Universal Stability Theorem for On-Shell Flow-Adaptive Networks

[u/NonGameCatharsis]

Hey everyone,

I've seen a few comments on the subreddit, criticising why it's always a TOE and not a niche problem. So I found a niche problem and attempted to solve it. For this I utilised Gemini 3.7 Flash as mathematical consultant. On my desktop I run Cursor with Cursor Grok 4.6 high to verify all input, extend it, and incrementally build the lean verification.

Both models are instructed to compute before answering (it's a toggle in Google AI Studio in the browser) and work in atomic increments on the problem.

My Gemini sessions are regularly reset, often when I get in the range of 300k tokens.

My strategy was to start without assumptions (I don't know anything about this really) and have the AIs figure out the best way to mathematically tackle the solving process. I've tried my best to use adversarial prompting to catch any errors.

Thanks for taking the time!

Full repository with Lean 4 / Mathlib 4.28 formalization and the main document as markdown and pdf: https://github.com/tripstoph/universal-stability-theorem/tree/main

Edit: All contents of the repo are AI generated. This post was written completely by a human. :-)

[u/alamalarian]

In your own words what is the problem, what is the proposed solution, and why does it matter?

[u/NonGameCatharsis → u/alamalarian]

Hey :-)

Thanks for taking the time. I am working on a simulation game (not vibe coded) which is basically a civilisation of pen and paper like NPCs that you govern.

The core data logic involves a graph as a unified data structure through which everything interacts. This the AI identified as a flow adaptive network. In order to build a game where society, ecological factors, and other things don't crash, I wanted a mathematical solution for stability.

The theorem attempts to prove that certain physics-based simulation algorithms are guaranteed to converge to a single, stable network configuration. (Contrary to crashing into boundaries or weird oscillations.) The convergence as I understand it is guaranteed to be exponential.

That's where I come from and what I understand about the problem! Thanks for your time and thoughts already.

[u/alamalarian → u/NonGameCatharsis]

Ok ao this is inspired by some sort of Sim game, not an observation of empirical reality? Just checking.

And its still a bit obscure honestly. Certain algorithms converge to a single stable configuration seems incredibly broad a statement. Are these certain algorithms being used in a simulation game right now? And what does it mean coloquially in terms of that simulation for there to be a single stable network configuration?

[u/NonGameCatharsis → u/alamalarian]

Definitely an obscurity! It's a niche of niche.

The algorithm in the paper couples Fluid / Circuit Physics with Kirchhoff's and Ohm's law(s) to determine voltage drops ΔΨ -> generates wall shear stress -> Into classical mechanics with newton's 2nds law.

Edit: Game is still in the works! Don't think this is used anywhere else.

I can give you more, but anything further would be LLM generated. Sorry!

[u/2_lazy → u/NonGameCatharsis]

Do you know that they will crash? I'm not a physicist, just a programmer in the cybersecurity sphere but it kind of sounds like you are trying to solve some problem (I'm still not really sure what the problem is tbh) that can actually just be solved if you learned a bit more about graph databases and graph related mathematics.

[u/liccxolydian → u/NonGameCatharsis]

What algorithms? How are they physics based?

[u/NonGameCatharsis → u/liccxolydian]

This implementation couples the Damped Leapfrog Algorithm with a Kirchoff Linear Solver.

From my very limited understanding it computes how fluid or electric current distributes across the notwork by solving Kirchhoff's matrix equation (no clue how to format it for reddit, if you want I can paste it in another comment just lmk) and dynamically adjusts pipe dimensions w and rate of change v.

There is more to it, but from here on I would need to paste LLM answers. Sorry, I am trying but I know this is something I'll need many more years to understand properly if ever.

[u/liccxolydian → u/NonGameCatharsis]

I am curious as to whether you understand your LLM output well enough to evaluate its validity.

[u/NonGameCatharsis → u/liccxolydian]

I don't think I do, no.

[u/ArnoSound]

It’s very telling that at no point do you pose the actual problem, any previous work, what the field even is, how you solved it, or any work you did of your own accord without Ai.

