Hi all Hi GandalfPC

I am referring to this topic https://www.reddit.com/r/Collatz/comments/1t20pje/the_geometry_between_collatz_segments/

There are three statements from you there regarding the Collatz structure.

>... but Collatz is a nonlinear 2-adic tree-like dynamical system with non-invariant, non-closed modular dynamics.

>Collatz does not form a self-similar geometric object, because its residue structure is not invariant or closed under iteration, and does not preserve consistent scaling across levels.

>Collatz is not a fixed piecewise affine system - the affine parameters depend on state-dependent 2-adic valuations, so the “mx + a per layer” model does not define stable scaling across iterations.

I had this translated and asked an AI to explain it to me in a bit more detail. But just to be absolutely sure what it all means, I thought I’d rather ask you directly.

non-invariant

As I understand it now, there is no equality of numerical ranges within the Collatz tree/space.
Nothing repeats itself in the exact same way.

non-closed modular dynamics (residue structure is not invariant)

Does this concern the infinitely many sieves that a number passes through on its way to 1, and the fact that one cannot say exactly which sieves one will encounter or when?

nonlinear 2-adic tree-like dynamical system

The AI ​​explained to me that the numbers in the Collatz system behave differently than usual, and are not arranged linearly within it.
This is both understandable and logical.
But does this also rule out the possibility that a linear order could exist within the 2-adic Collatz system?

not a fixed piecewise affine system

As I understand it now, there are no fixed residue classes or domains into which one could classify the numbers to describe the whole system with linear formulas. Am I understanding this correctly?

does not define stable scaling across iterations

Does this mean that from a classical perspective, there are no formulas to describe consistent scaling coefficients because of this 'chaotic' behavior?

I hope I am not bothering you with the questions of a hobby mathematician, and I am very much looking forward to your response and insights.

Hi everyone^^

Some time ago, I saw advice in this community along the lines of:

“A meaningful contribution to Collatz would be to transform it into a sharper and more attackable obstruction problem.”

That idea stayed with me for quite a while, so instead of trying to “prove Collatz,” I tried to see whether the problem could be reduced into a more explicitly arithmetic form.

To be clear: the remaining difficulty still feels essentially Collatz-level to me. The goal was not to solve the conjecture, but to isolate more precisely what the remaining obstruction actually is.

The framework I ended up using is a Δ-core reduction framework built on the accelerated odd-to-odd map.

I define a correction cocycle Δ_k by

Δ_{k+1} = 3Δ_k + 2^{A_k},

and track trajectories via

n_k = (3^k n_0 + Δ_k)/2^{A_k}.

I then define the centered residue of the correction term as

s_k = cent_{2^{A_k}}(Δ_k).

Within this coordinate system, the main deterministic result is:

{ k ≤ T : |s_k| ≤ B } = O_B(log* T).

The key mechanism is a fixed-residue recurrence gap.

If the same centered residue repeats,

s_i = s_j = r,

then one obtains

3^{j-i} ≡ 1 mod 2^{A_i},

and applying LTE yields

j - i ≥ 2^{A_i-2}.

So any infinite bounded-strip recurrence would necessarily become tower-sparse, eventually inducing a kind of 2-adic congruence lock.

At that point, the remaining obstruction appears to reduce to something like:

“Can a nonperiodic tower-sparse 2-adic congruence lock actually be realized by a positive-integer Collatz orbit?”

The main conceptual distinction that emerged for me was:

local admissibility
vs.
global realizability.

In other words, local congruence constraints may continue to remain compatible at arbitrarily deep 2-adic levels, while it is unclear whether such structures are globally realizable as actual positive-integer orbit sections.

Again, I am not claiming this proves Collatz.

My feeling is closer to:

“the orbit-dynamics problem may be reducible to an arithmetic-dynamical realizability problem.”

So my question is mainly whether this direction itself looks mathematically meaningful, or whether it is ultimately just a reformulation of the original difficulty in different language.

I would especially appreciate criticism/comments on:

1. the fixed-residue return gap argument,
2. the O(log* T) sparsity step,
3. the local admissibility vs global realizability framing,
4. whether this is genuinely a sharper obstruction or merely a repackaging of the original problem.

