Bad at Arithmetic, Promising at Math

[-]gwern3y*5731

Some observations:

the use of ChatGPT is a bad idea. ChatGPT greatly underperforms what we know GPT and similar models are capable of in the inner-monologue papers on many math-related questions (particularly after instruction tuning), almost all of them far harder than the CRT. You are wasting your time doing math with ChatGPT unless you are interested in specifically ChatGPT, such as looking into its mode collapse. Which you aren't. (Remember: "sampling can prove the presence of knowledge, but not the absence"---especially after RL or finetuning intended to change drastically what samples come out!)

You would be better off tinkering with davinci-003 or possibly a codex model (and making sure to try out best-of=20 to avoid being fooled by stochasticity of sampling and see what the 'best' results really are).
these systems are already combinable. GPT-f (and later Lean work) benefits from its pretraining on natural language math corpuses, which provides a powerful prior for its ATP. The inner-monologue papers show benefits from a LM thinking about a problem and writing Python code to solve the problem for it (exploiting increasingly-amazing code-writing capabilities). And there's the autoformalization with LMs which I haven't yet caught up to in my reading but looks pretty exciting, and Szegedy is excited too.
the 'math alignment' problem is about as hard as the 'image generator alignment problem', which is to say, little at all like 'the AI alignment problem'.

The great thing about it is, if you don't like a result from GPT-f or Minerva or the auto-formalizing models, you just throw it away & try again, no harm done beyond some wasted electricity. (That's not the case for autonomous agents.) As the joke goes, mathematicians are great - all they need is a chalkboard and a waste basket. So if they build up castles in the sky, that's not a problem as long as it still proves useful for end applications: does access to those theorems shorten proofs of theorems we care about? Do models trained with that produce better proofs? etc. You can feed them back in and keep going. (This is a common trick now with image-gen: as long as you can generate some images that you want, you can bootstrap by training a DreamBooth or Textual-Inversion or DreamArtist or the finetune du jour on those, generate, select, re-finetune etc.) You can simply keep adding rewards for human-directed uses and penalties for irrelevant tangents and adjusting and doing end-to-end training, RL training if necessary. What they 'want' will be defined by the rewards/penalties.

So I don't see much of a 'math alignment' problem beyond a relatively easy engineering problem - the real challenge remains getting systems whose 'math alignment' you'd care about at all. (Any threat comes from integrating them into self-improving systems such that they might be doing mathematical research into how to make themselves more powerful, but that requires some degree of agency or RL setting, not just a model in a GPU tracing out a tree.)
from a RL perspective, it's not absurd at all to suggest that exploration and novelty drives can be meta-learned from a few positive outcomes that yield ground-truth rewards. It's meta-learning all the way down/up and reward is enough. I can totally believe that math models can do the same thing, learning preferences/esthetics towards work that will one day be useful, based on techniques like PBT selection for models which learn drives towards useful work or training on past human work to predict utility etc.

[-]cohenmacaulay3y80

Thanks for the pointer to davinci-003! I am certainly not interested in ChatGPT specifically, it just happens to be the case that ChatGPT is the easiest to pop open up and start using for a non-expert (like myself). It was fun enough to tinker with, so I look forward to checking out davinci.
I had not heard of GPT-f - appreciate the link to the paper! I've seen some lean demonstrations, and they were pretty cool. It did well with some very elementary topology problems (reasoning around the definition of "continuous"), and struggled with analysis in interesting ways. There was some particular theorem (maybe the extreme value theorem? I could be forgetting) that it was able to get in dimension without too much trouble, but that it struggled hard with in dimension 2, in a way that a human would not really struggle with (the proof of this particular in dimension 2, or dimension $n$ for that matter, is very nearly identical at first reading). Breaking down it's failure, the demonstrators argued pretty convincingly that perhaps there's actually just honestly more going on in the dimension 2 case than the dimension $1$ case that a human prover might be glossing over. The machine can't say "this set is compact, so the extension of the previous argument is completely analogous/obvious/trivial," it has to actually go through the details of proving compactness in detail, carrying out the extension, etc.. The details may not be deep (in this particular case), but they are different. I think it would be really cool to see lean-like systems that are capable of making possibly illegitimate logical leaps (canonically: "this is trivial," or "we leave this as an exercise") to escape some nasty, tangential sub-sub-argument they're stuck on, even if they diverge from that point and ultimately get the conclusion wrong (as you say in bullet 3: just run 500 more randomized instances, and see where they go, or inspect the results manually), just to see where they might go next. Regarding code-writing capabilities, it doesn't seem like there's anything in principle special about python, and it would be cool to start seeing variants pop up with access to larger mathematics languages (think sage, matlab, mathematica, magma, etc.) that might be more directly useful to automatic proof-writing.
I partially agree and partially disagree on point 3.

