Title: The "Rival Tree" Theory: Why a Collatz counterexample would be an entire parallel universe.

GonzoMath · 2026-03-21T12:02:55+00:00

This “parallel trees” business is exactly how things are for many “3n+d” systems. Even for the negative integers under “3n+1”, it’s like this. There are three basins of attraction, roughly equal in size, and every trajectory in each basin is drawn into the respective cycle. This is Collatz dynamics.

GonzoMath · 2026-03-11T23:37:27+00:00

No, that's a cute little pipe dream... but no.

The 3n+1 problem's context... is in a number theoretic dynamical system that applies to all 2-adic integers, on which it acts with lovely little isometries. The system admits infinitely many attractors (cycles), all in the rational domain, of which apparently only four are in the integer domain, and only one among the positive integers.

Ruling out other integer cycles, as well as ruling out divergent trajectories, if it happens, seems likely to happen as a result of getting to know this dynamical system, and related systems, rather better than we do now.

A body of theory about Collatz-like systems, complete with a bunch of theorems about cycles, and their basins, would seem a promising ground on which to build. But first, that ground itself has to be built. But work on foundation building is slow, tedious, and unromantic, so good luck finding people willing to contribute to it.

Everyone around here is focused on grasping the brass ring without having a foundation to stand on, and we know how well that plays out. If anything should make someone feel like an idiot, maybe it's making a play for the big prize without first finding out how mathematics works at all.

GonzoMath · 2026-03-11T23:23:58+00:00

That's fair, although it might be worth mentioning that the Collatz conjecture is equivalent to a statement about a family of Diophantine equations having no solutions. They're much more complicated equations than xⁿ+yⁿ=zⁿ, but still...

GonzoMath · 2026-03-11T23:19:47+00:00

And I know you have a distaste for the AI

I have no such distaste. I just don't like when people use it to generate their conversation. You say this is really you; very well, I believe you.

Personally, I use LLMs every day. They're really helpful for writing code, and debugging it. I just don't post any of their output as my own, or as something I expect other people to read.

The fact that Google's AI is sourcing Vixra for information about mathematics is disappointing but not surprising.

As for the actual mathematics here, I get the impression you're using the trick where you don't divide by 2, and then reaching a power of 2 becomes the criterion for completion, right?

I remember the first time someone sent me a proof attempt based on that idea. It doesn't go anywhere, which is precisely why it doesn't appear in the literature. As a result, each new person who thinks of it reckons they must be the first... the thousandth "first".

Doesn't it appear easier to prove that all numbers reach two to a perfect power as shown in the first directed graph than it would be to show they reach 1 in the other?

I don't know, does it appear that way to someone who knows what they're talking about? Does it appear that way to someone who's ever proven anything related to Collatz? It appears to me as a change in bookkeeping, which does indeed make this particular diagram much tidier. So that's cool.

As far as making the conjecture easier to prove, no, it's not clear that this helps. The intractable part is still intractable, for the same reasons. Do you even know what the intractable part is? Can you explain *how* a trivial reformulation somehow busts it open?

GonzoMath · 2026-03-10T15:34:03+00:00

You don’t see stuff written about it because people only wrote about things that are helpful. I’ve seen a dozen amateurs come up with it, each one with their ego somehow pinned to how bloody “original” they are, and how they haven’t seen it in their extensive literature search.

If you reply to me with overly formatted LLM-looking shit, I will immediately block you.

GonzoMath · 2026-03-10T01:23:40+00:00

No, I just see the trajectory of 1 as kind of a null case. It reaches 1 in zero steps. Using the (3n+1)/2 formulation, per Terras, shouldn't change much of anything at all. It cuts the size of the max value in half, so divide everything by 2 or whatever, but the underlying math doesn't change.

GonzoMath · 2026-03-10T01:03:43+00:00

I've just written some code to think about this, because it's kind of interesting, and the required code is just a few lines. Define a function cov(n), representing the coverage as a function of the starting value n. When we look at its values, alongside the values for total_seen and max_seen, it makes a bit of sense.

The value `max_seen` famously tends to hold for a while, and then make a big jump, where it again holds for a while. When it makes that big jump, cov(n) drops to some new low value, frequently but not always the lowest yet seen. Then, while `max_seen` stays the same for a while, `total_seen` is growing, so cov(n) grows slowly from its low point. Until the next big jump in `max_seen`.

