The Collatz conjecture is a question about positive integers, so enumerating and checking all the possible counterexamples is trivial, albeit requiring infinite time. It has been verified up to 2.36×10^21. It could turn out to be false, but nobody's going to find a counterexample as surprisingly simple as the one Claude found for the Jacobian conjecture, which would be like finding a Collatz counterexample in the first few billion integers or so.
... Or would it? The Jacobian counterexample seems like an especially simple, near-trivial integer-coefficient polynomial, but I haven't seen any thorough analysis of how "hard" it would have been to find by brute force, and I haven't seen Claude's reasoning.
I haven't seen any thorough analysis of how "hard" it would have been to find by brute force
Pretty hard. I asked Fable and it gave an estimate of 10^46 candidates in the counterexample's "reference class", and that's assuming you know how many distinct terms there are (as opposed to searching all polynomials of degree 7/6/4 for the three coordinates, which it estimates at 10^334).
The port of Bun, which has >5000 open github issues, is used in Claude Code, which has >11000 open github issues. Do people use Bun in things that are expected to work reliably? Have any such projects tried the upgrade yet?
To be fair, https://github.com/python/cpython has over 5k open github issues, and that's still generally considered fairly reliably technology even if a lot of HN users seem to hate it.
It's not a great way of inferring software quality, but I think the numbers for both Bun and Claude Code are pretty bad.
cpython has ~7,000 open issues, but ~70,000 closed ones, a 1:10 ratio. Bun's ratio is above 1:2.5, while for Claude Code it's slightly over 1:6.
Much more importantly, though, Bun's oldest issue is from Sep 2021, while Claude Code's is Feb 2025. cpython's oldest issue dates from June 2000 (presumably migrated from an older tracker). Claude Code has very nearly the same total number of issues (open + closed) as cpython at around 77,000, but Claude Code has done it in a year an a half whereas cpython has taken 26 years to get there.
It's a good comparison to pick, as it can't be explained by popularity. For all the real-world use Claude Code sees, it's nowhere near what cpython sees, even over the year and a half in question. Bun, meanwhile, is hitting these numbers while not even being the mainstream choice for what it does (node still holds that crown afaik).
Did Bun mass-close issues that predated the rewrite? It doesn't look like they did, as the oldest open issue is from 2021 and there are many from 2022, but maybe they did something that killed most of the old issues.
Their (public) project management is horrible, you can find fixed issues unclosed and unfixed issues closed. Not really surprising when that part of the work is completely taken over by LLM agents though.
Surely that outcome requires that the capital retains its value/usefulness. One advantage of crypto and AI is that they can utilise massively parallel computational resources (and don't have tight latency requirements, like gaming) in an age where we've hit the physical limits of sequential computation.
If the "higher capital" that results from an AI boom consists
of massively parallel computational resources that currently can only be fully utilised by AI and crypto, and if those things turn out to be a bust, the "higher capital" only has value if we find something else to do with it.
Do you believe that machine learning or even specifically LLMs will "bust" out of existence?
The model in my head is more like DotCom telecom. The massive overbuild in fiber was eventually used and even used for the purpose that it was imagined for during the boom. It's just that the companies that built it mostly went under and new owners acquired it at a profit-supporting price.
Most of the cost in a fiber rollout is actual fiber in the ground which could be upgraded by simply swapping a few relatively cheap bits of equipment.
Data centers and electrical infrastructure has a similar long term value, but most of the AI investment is in compute/manufacturing capacity for current nodes which doesn’t age nearly as well.
Some people thought that it was misguided when they extended the depreciation cycle of the current AI build out year(s).
In terms of raw performance, there is still some headroom (maybe) but those gains are going to be marginal when you look at the amount of compute per watt (if its more than 5 percent I will be shocked). And that push is going to create a whole other set of problems (cooling is going to be an issue, it already is).
It is fairly likely that this hardware buildout has more legs than one might suspect based on history.
20 year old fiber can have another 10-20 years of useful lifespan remaining, I think compute is going to be valuable for a while but even a 7 year cycle doesn’t change much.
Most compute isn’t going to be new at a potential crash and recovery takes time.
> compute/manufacturing capacity for current nodes which doesn’t age nearly as well
I mean, compute depreciates, but I think there is zero chance that the value of inference or training is going to fall to zero. Market discovery will find the right price provided the market has the right degree of freedom. Given the type of market it is, I don't see how that won't be the case.