Partial agreement: For an application like "I am trying to prove [Theorem X] that I have good reason to believe is completely provable with nothing more than the toolbox already in this paper ([Lemma 1], [Lemma 2], [Lemma 3], ..., [Lemma n]) plus standard results in the literature, then I agree, reward it for directions that seem productive and seem likely to shorten the proof, and penalize it for irrelevant tangents. Build a corpus of training theorems that are similar to the things we care about, where we know of the existence of a proof using tools that we know the system is aware of (e.g., don't train a system on only analysis texts and give a problem that you know to require Galois theory). Some models will produce proofs nearly as short or shorter than the proofs we know about, others will go off on tangents and build castles to nowhere, never really getting around to the point. Clearly we prefer the former. This is likely to be the main near-term application of these systems.

Partial disagreement: For a longer term project, where the system is allowed to make more open ended moves than "help me finish the proof of [Theorem 3.2.6]," I don't quite agree that it is entirely obvious how to "align" the system to do things that are productive. I expanded a bit on this in another comment in this thread, but imagine a scenario where you have a long (which may mean indefinitely long: maybe 5 years, maybe 20, maybe 200) term project like "prove Fermat's last theorem" (in a counterfactual where we don't already know it) or "prove the abc conjecture" or "prove the Collatz conjecture" or something. Some easy to state theorems that are easy for any system competent in basic algebra and number theory manipulations to get started on, easy to generate and test examples, clearly interesting to all parties involves, and of very high interest. We cannot evaluate a system based on whether it produces shorter proofs of FLT in a hypothetical scenario where we don't already know what the proof of FLT will involve. In the short term, the relative value of "productive-seeming" work (mass volumes of direct manipulations on the equation $x^{n} + y^{n} = z^{n}$ using standard methods that existed in Fermat's time) vs "irrelevant tangents / castles in the sky" (inventing a theory of primary decompositions, modular forms, schemes; 95% of which is clearly irrelevant to FLT, but access to the broad body of theory appears necessary) is not entirely straightforward, since there is no easy way to test an irrelevant tangent on whether it makes the proof shorter or not. It's very hard to evaluate whether a research direction is "promising," and an unfortunate truth is that most of our evaluation comes down to trusting in the intuitions of the researcher that "this is important and relevant to the heart and soul of FLT."

I think it's a problem that grant application evaluators face. I want to fund the research team that will most strongly contribute to long term progress on FLT (or pick a problem that seems likely to have a 30 year horizon, rather than a 200 year horizon), and both teams are motivated to convince me they deserve the funding. Even for humans, this is not an easy task. Generally speaking, given a sufficiently technical mathematical paper (or cluster of papers in the authorship graph), there may exist no more than 20 people on Earth that really understand the subject enough to honestly evaluate how relevant that paper is to the grant subject: all of those people are friends of the author (and may be motivated to advance the author's career by helping them secure a big grant, irrespective of the abstract relevance of the research; one author in their network's success contributes to their own success, via collaborations, citations, etc.), and none of whom are evaluating the grant applications. That is, someone is building a tower / castle in the sky, and the only people who really understand the castle are personal friends of the builder. We are placing a lot of trust in the builder (and their authorship network) when we choose to fund their project, like it or not. So, here the research teams are two (or a thousand) competing parameterizations of some model in a training environment, I may strongly suspect that massive towers will have to be built, but I don't want to trust my own intuitions about which towers are useful or not. I want to at least leave open the possibility that we (like Fermat's contemporaries) have just no idea what needs to be done to make progress, and that we should allow "well-aligned" crazy ideas to flourish. I have to (at least partially) trust that the tower-builders genuinely "believe" on some internal level that they're building in the right direction.

