u/wellomello

Hi all. Some time ago (maybe a couple months ago), I posted some screenshots of this personal project of mine, a so called "simulation of a living world", where I dreamt of a little game that had a least one toy model of the complexities of a living earth?

Well, I didn't abadon it. It is very interesting to see where a discipline, a tenet, how far it can take you if you take it to very very end.

So, I'm very stubborn, right? My axiom was that noise maps would ONLY be allowed as initial conditions for the causal systems (remember? A dynamic system is something like: x_t = f^N(x_0)? Well, x_0 is free, a priori. So that's the only affordance I have for noise), and everything else must come from real simulation?

Well, this makes progressing extremely difficult haha. You have to reinvent things from zero. I don't even have an inventory! Nor even a torch! As a player, because, to have a torch, we still have to rederive society!

Still, I want to make you notice a lot of tiny details. I put them in the following images.

For example you may notice the wake entities leave as they travel through water? It is not a rendering trick. Again, following the philosophy to the end, I spent many weeks implementing a proper fluid simulation field for the water.

The same went for the fauna. It is evolved twice: A single evolutionary system. I did not really design any of the fauna (nor the flora) you see in the video. They were evolved in the world itself, in a laboratory, where surviving and reproducing evolve both the body and the brain.

In any case, too many things to write, I leave the video and the screenshots in case it is of any interest to you guys.

The coastline

Part of the world map. A 64km2 living, globally simulated world.

Another slice of the world map, with glaciers and snow. As I said many months ago, there are no biomes at all, just the consecution of 22 different earthlike systems

Meteorology, for example, is fully simulated, globally

And wherever it snows, it snows, whether you're there or not

And when it rains, it also rains (over bodies of water that *are* a field fluid simulation)

In a sense, there are no \"river systems\" as a proper subsystem, it's just a closed water cycle. Rivers are just where water flows.

Here, for example, you can see both the wake entities leave when moving though water, and those small wavelike phenomena in the body of the water? They too are part of the physical simulation: They come from the effect of the wind velocity field acting over the water field. The small \"white cells\" in some of the coastlines of this little lake are, too, physical, they come from the simulation of the water breaking there. Notice it's just the sides that are against the windwards?

In any case, I still started working on more gamelike stuff like building, the difficulty is clearly that even these things must be subject to the simulated natural world, and so must suffer from wind and rain and everything, so, progress is very slow.

Last time I didn't feel it was complete enough to show a video on, This time I do. Notice that the game is turn based for now, the hiccups are me doubting what to do. Sorry also for the lack of video editing, I'll too work on that.

https://youtu.be/m0I1bqd0O-U

Hi all, I present myself. I am a masters student of Mathematics. These last months while working on my proper thesis, a bit overcome with a lot of stress, my supervisor suggested that I look into some other subject to pass time, so I chose this one.

Now, I want to say that I have nothing nowhere close to a proof to the Collatz conjecture. Instead, I have been sketching a proof of a property of the Collatz map over 1 that I think is mostly completely expected to be true by any serious researcher but it has not been really formally proven, at least for what I could gather from reading the literature, and I think may resonate a bit with all of you that read a lot of the investigations thrown around here:

There are no modular obstructions in the inverse Collatz tree. (if the sketch is indeed correct)

Now, this is, I have to say, an infinitely weaker statement than the full Collatz conjecture (in fact, it can't even distinguish between the 3x+1 and the 3x-1 map!) but it's a nice, constructive and global sketch of a proof.

So, for those that don't exactly get what this means, this means a very simple thing: Choose any finite modulus you want (say, 7, 21, 9824987, ...), and a residue (say, I want residue 1, this means, that when dividing a number against this modulus, I have a rest of 1), then, somewhere along the line, in the Collatz tree, there will be a number that witnesses that/has that residue under that modulus.

Why is this important? Well, in a sense, this would be a nice positive and global result that'd give more "evidence" to the conjecture being true. But in a sense it is also a kind of a negative result for those that look for some kind of magic to happen using 'mod-tricks' to find some kind of kink or counterexample. The tree saturates every admissible residue class. If there is some counterexample it has to be something different from congruences.

The sketch has a simple idea. Sorry if I explain it like this to the mathematicians but I'd like to clearly explain the motivation.

We start from the accelerated inverse tree right? And with a simple observation: any modulus can be decomposed as M = 2^a * 3^b * N, where N is coprime to 6 (this just means it's coprime to 2*3, the other two bases, at the same time).

Instead of looking at the usual map:

x |-> (3x+1)/(2^h),

we go backwards

y |-> (2^h*y-1)/3^(,) right? This just asks "who maps to y"? Well, in this case it is any x that satisfies that formula and comes out to be a positive odd integer. h is the height (how many factors of 2 we divide out in the forward step) and different heights give different predecessors.

Now, the forward map itself is extremely complicated, and the question "can we construct any number from this mapping starting at 1?" is exactly equivalent to Collatz, but we can take a much simpler object:

y |-> (2^h*y-1)/3 (mod M)

And ask the same, but infinitely simpler: Can we find all the admissible M classes in these "compressed trees"?

Now, the idea is simple right? If we could somehow take the map, which in a sense is a very "simple" map (an affine map, just multiplications and additions, at least with respect to y) and show that we can construct an example of any class in this finite set of residues using the mapping, given that the map itself gives a "certificate" that the number found "started" from 1, then the theorem would be proven.

And affine maps are very flexible right? You can compose them, invert them, and so, and the results are also affine maps, so you never end up in a "complex" place in mathematical theory.

And you notice? The decomposition M = 2^a * 3^b * N very much looks like the main numbers in the Collatz map. If somehow we would be able to "decompose" the problem into transformations that control the 2-part (the dyadic part), the 3-part (the triadic part) and the N-part (the coprime-to-6 part), and then recompose it together via some Chinese Remainder Theorem magic, then the proof would be done!

There is a catch though. Not every "height" h is valid at every y we generate. What "heights" are valid will depend on y mod 3. (I call it its color, just not to repeat things too much). For residue 1, only even h is allowed; if the residue is 2, only odd h is allowed. And if the residue is 0, then it is basically a "dead branch". Then, the color of the output determines itself what heights are "legal" in the next step. This is the 2-adic structure interacting with the 3-adic structure. We can't just freely choose a sequence of "heights" and remain under the same residue class. The sequence has to be compatible with the class we want to create.

Well, after a truckload of computation, writing proofs, checking Lean, making GPT roast me and crying before sleep. I think I got the correct method, specially because I found what I call a family of "commutators": two inverse compositions that share the same multiplicative part mod N but whose constant terms differ by a unit. That unit translation is what gives the algebraic freedom to sweep through every residue class. Getting all the other bookkeeping right is where the pain lives.

At this point the machinery gets a bit involved (and many parts are rote bookkeeping so not that interesting), but I think I have approached something, maybe.

I have presented this to my supervisor, who is set to read it sometime (many of you probably know that supervisors hardly have time, even for their supervisees' main work haha) and come back with some comments, I hope soon.

I wanted to ask for your opinions, if you do think you have some free time to read it or at least skim it.

Thank you.

The sketch is here: Paper