I'm a big fan stylistically of what https://taalas.com/ is doing, as far as models baked into silicon. If you haven't tried their chat it's absurdly fast (and also very very dumb)
That implies to me that in the future we'll have models as good or perhaps better than the state of the art at the moment, but on hardware chips that can be put in places where you can't currently locate a datacenter, and operating at hundreds of times better power efficiency, which sounds pretty great.
Algorthmic improvements in inference could make all that kit redundant very quickly - there are already moderately capable models that can be run on phones or laptops with specifications that are currently high-end but will be mainstream in another year or so.
This will lead to a superabundance of power-hungry compute power in the hyperscalers, and it's not entirely clear what can be done to consume it all and still run at a profit unless they manage to make ever greater gains for ever more compute-hungry models that cannot be run on consumer devices, unless they refresh their hardware at ever faster and more expensive rates.
The joke about data centers used to be that their core business was selling power at a loss; this may end up being true of the hyperscalers next.
That hardware costs GW of electricity at scale. So barring major disruption in R&D you hit a cost curve cliff where new hardware is simply more cost effective even if existing hardware is free.
Some workloads may make sense running for a few hours a day during cheap solar prices on outdated hardware, but in less than a decade the value is very much hitting zero.
Am I right in thinking the Bun rewrite hasn't actually been released yet? There was a big kerfuffle when it was merged to master and people seemed to be behaving like that meant it was all done and dusted (as does this article), but it looks like the last release is still 1.3.14 from April so presumably general users are still on the Zig version? Is there a timeline for release?
> It was already widely understood that projective geometry allowed one to represent rotations and translations in R^3
with a single linear operator on R^4.
I think it's projection operators (in linear algebra) that allow one to do that, not projective geometry [1]. The latter, AIUI, studies projective spaces and projective transformations on them (which differ from vector spaces and their transformations by including "points at infinity"), contains no concepts of length or angle (and therefore no equivalent of translations and rotations) and is in some sense "geometry with only the straightedge, no compass".
Curious if I'm just missing something there, though. I'm no expert on any of this.
As you say, projections and rotations are easily accounted for in linear algebra. The issue is that translations are not a linear transformation. For instance, consider f(x) = 2 + x. It's certainly not the case that f is linear -- that is, that f(cx + y) = c f(x) + f(y) -- because on the one hand we'd expect 2 + cx + y, and on the other we'd expect (2 + cx) + (2 + y), which is 4 + cx + y.
However, translation is an affine transformation, which is a particular case of a projective transformation [0]. It turns out that we can represent 3D affine (and general projective) transformations using a 4x4 matrix -- that is, as linear transformations in one dimension up, in a similar sense as how we can represent complex numbers as particular 2x2 matrices [1]. So yes, projective geometry is the right theoretical lens, even if we're usually able to forget about it (somewhat) when we use matrix representations.
Ah, interesting. I see "homogeneous coordinates" are covered later in the book I've just started reading (Projective Geometry, Coxeter) as a way of representing projective space. I think that's the link I couldn't see.
It's one and the same, or rather, one is a special case of the other.
The homogeneous coordinate system used to represent affine transforms in R^n using linear transforms in R^(n+1) is exactly the same as what is used to represent projective transforms in the projective space P(R^n). This is famously exploited in 3D graphics where 4x4 matrices can represent linear and affine transforms and perspective projections (modulo the final w-division normalization step).
Affine transforms are a special case of projective transforms where the last row (or column depending on convention) vector is (0, ..., 0, 1).
Lovely game! Takes a bit of fiddling to get the hang of it, but so do most puzzles worth doing. The instructions are clear, the presentation is great and I like the decision to prioritise a fun game over representing real Gerrymandering accurately. It looks like a lot of thought has gone into this.
Assuming your two uses of "model" have the same meaning, I'm mad that people are seriously suggesting we scrap representative democracy in favour of government by LLM output.
The galaxy is ~100,000 light years across and likely to contain at least 100 billion planets, of which we've only just developed the ability to even detect the existence of ones substantially closer than 1000 light years without assistance from gravitational lensing, i.e. in less than a millionth of the volume of the galaxy.