I of course agree entirely that systems that are advanced enough for the above to actually be a problem worth intensely caring about are quite a ways off, but they will come to exist sooner or later, and I don't think it's obvious how to get those systems building in productive directions.

[-]gwern3y*193

Yeah, that's the problem with ChatGPT: it's so easy to use, and so good within a niche, we're right back to 2020 where everyone is trying the first thing that comes to mind and declaring that GPT is busted if it doesn't (won't) do it. Heck, ChatGPT doesn't even let you set the temperature! (This is on top of the hidden prompt, RLHF, unknown history mechanism, and what seems to be at least 1 additional layer of filter, like the string-matching that the DALL-E 2 service uses to censor and decide when to apply 'diversity prompt'.) 'Deep learning is hitting a wall' etc...

Just remember anytime anyone uses ChatGPT to declare that "DL can't X": "sampling can show the presence of knowledge but not the absence."
Python is special in that there's a ton of it as data, and so it's probably the single best language in any code-generating model. Using the special-purpose languages will cost you, making it a classic bias-variance tradeoff. That data richness won't go away until you have full bootstrapping systems - not the fairly limited one-step self-distillation or exploration that we have now (eg they can generate Python puzzles to train themselves but the bootstrap seems to only work once or twice).
It's not easy, but I think we have lots of ideas and I already covered them in my final bullet point.

There's a rich vein of ideas in reinforcement learning & decision theory & evolutionary computation about how to do novelty search or meta-learn exploration/intrinsic-curiosity drives which balance explore-exploit. If you train models based on reward and do things like evolutionary selection, you can develop models which have 'taste' in terms of making short-term actions which obtain information or useful states for eventually maximizing final reward across a very wide set of environments. There are a lot of links about that in my links there, like Clune's manifesto.

We have plenty of ideas about how you would do this; for example, here's an evolution-strategies approach that would meta-learn exploration: 'For each GPU, initialize a random set of axioms in a formal system(s), specify a random distant target theorem(s); apply random mutations to a large language model pretrained extensively on math corpuses to do tree search (or whatever ATP approach you prefer); do that tree search for a fixed time budget; keep the X% which successfully find their target theorem; repeat until toasted golden-brown and delicious'. This would gradually select for models which embody optimal time-limited search in a very large space of possible formal systems.

To show that it's not merely theoretical Schimdhuber-esque musing, consider the recent very impressive success in meta-reinforcement-learning I don't think has been discussed on LW yet and which many people probably still believe impossible in principle*: "VeLO: Training Versatile Learned Optimizers by Scaling Up", Metz et al 2022 (Twitter)

While deep learning models have replaced hand-designed features across many domains, these models are still trained with hand-designed optimizers. In this work, we leverage the same scaling approach behind the success of deep learning to learn versatile optimizers. We train an optimizer for deep learning which is itself a small neural network that ingests gradients and outputs parameter updates.

Meta-trained with approximately four thousand TPU-months of compute on a wide variety of optimization tasks, our optimizer not only exhibits compelling performance, but optimizes in interesting and unexpected ways. It requires no hyperparameter tuning, instead automatically adapting to the specifics of the problem being optimized.

We open source our learned optimizer, meta-training code, the associated train and test data, and an extensive optimizer benchmark suite with baselines at this URL.

https://twitter.com/ada_rob/status/1593702507422912516

Here is a real-world 🚲 example (not in the paper) for T5 Small (~60M params). The VeLO-trained model reaches the same loss as >5x steps of Adafactor and takes only ~1.5x as long per step in wall time 🕓...a >3x speedup⏫!! Pretty amazing for such an OOD task!