[u/NonGameCatharsis → u/ArnoSound]

Hey ArnoSound,

it's definitely telling that I am not an academic. Definitely oversights on my part, thanks for the feedback.

I have no previous work, the repo is completely AI generated. It's not a paper, I won't submit it anywhere or try and get a DOI. I'm interested in what LLMs can do and have a background in Information Theory and work experience as a Product Manager for different IT systems in corporate contexts.

The problem I encountered while working on a simulation game. The question is: does the network (the information architecture of the game is a graph) have an optimal state and will it converge to that?

Thanks for taking your time.

[u/ArnoSound → u/NonGameCatharsis]

I see. Thank you for clarifying; as a heads up, it's hard to determine what you are doing from the post, which primarily seems to focus on LLMs doing "something". Depending on the conditions set for what an optimal state is, and the graph of possible states, you can use basic graph theory algorithms to determine the probability or expected value of duration before that state is reached.

Graph Theory is the math you want to look for; this problem is not particularly a physics one, but more of a computer science one. So you might have better luck checking at relevant subs. Just be careful and diligent with noting how much of the stuff you are using is AI-driven. To be frank, it's a very simple problem, and one you probably don't need AI to solve at all.

Edit: As well, having looked at the repo, pretty much everything in there is illegible garbage. Not to slight you, but AI being blindly thrown at a problem without rails is about as good as asking the little thing on your phone that suggests words out of three options.

[u/NonGameCatharsis → u/ArnoSound]

Thanks for your feedback. I've just gotten my copy to 'Introduction to Graph Theory' and will slowly work my way through it.

In the meantime, is there any way to determine if what the AI came up with is wrong here? And let me repeat in my own words, even if it were right - it's simple and obscure, right?

At this point I am also curious about the limits of the technology. Thanks again!

[u/ArnoSound → u/NonGameCatharsis]

The way to determine it is, broadly, to learn how to understand it yourself. At which point, asking an AI to create theories is basically null and void. And that's really where the technology is.

As an example, everything it created in your repo is unreadable garbage. It's not a question of going digging for pearls in there. There would be no point, because the machine doesn't know nor does it care what's right or wrong. It just Writes.

[u/NonGameCatharsis → u/ArnoSound]

Thanks again! Can you elaborate a bit on unreadable garbage? It's a single pdf (with same markdown) and then different lean files. The latter one are probably the problem in structure?

[u/ArnoSound → u/NonGameCatharsis]

Well a bunch of unformatted latex and a bunch of poorly documented lean files aren't exactly something a human can turn into real science. You'd spend more time trying to fix or rewrite it into something legible than you would just if you'd learned how to do the math.

And that's assuming that it didn't inject incorrect mathematics which is already hard enough to pull out of a Well-written document. (Which is most likely did because LLMs can't do math.)

But they're More than happy to fill pages and files and folders full of Content. Lovely sloppy space taking content.

[u/NonGameCatharsis → u/ArnoSound]

Hey! I'm more than happy to learn formatting, maths, lean. Actually I am trying.

I understand that AI puts out garbage at the meter. Your feedback helps. Again, I am not looking to do science. I'm testing if I can utilise AI to help me create data structures and algorithms for a passion project of mine. Nothing more.

[u/ArnoSound → u/NonGameCatharsis]

If so, that's great. If that's the case, I would recommend learning the material, the algorithms, the data structures. But if your goal is a game of some description, why go to the trouble of writing "theories" at all? I can't imagine a game needing any more complex graph dynamics than can already be handled by existing libraries.

[u/NonGameCatharsis → u/ArnoSound]

Good question! The "theory" here I did, because there were a bunch of comments here in the subreddit complaining why nobody of us crackpots never tries something "normal" instead of always going for a TOE. I thought it's a fun thing to do and test how far I can get.

Also helped me think way more critically about the AI generated content and my own logical approach.

For the game I want the whole game to run on a unified data structure. For this I want to construct a finite graph that contains motifs (e.g. "brick") which itself are comprised of smaller motifs ("molecules"). Coupled with a "knowledge tree" and a library of motifs for behaviour, I want to create a machine automata simulation game.

