Does a grown geometry remember how it was made? Turns out yes, and not in the way I expected
I'm an independent researcher. No affiliation, no funding, no credentials that matter here, so take this as what it is.
The whole thing started with a question I couldn't get out of my head. If space isn't something that's just there, but something that gets made, one piece at a time, and the connections you form when you're born stay with you while everything keeps growing around you... does the final geometry carry its own history? Or is history just a fancy way of picking a random configuration, and once the thing is built you can't tell it apart from one that was thrown together at random?
I built the dumbest possible version of this. A ring that grows by inserting new sites between old neighbours, where the old neighbours stay connected to each other after the insertion. That's it. No parameters, nothing to tune. And the first thing it did was land exactly on a known critical point from long-range percolation, by itself, without anyone asking it to. That got my attention but it wasn't the interesting part.
I'm not claiming that growing graphs is a new idea. Preferential attachment, random recursive trees, the sequential growth of causal sets, all of that has been around for decades. What I haven't been able to find anywhere is someone taking the history itself as the object. The usual thing, when you measure geometry on a graph, is to draw the graph from a distribution and measure it; and when the graph grows, what people care about is which distribution it ends up at. Nobody, as far as I know, takes the thing that grew, builds it a twin with exactly the same statistics and no history, and asks the two of them whether they're the same object. If someone knows of that experiment being done before, I'd honestly like to read it.
So that's what I did. Take the grown ring, keep every single connection it has, same lengths, same number, same everything you can count, and just put them back at random places. By any statistic you'd normally look at, the two are identical. But if you let a random walker loose on them, it feels a different dimension. The grown one is more "spacious", for lack of a better word. The difference is small, it's consistent, and it's not in anything you can read off a histogram. It's in how the connections are organized together. That's the first paper.
Then I asked what happens if you shake it. You take the grown geometry and run it under a dynamics that had nothing to do with building it. If the dynamics can move connections anywhere, the whole organization melts in a handful of steps, without any energy cost, like ink in water. But if the dynamics can only move things to the neighbouring site, it's the random version that behaves like a glass and gets stuck, while the grown one doesn't age at all. It just forgets slowly, from the smallest scales up, with the large-scale structure protected for enormous times simply because moving things locally can't undo it. So the memory isn't locked behind a barrier. It's locked behind locality. That's the second paper.
And then the thing that forced me to change my mind. I had been telling myself the grown geometry was "memory, not a state", something no equilibrium description could capture. I was wrong. The probability of each grown configuration can be written down exactly, and it turns out to be an ordinary equilibrium measure for an energy that nobody wrote and that only the growth process points to. The grown geometry is a state after all. It's fragile under any dynamics that didn't make it, and perfectly stable under its own. My founding hypothesis died, and I kept the death in the paper because that's the only honest thing to do with it. That's the third.
I should say how this was done, because it matters in this forum. The simulations, most of the derivations and all the editing were done with heavy LLM assistance, and the refereeing was also done by LLMs, over many rounds. No human has peer-reviewed any of it. Every prediction was registered before running it, and there's a private list of the ones that failed, the claims I retracted, and the design mistakes I made, including a recent one where a reviewer found that my supposedly exact control wasn't exact, and fixing it made the effect smaller. The smaller number is the one in the paper.
I'm not asking anyone to believe this. I'd rather someone tried to break it. The first result in particular can be checked by brute force on a laptop.
Papers:
- Same marginal critical exponent, different spectral scaling — https://doi.org/10.5281/zenodo.22539518
- Memory as organization: two-tier relaxation under local and nonlocal dynamics — https://doi.org/10.5281/zenodo.22539664
- The growth measure solved exactly — https://doi.org/10.5281/zenodo.22539562