(I'll note in passing that "Here is an amazing DL PoC that neither you nor anyone else has heard of yet, but already does the thing you are speculating that AI might some day do in 2040 after another dozen paradigm shifts" describes a lot of DL research in 2022; there's so much going on right now, and the people who should be pulling 16-hour days reading Arxiv are often distracted by things like FTX or Elon Musk, or just the image generation stuff, so there are many things falling through the cracks. But 5 years from now, you won't care how much time you spent following Twitter drama, and you will care about missing out on things like Dramatron or CICERO or U-PaLM or RT-1.)

* Given some of the comments on my Clippy story, anyway...

[-]Yitz3y20

That paper is insane…you’re finding this stuff just by trawling through Arxiv, or through some other method?

[-]gwern3y102

Twitter + Reddit + Google Scholar Alerts on key researchers. Honestly, it's too much, I am many months behind in my reading.

[-]Mitchell_Porter3y130

Seems a good time to link to the thoughts of a Fields medalist (Akshay Venkatesh) on how a hypothetical AI-based "power tool" called AlephZero could transform the discipline.

[-]Yitz3y20

Fascinating article!! Will definitely be thinking about this for a while…

[-]cohenmacaulay3y10

Thanks for the link, this is great!

[-]cubefox3y70

How about making a follow up with GPT-4, and testing how it improved? From OpenAI GPT-4 is only available via ChatGPT+, but Bing has also a free variant of it. Though the latter is still a bit limited (15 model replies per conversation) and currently based on a waitlist.

[-]Viliam3y71

Just guessing here, but I think that interesting mathematical statements are those that are easy to understand, but difficult to prove. (Of course "easy to understand" is relative to one's education.) Such as Fermat theorem, for those who know what is exponentiation.

Even more impressive are things that are useful for many purposes. Like, suppose that some statement S is difficult to understand and difficult to prove, but later you ask a question that is easy to understand but difficult to prove, and someone says "yes, that is a consequence of S, where you substitute x for 7". And you ask another, seemingly unrelated question, and someone says again "that is also a consequence of S, where x = 1/4".

The uninteresting example you provided is something that is difficult to prove, but also difficult to understand what it means. And therefore unlikely that we could use it to prove further interesting things. Unless AI could show us how to solve interesting (for us) things using that theorem, in which case humans would become interested in how exactly that works.

[-]cohenmacaulay3y40

I think I largely agree with this, but I think it's also pretty hard to put to practice in training an AI proof system.

Fermat's theorem is actually a great example. Imagine my goal is to "align" a mathematician to solve FLT. Imagine a bizarre counterfactual where we have access to advanced AI models and computational power but, for some reason, we don't have algebraic number theory or algebraic geometry -- they were just never invented for some reason. If you want something less far-fetched: imagine the abc conjecture today if you're of a camp that believes the inter-universal Teichmuller theory stuff is not going to work out, or imagine perhaps the Collatz conjecture (very easy to state, very hard to make progress on), if you believe that we might need some truly out-there stuff to resolve it. I'll say "FLT" as a stand-in for "easy to state problem that everyone agrees is important, but is completely resilient to methods that exist(ed) in it's day."

So Fermat just died, we opened his notebook, and read his famous "the margins aren't wide enough" note. We decide that a major goal of the world's mathematical enterprise is to regenerate his proof, and we set our anachronistic advanced AI systems to work. Most of them, trained on a corpus of Fermat (and Fermat-like authors) text perform a bunch of weird factoring tricks, equation rewrites, reductions mod n, etc. Fairly standard number theory at the time (I think; I'm not a historian).

We have the benefit in our timeline of knowing that a proof of FLT (at least very strongly seems to) require a rather gigantic castle in the sky to be built first. It just doesn't look like there's a road to FLT that doesn't pass through number rings, elliptic curves, modular forms, cohomology theories, schemes, etc.