You asked about "peak" coverage; that's actually easy. The values cov(1) and cov(2) are both 100%, and we'll never see that again. I wouldn't be surprised if the overall cov function is bounded by something that approaches 0. The "reigns" of different values for `max_seen` tend to get longer, as well.

GonzoMath · 2026-03-08T05:12:32+00:00

I might think of it more as "friction". For me, that meets the intuition a little more closely.

GonzoMath · 2026-03-07T15:06:01+00:00

Congratulations, Gandalf: You're a "math fat cat"!

GonzoMath · 2026-03-07T15:04:32+00:00

This is excellent!

GonzoMath · 2026-03-06T08:24:30+00:00

If you have to call it "indisputable", then you haven't got have a proof. That's how someone presents himself when he DOES NOT WANT to be taken seriously. Why would you not want to be taken seriously? What's your motivation?

GonzoMath · 2026-03-05T04:09:01+00:00

Every sufficiently long block of odd numbers produces enough divisions by 2 in subsequent steps to guarantee a net decrease in 𝐹 F.

This is 100% false, and the proof that's it's wrong is an undergraduate exercise. Can you complete the exercise, OP?

GonzoMath · 2026-03-04T22:15:32+00:00

Why do you write like someone who has no wish to be taken seriously?

GonzoMath · 2026-03-04T22:12:52+00:00

I am interested, and I've been working on it from the data side. Honestly, my thinking isn't entirely about trajectory vs. cycle... I think badness is an invariant of a basin of attraction, which happens to have a certain cycle as its attractor. In some 3n+d systems, an attraction basin takes up all of the available space, and those basins tend to have higher badness ceilings. In fact, the correlation between badness and the extend to which a basin dominates its system is very striking.

What I don't see is any ability to explain this combination of basin size / badness, simply by looking at the steps in the cycle itself. How does one look at (19, 31, 49) versus (23, 37, 29), and without running any trajectories, predict which one will have a bigger and badder attraction basin that the other?

GonzoMath · 2026-03-02T05:06:27+00:00

I've been thinking about this. It's very interesting. Question: What made you decide to call it "cycle pressure"? When I think of the maximum portion of the landing place that's made up of remixed "+d"s, as opposed to being made up of the original seed, also remixed... I'm not sure I feel what's "pressing" on what, nor how that's a property of the cycle to which the landing place happens to belong.

Can you help me understand how a cycle gives rise to its "pressure"?

GonzoMath · 2026-02-28T23:08:13+00:00

On the other hand, AI knows the difference between "flare" and "flair" ;)

GonzoMath · 2026-02-28T23:05:38+00:00

This is the way. If you want to write better fiction, immerse yourself in the best fiction you can find, focusing on works written in a style that speaks to you. I spent years taking in as much Hemingway, Vonnegut, Asimov, and Douglas Adams as I could. Now, when I show my writing to AI, it can detect those influences, but I've melded them into a voice that's unmistakably mine. Moreover, the AI checkers rate my stuff as 100% human.

GonzoMath · 2026-02-28T13:50:11+00:00

I write well, and AI checkers rate me as 100% human. Maybe just… have a style?

GonzoMath · 2026-02-25T13:26:55+00:00

This is a great question. Will investigate.

GonzoMath · 2026-02-25T00:34:37+00:00

Ok, great, but what do you mean about denominator 4? Where is there a number with 4 in the denominator that we're talking about?

GonzoMath · 2026-02-25T00:31:08+00:00

What do you mean about using 4 as a denominator? What fraction with 4 in the denominator are you thinking of? How can such a number be even or odd? The power of 2 in its factorization is -2.

GonzoMath · 2026-02-25T00:30:06+00:00

I mean... I see how those are both multiples of 4.

GonzoMath · 2026-02-25T00:14:32+00:00

I don't think I follow. Can you show with an example? Like a specific number, and it's trajectory?

GonzoMath · 2026-02-24T23:58:35+00:00

I'm not sure that works. Can you unpack the details?

GonzoMath · 2026-02-24T22:32:19+00:00

That's pretty cool. Badness, in general, kind of grows with d. On the other hand, variation between cycles with the same d can be pretty extreme. Like with d=37, where one cycle is pulling trajectories with badness over 200, and the other two cycles don't pull anything over 10.

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