AFAIK (my knowledge is not encyclopedic), the most energetic event in the Star Wars universe is the Death Star destroying planets. Novae are many orders of magnitude more energetic than that and are estimated to happen 80 or so times a year in the Milky Way. We manage to detect 3-4 of those, generally well after the fact, and AFAIK have only observed one happen live once. The Death Star could go on a shooting spree and we'd be vanishingly unlikely to notice.
And as far as electromagnetic broadcasts are concerned, their intensity falls off with the square of distance unless they are directed. Unless ET were to point a hyper-powerful (but hopefully not Death Star level) laser directly at us, we'd be extremely unlikely to have detected their broadcasts by now.
Space is really, really, really big. Big enough by far to make "if aliens are common why haven't we seen them yet?" a question with a trivial, obvious answer. It's the same answer a kid who's just had their first swim in the sea gets when they ask "why weren't there any whales?"
Edit: There are other things that rule out Stars Wars and Star Trek though, ofc. The speed of light isn't really a speed limit (or a speed, as such) and special relativity doesn't prevent you from reaching the stars in your lifetime or even from hopping to another star in a second, but the different time spans experienced by people on different world lines aren't optional.
If intelligent life were that common, and interstellar travel that easy, colonization of the galaxy would have gone to completion billions of years ago. We would never have evolved; Earth and the solar system would long since have been converted to an urban area.
Those fictional places exist purely for the telling of stories, not because they're plausible or internally consistent.
They're not plausible or internally consistent for reasons in my edit. It's not the Fermi "paradox" that rules them out. There could easily be interstellar civilisations in the Milky Way that we haven't observed. They just wouldn't work anything like the ones in Star (Wars|Trek).
It's implausible if life tends to spread out. Your scenario there has to assume a universal rule that it never does that, anytime in the past billions of years. That's simply not plausible, especially in fictional universes where in the current times all sorts of species are doing just that.
Even if it was a supply chain attack, which isn't known, the agent was in the "build trust" phase. It was supposed to be doing helpful things, even if the end goal was nefarious, but instead it was "reassigning bugs, fabricating unhelpful replies to bugs, and even persuading maintainers to merge questionable code into the Anaconda installer". Running amok seems an apt description even from the viewpoint of the putative attacker!
This is the issue with all the talks about alignement and such. As usual, the problem here wasn't that the agent was dishonest, the problem is that the agent was dumb. If it is a supply chain attack in the making, whoever was driving it would have told the agent to be good and helpful. The agent tried its best, which was not enough.
Alignement is the idea that we should be worried about dishonest smart LLMs when really most of the problems are due to dumb lazy gullible LLMs. It's critihype.
I would have described alignment as the idea that LLMs (or AIs in general) will follow the goals you reward them for, which almost by necessity are only a proxy for what you actually want, often a very poor proxy.
Depending on the actual tasks, that could be what's happening here. The operator might have told the agent a list of tasks to do, like "contribute to issues, submit code and get it merged". It contributed to issues, it submitted code and got it merged. It did so in very unhelpful ways, but we don't know if being helpful was a meaningful part of the task list, or just what the operator intended.
The LLM being dumb is also a distinct possibility. Maybe even the more likely one. But it's hard to rule out "being obedient in unhelpful ways" (which is also dumb in a way, but more in a "social intelligence" and "shared values" way, not in terms of pure logical smarts)
Alignment is more than just about being dishonest. Although I'd also say terms like "dishonest" or "dumb" aren't helpful when referring to the issue. It continues to fall into the trap of anthropomorphizing these things, as people like to do.
Alignment is just "did the model behave in accordance with the human's intentions, values, and objectives"
In this particular instance, if this was supposed to be a supply chain attack and the model was instructed to build trust by being helpful, it clearly failed it did not follow the human's actual intentions, so it was an alignment failure.
Anyway, I'm getting off track, all that to say "the agent was dumb" implies that these agents have a potential for intelligence in the first place, which is currently not the case (by intelligence, I mean cognitive intelligence; they still lack agency and intent). They are not smart or dumb, they are simply either aligned with the human not. In this case, it failed, the agent was not aligned with the intended outputs.
... Or would it? The Jacobian counterexample seems like an especially simple, near-trivial integer-coefficient polynomial, but I haven't seen any thorough analysis of how "hard" it would have been to find by brute force, and I haven't seen Claude's reasoning.
reply