Suppose that one of our AI systems (one of the ones that's allowed to try to be creative) starts drifting out of the factoring/rewrites/reduction mod n realm into doing more pie-in-the-sky stuff: it starts inventing words like "primary decomposition" and "Noetherianity / ascending chain conditions" and produces a massive volume of text on something called "Krull dimension." Should we punish or reward it? A similar AI goes off and produces stuff about "cohesive rings" and "fusible quasitowers" and "entwineable bundles" and such. Should we punish or reward it? Of course, in reality, they'd be taking baby steps, and we'd be punishing/rewarding far before they can start building towers. Perhaps this means we have even less information about where they're going. (If the math police were in my office punishing/rewarding me after every line I write down, I'd certainly never get anything done, that's for sure!)

The kinds of people who invented primary decomposition, and the ascending chain condition, and Krull dimension were "well-aligned." They cared a lot about problems like FLT. In fact, primary decomposition (more or less) directly arose out of various failed proof attempts of FLT, and Noetherianity/ACC was (largely, initially; disclaimer that I am not a historian) a technical tool to get primary decomposition working.

It's not so easy to run a test like "does this proof shorten the distance to things we care about" because primary decomposition, while a necessary part of the tower, was still upwards of a century away from a proof. We can't, from Fermat's time, look at the statement and "obviously" tell that 'it's going to make a proof of FLT shorter. The farther you climb up the tower (someone in the corner inventing cohomology theories on schemes) the more strained the link to FLT, but all those links do appear to be ultimately necessary. If FLT is the main thing we care about, we are sort of stuck relying on incredibly vague intuitions from people like Noether and Lasker and Krull (and much later Grothendieck and Tate and Shimura and Wiles; and all the people in between) that the tower they're building might actually be the "shortest bridge" to the elementary-to-state statements we care about (like FLT), even though (pre-Wiles) they have absolutely no way to prove that. They can in no way absolutely prove to their punish/reward trainers that "primary decomposition" and "sheaf cohomology" and "modular forms" aren't a mis-aligned waste of time. They can argue how it's related, but they cannot know for a fact that these things will shorten the proof.

So, how do we foster the kinds of thinking like from the Noethers and Laskers and Krulls, who were building a tower based on vague intuitions like "FLT is important and I think this is a part of understanding what's going on," while dis-incentivizing the time-wasters who want to invent a random theory of "cohesive rings" or "entwineable bundles" or whatever? Listed side-by-side, there's not an obvious sticker that says "this one is important for FLT, and this one is useless." Plausibly, a sufficiently persuasive mathematician (who is sufficiently motivated to obtain grant money) could spin a tale that could convince us that what they're doing is the most important thing in the world for (*grabs random scrap of paper from a hat*) Fermat's Last Theorem. If we can't trust our ability to resist persuasion by mathematicians hungry for grant money, how do we train them to have motivations "aligned with" building towers that contribute to FLT, instead of building towers to nowhere that trip random reward centers.

[-]Richard_Kennaway3y70

Meanwhile, you can quickly identify gibberish like "A Riemannian group is a universally flat manifold ring in the Lagrangian graph category."

Well, you might be able to. :)

[-]gjm3y152

Just confirming that I, another mathematician (and a pretty rusty one; I've been working in industry for decades) would also look at that and think "that sure looks like word salad to me" in a way that I wouldn't for "an excellent Cohen-Macaulay domain is normal".

Unpacking the thought process a little, though I still expect this to be incomprehensible to people who aren't mathematicians: I don't know what the "Lagrangian graph category" might be, but I couldn't swear there isn't one; but if there is, it doesn't seem like a thing that would have things called "manifold rings" in it. "Manifold ring" is not a phrase I have ever heard but I could imagine someone using it to mean something along the lines of a Lie group. Maaaaaybe you might then be able to define manifolds and rings "over" some weird category (though "the Lagrangian graph category" doesn't feel like it could be the right sort of category, even if there were such a thing) but never "in" it: a thing "in" the "Lagrangian graph category" would surely have to be Lagrangian graphs, or graphs understood in some Lagrangian way, or something like that. I have heard "flat" applied to kinda-algebraic-kinda-geometrical objects so maybe there could be "flat manifold rings" and even "universally flat" ones, but I'd be a little surprised. Usually anything of the form "An X group is a Y ring" is going to be nonsense; rings have extra structure that groups don't, and if you have some special kind of ring you're not going to call it an anything group; but in mathematics a red herring is not necessarily either red or a herring, and e.g. a "quantum group" is a kind of algebra, and an algebra is among other things a kind of ring, so meh. But "Riemannian group" feels very strange indeed. I am too rusty to be more than, say, 95% confident that this can't all be made to fit together somehow, but it's very much not the way I'd bet.

