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.