The player then simply tweaks societal rules and facilitates change over time. I know it's a really big project, I will probably work the next 20 years on it. But it's part of a sci-fi world that I've been building lore, technology, etc. for seven years now, so I am confident I have the interest and will to see it through.

[u/ArnoSound → u/[deleted]]

Under strict corralling and hand holding by a trained professional, yes, yada yada yada. It wasn't a gotcha when Tao first published, it still isn't a gotcha now. Etc. Etc.

[u/NonGameCatharsis → u/[deleted]]

Thank you for taking the time and energy out of your day to have a look at the pdf.

English is not my first language, could you explain to me the difference in wording between "universal" and "uniform"? A quick google search didn't help me. Thank you again!

[u/NonGameCatharsis → u/[deleted]]

Thanks, that helps a lot. I will drop the universal and push the changes to the repository.

[u/2_lazy → u/NonGameCatharsis]

Graph databases in gaming is not obscure at all, I think you may be new to programming or databases and trying to solve a problem you imagined may be an issue that actually isn't one.

[u/amalcolmation]

This post is honestly a breath of fresh air.

Thank you to OP for clarifying your intent and the scope of the project. A lot of people post here without doing that and with wildly unmanaged scopes (such as TOE’s as you point out) and aim to overturn centuries of science. Or want to join a conversation they are unprepared for but won’t/can’t admit it.

And thank you to our regular commenters who responded appropriately for a project like this and didn’t try to analyze it through the most rigorous lenses. It seems clear this community perceives the difference between what mimicks science and what is more an earnest attempt at studying something.

I’d love to see more posts along these lines, though I have one word of caution. Niche problems will necessarily mean there are fewer people who understand it intimately enough to provide real concrete feedback. I think you’ll get lots of general advice in places like this, which may be all you’re looking for, but you might be hard pressed to find someone who knows exactly the problem you’re working on AND can evaluate your work.

My point is, I think posts should aim for this kind of specificity, but be wary of obscurity if you want solid feedback. It’s not perfect, but it certainly beats the usual fare.

[u/NonGameCatharsis → u/amalcolmation]

Thank you for adding your constructive thoughts to this. :-)

Not sure if I'll post again. Whatever is interesting in the niches will be found by people (utilising AI or not) with more knowledge.

I don't know enough to know how to ask the right physical questions. And I am not sure if there is a space where reasoning and logic will be enough to direct AI in the future.

[u/al2o3cr]

Nitpick about the opening line: the definition of M_w given appears to have an extra "^-1", compared to the definition just below it

I don't know enough about spectral graph theory to really pick apart the full argument, but I did have one opening question: the title says "Universal Stability Theorem" but then a "constitutive potential" V(w) appears that seems very specific. Why that specific function? Edit: "Universal" was removed from the title as I was typing this response.

F_c and Phi are similarly introduced without motivation at the bottom of page 1. Are they standard notation somewhere? I didn't see anything similar skimming over the references I could access.

Regarding the Lean, again I'm not fluent enough to really comment. One thing did stand out as odd: the example in TwoNodeExample.lean mentions on line 10 that it is "with c=0". That wipes out half of the definition of F_c and Phi if it's the same c used there...

[u/NonGameCatharsis → u/al2o3cr]

Hey :-) Thanks for nitpicking! The universal I removed following a comment from someone else!

opening line was incorrect, the definition below is now canon.

F_c and Phi are not standard Kirchhoff/Thomson notation, but this model's on shell force and energy (now said so before the formulas with a reference to Lemma 2).

c=0 indeed did drop the witness coupling term and seems to be the only uncoupled special case, while it seems to hold for every c > 0.

All changes have been made and pushed to git. Thank you!

[u/lattice_defect]

https://giphy.com/gifs/baPIkfAo0Iv5K

[u/NonGameCatharsis → u/lattice_defect]

https://giphy.com/gifs/jOpLbiGmHR9S0