[-]gjm3y61

I feel that "promising at math" is a bit too optimistic for ChatGPT in its current form, and "bad at arithmetic" understates its deficits.

(I should also acknowledge that it's an astonishing achievement to have created something good enough that that sentence, as opposed to something like "of course all it produces is meaningless word salad", is one worth saying at all.)

"Bad at arithmetic" understates it because ChatGPT's failures of mathematical reasoning aren't just about being rubbish at arithmetic. It's bad at logic. For instance, I asked it to give examples of two non-isomorphic groups of order 5 (which of course it can't; there is only one isomorphism class of groups of order 5). It said something along the lines of "Yes, I can do that. One of the groups is the cyclic group of order 5, which [... blah description blah ...]. The other is the dihedral group D_5, which [... blah description blah, including mentioning that it has order 10 ...]. These groups are not isomorphic because they are of different order." I pointed out that these groups are blatantly not both of order 5, and it said oops and provided another example of exactly the same type: two groups, one of which is C5 and another of different order. After repeating this process maybe four times, with different bogus examples each time, I pointed out the pattern and asked whether it was sure that an example exists, and it just repeatedly doubled down. This isn't about being bad at arithmetic, it's about piecing together fragments of reasoning without any sort of higher-level coherence between them. I am more sympathetic to the "stochastic parrot" view of LLMs after trying this experiment.

As for "promising at math": maaaaaybe, at least in the sense of "produces high-level word salad of a sort that might occasionally happen to contain interesting mathematical concepts". But I'd want to see more examples, and preferably better ones than the one here (I dunno, maybe n-coherence is more interesting than it sounds? It sounds pretty unlikely to be worth studying, to me). I am aware that "can produce genuinely interesting new mathematical ideas at non-negligible rate" is a really high bar to clear, of course...

[-]cohenmacaulay3y20

I think what I mean by "promising at math" is that I would assign a reasonably confident belief that a scaled up model with the same basic architecture as this one, with maybe a more mathematical training set, would be able to accomplish tasks that a mathematician might find helpful to a nonzero degree. For example, I could imagine a variant that is good at 'concept-linking,' better than google scholar at least, where you ask it something like "What are some ideas similar to valuations rings that people have explored?" or "Where do Gorenstein rings come up in combinatorics?" and getting a list of some interesting leads, some nonsense, and some lies and fabricated sources. If you try this with ChatGPT, you get mostly either (1) very well-known things that you could find by searching mathoverflow or stackexchange or (2) made-up sources that sound really cool and believable, until you figure out they aren't real. I would imagine that prompt-engineering could weed out type (1), and that failure mode (2) is something we'll see gradual progress in. It seems that there is a lot of interest in fixing the "making stuff up" issue, and I tend to believe that if a lot of smart people gravitate towards an issue, it will probably get at least somewhat better over time.

As to "bad at arithmetic," I agree, it definitely goes quite a bit farther than not being able to reliably compute 416*31 or something. I was able to get it to solve some very simple Sylow theorem type problems with a lot of hand-holding and pointing out its answers, but the failure you describe definitely sounds about right to me. It can say something like [P and Q and R] therefore [S]. Then you convince it [P] is false, and it will say [not P and Q and R] therefore [S], as though correcting [P] to [not P] fixes the argument (rather than breaking the implication). You may prompt it again, and it says "I apologize for the mistake! Here is a corrected version: [Q and R] therefore [S]" when [Q] and [R] are not nearly enough. I do not expect these things to be good theorem-provers or logicians without significant changes. I'm not even sure how you would coax a natural language model into doing formal reasoning competently. At one end of the spectrum, you could ham-string it to only produce strings that are well-formed in a formal system, only using valid rules of inference, but then it's going to be so limited in the leaps it's able to make that it will never be able to discuss high-level concepts with you. I don't know how you get a stochastic parrot to map the word salad it's generating onto something that resembles valid formal reasoning.

