)

In each of the above, LLMs come up with the lemmas / strategies but then use Monte Carlo search ("brute force") or likewise to investigate what they came up with. These are a minority. In the "AI Standalone" category, these "hybrid" tools made up only ~20% of attempts and successful proofs. Hybrid tools actually made more of a contribution in the "AI Alongside Literature" (related literature found afterward) and even more of the "AI Building On Literature" (related literature known beforehand) categories, which is the opposite of what people like you expect.

And even with the hybrid tools, it's still the AI doing the heavy lifting when it comes to strategy. Non-AI theorem solvers, again, don't have a spectacular record for churning out novel proofs to unsolved problems. Tools like Lean are more about mathematical rigour - a passive environment that requires a driver (a human or AI) to feed it actual strategies, lemmas, and proof steps. And no, you cannot brute force "strategy" in the vast majority of cases, which is, again, why automated theorem solvers don't have much of a track record with unsolved mathematical problems.

Let's take a random example: the disproof of the unit distance conjecture. It was solved purely by a general purpose commercial GPT model, not custom-trained to mathematics, with no external tools. Read what the various mathematicians reviewing / commenting on it have to say (sections #3 and onward). Seriously, don't skip reading them, actually read them. This was one of Erdos's favourite problems. He mentioned it commonly in his lectures. Essentially every mathematician working in complex geometry has thought about this problem. The approach that the model came up with was highly novel approach, based on CM-fields and class field towers.

I know you don't want to accept this reality, but it is the reality, so you better improve your ability to accept it,. The field of mathematics is already doing so.

Dang, link didn't post.

Ah, good 'ol "Model Collapse" theory that people have been pounding on for years now, predicting an imminent collapse in model capabilities. How has that been working out for you?

Except what the US is actually facing, at least in the near term, is just the opposite, a worker shortage.

You can't pay royalties to the entire internet. That's not realistic.

What is practical are things like taxes to fund public benefits, or requirements of returning things to the commons (for example, open models).

In all seriousness, though, if bankruptcy is a real possibility, the idea of a public buyout of some of these old companies isn't a terrible one. Maybe even have the government buy it and make it free for U.S. citizens, but continue to make money on the property abroad. :-)

Assuming M4A ends up being a single payer system, that would, in fact, make it very possible to force providers to lower costs.

Research should be funded directly, not by paying more for unrelated prescription drugs. That's the whole point of having grant programs from agencies like NIH.

One of the biggest problems, IMO, is healthcare consolidation. When most of the hospitals in an area are owned by big chains, it really doesn't matter if they are nonprofit. Big organizations just naturally tend to bloat and waste tons of money at every level of the system, because they don't have the same incentives to keep things lean. Consolidation has generally resulted in higher prices and lower quality of care, from what I've seen.

Every other entry? Try every entry. Something like 1% of my Facebook feed is actual organic content from friends. 14% or so is from groups. The other 85% is ads. And I'm being optimistic when I say that it is only 85%. When I see about the first or second ad, I close Facebook, because it's just going to be ads all the way down after that.

I'll make an offer right now to buy it for $1,000 with the intent to release the entire catalog into the public domain (CC0) 28 years after its creation.

https://en.wikipedia.org/wiki/...

https://en.wikipedia.org/wiki/...

https://www.theguardian.com/us...

Even the modern supermarket was a tool in American anti-Soviet propaganda:
https://www.youtube.com/watch?...

There's no need to use the same recovery system for every stage either. That includes the space-based ones. SpaceX's plans for Starship are really based on their goal of going to Mars the first time, and that might work with some tweaks for a few trips to the moon too. The design of the super heavy booster is dictated by the need to refuel Starship in orbit.

The net is more efficient for the first stage and probably a second stage as well. Landing on the moon is best done with a lander designed for that, and if you're serious about regular tirps the transit in between would be best in a space-only vehicle. Eventually you're going to want to build a return booster that runs on stuff you make on the moon too.

I doubt the Chinese will use a reusable booster at all for their first few moon trips, just like the Artemis astronauts aren't going to be launching on Starship.

You're the one making the claim that modern frontier models still make this mistake (they don't), so how about you go to a frontier model, reproduce what you're claiming happens, and click the "share" button and post the link here?

Don't worry, I'll wait.

Your inability to reproduce the thing that you claim happens is duly noted.

You are speaking irrelevant nonsense. LLMs are trained in words

They are not. They are trained in tokens. Tokens do not align with word boundaries, and an arbitrary word can be tokenized in many different ways.

and they think in words

They do not. They don't even think in tokens. The process is: words are split to tokens, tokens point to an embedding position (latent space) while RoPE encodes a relative position, and all reasoning is done within latent space, which is not at all verbal (concepts are directions in latent space, and math is done on concepts, not words).

Also, a note: when spec'ing a generator, you need to know how much you're planning to use it vs. batteries. If it's only going to be used rarely, you prefer low mass, low volume, low cost, and low maintenance when unused (at the cost of low efficiency and higher maintenance in use), whereas if it's going to be used a lot, you prefer high efficiency and low maintenance cost in use, even if at the cost of higher mass, volume, cost, and maintenance when unused. In the former case, you'd prefer to allocate that extra mass, volume, and money into a larger battery pack.