[-]gjm3y30

An idea that feels promising to me for proving things using LLMs, though I don't know whether it actually is, is something along the following lines.

Completely rigorous automated proof-checking system along the lines of Lean.
Language-model-y translator from informal mathematical statements plus purely-formalized context into formalized equivalents of those statements.
Language-model-y system generating highly-unreliable proposals for the next thing to try in a proof.
(If necessary given the foregoing:) Traditional good-old-fashioned-AI-type search algorithm attempting to build proofs using LLM 2 to generate candidate steps, using LLM 1 to translate them into formalism, and using Lean-or-whatever to see if the candidate step is easy to prove from what's been proved already.

This is not so very different from what human mathematicians do, and it doesn't seem super-improbable that a modest amount of improvement to the models might be sufficient to make it work pretty well.

[EDITED to add:] One thing I don't like about this sort of separate-models-and-duct-tape approach is the lack of integration: the language model doesn't really know what the proving-bits are thinking. I don't know how much difference this actually matters, but it feels a bit iffy.

[-]Nicholas Kross2y30Review for 2022 Review

Still seems too early to tell if this is right, but man is it a crux (explicit or implicit).

Terence Tao seems to have gotten some use out of the most recent LLMs.

[-]Algon3y10

Are you sure that n-Cohesive Rings are actually a novel concept? There's such a huge amount of work on algebra that I'd bet -- a little, at 50-50 odds -- someone has come up with the concept before.

[-]cohenmacaulay3y40

If anyone knows where to find an equivalent concept, definitely let me know! It's hard to prove that a concept has not been investigated before, so I'll just give my reasoning on why I think it's unlikely:

I believe that the name "-cohesive" is untethered to the literature, even if the underlying concept exists, since it was specifically told to change the name to something new.
The original name, " $n$ -smooth," appears in number theory, but the number theory concept is genuinely unrelated. The word "smooth" is heavily overloaded in algebra, but generally points towards a "main" concept called "smoothness of a morphism," and smoothness of a morphism is incapable of caring about which power of a prime divides something. So it is at least unrelated to the main uses of the word "smooth."
When you start doing math research, you have to develop something resembling an "originality" radar. Mine is far from perfect, and only really works well in a narrow, narrow corner of ring theory. But for what it's worth, it's set off by the concept. It's hard to communicate this sort of intuition, but subjectively to me, it contributes to my belief.
Cultural exposure is part of it. If you tried to pass off a made-up geometry word to me, it would be pretty easy for you to succeed (I'm not a geometer), but, at least I would like to believe, my "fake-detector" for ring theory is a bit more calibrated, and not only does this not trip the "I've heard of this alarm," it doesn't even really sound similar to other points in the cluster "unit group stuff" out in conceptspace. I know that generators of unit groups are important, and orders of generators are probably important by proxy, but why care about order related to the residual characteristic? It sets off my flag as "weird" and "I've not heard of this sort of thing before," which triggers a subjective belief "I don't think people have done something like this."
In general, concepts in ring theory about a prime dividing (or not dividing) the order of something exist, and concepts where the prime has something to do with the characteristic are certainly important, but almost always when the characteristic is prime. People just don't really talk about non-prime characteristic that much. And the same power for different primes? Even if those primes divide the characteristic with different multiplicity? That's just odd.
Stepping out of what the AI can do, and what a human might realize: in $Z$ , the unit group of a quotient $Z / n$ has order equal to Euler's totient function of $n$ , which not only involves the prime factors of $n$ , but those prime factors minus $1$ . $n$ -cohesiveness then asks about powers of a prime dividing the totient, which implicitly requires you to understand the prime factors of various primes minus $1$ , which is generally very hard. I am genuinely not sure that I could classify $5$ -cohesive ideals of $Z$ . It doesn't feel like a problem I've seen people try to work on before. And that's just $Z$ , forget $Z [\sqrt{- 19}]$ or some cyclotomic extension.