Thanks to the community for all the discussions and advice over time ^^

Preprint: https://zenodo.org/records/20322532

Numberphile recently posted a video about moving red and black knights around an infinite chess board. It has basic rules like collatz in that knights can't ever attack each other upon moving to a square.

Interestingly in the basic version displayed in the video, at small steps it appears random(sounds familiar right) but as they increase the amount of steps and therefore the size of the structure. It forms pretty reliable growth in the 4 quadrants of the 2D plane they used to represent it. In these 4 quadrants there are large swaths where ONLY red knights or Black knights can exist.

How this pertains to collatz is this. What if we had a similar way to map the conjecture. Would visually seeing these predictable regions(if they existed in a proper collatz mapping) help us with identifying the underlying cause of them?

I'm only wondering this, because watching that video. It reminds me fully of the game of life. Which in turn reminds me of collatz.

But seeing those predictable regions existence feels like we should be able to identify it's cause easier than without the visual representation. I have a feeling we would still have the issue of small regions that are unpredictable, even if structured and entirely predetermined by the starting conditions.

It's Just a thought, but maybe someone can find a good mapping for collatz visually that would identify it's complete structure visually, even if we would have to analyze it to find out the mathematical reasoning.

Quite possible this method of visual analysis can be easily extended into other 3x+y systems, or even the broader class of zx+y systems. Since each system would give it's own mapping independent of others.

their structures would be much more similar than those produced by the various motions of different chess pieces on their infinite chess board.

And maybe, just maybe, this visual route would identify what we are looking for. Especially if we are able to understand the structures we are viewing for various zx+y systems.

I spent around 3 years on the collatz conjecture, what I saw is that at its core lies randomness, which can also be understood as probabilistic nature.

So at the moment I have developed a technique to work around or reduce this randomness.

The technique involves taking information in a certain dimension “A” then convert it into another dimension “B”. This has removed some of the randomness somehow.

It turns out that the collatz sequence was not treating numbers as quantities of even or odd numbers, instead it’s about the binary structures of the numbers, not about being even or odd numbers this whole time.

My sources are telling me that Collatz is example of a cohomology style obstruction problem related to the relationship between addition and multiplication computed over local to global scales.

If this is true, my question is if anyone has done any research around lifting the Collatz universe to a domain where multiplication becomes addition, and addition stays as addition. I assume log is involved.

I can see how multiplication and addition might not exactly get along over unbounded stretches along the number line, but it seems like addition isn't going to obstruct addition. Maybe I'm wrong.

Anyone know of any papers specific to Collatz and this line of inquiry to review? I know how to lift the problem statement, but then it becomes a weird trig impossibility argument.

I have an i5 just sitting here doing nothing worth a darn. So I coded up a program to generate the largest stopping times. I don't know what I will do with it, most likely nothing, but hey, I'm doing something. These are the last 5 so far. As you can imagine, it takes longer and longer for each number. Its max will be 18,446,744,073,709,551,615, then I will have to switch to big numbers. I don't expect to get close to that number.

Count: 881,715,740,415 Steps: 1,335

Count: 898,920,104,505 Steps: 1,335

Count: 989,345,275,647 Steps: 1,348

Count: 1,122,382,791,663 Steps: 1,356

Count: 1,444,338,092,271 Steps: 1,408

Hello guys someone recommended me this subreddit, I am very passionate about the topic of the collatz conjecture. Happy to share my work with you guys sometimes. And also learn from your works too. I am genuinely happy to be here. Networking with real mathematicians who enjoy exploring the beauty and the art of mathematics is something I truly enjoy and love having discussions about with other passionate mathematicians.

If 3N + 1 results in a random even number we have the formula:
(3N + 1) / 2^K where K is the number of factors of 2 factors that 3N + 1 results.

After applying this formula to N we can have 1 or an odd number and this process will only stop if N reaches 1

If all evens have factors of 2 but the quantity of those factors varies but have a probability which is:
50% of evens have 1 factor of 2
25% of evens have 2 factors of 2
12.5% of evens have 3 factors of 2
6.25 % of evens have 4 factors of 2
...

so we assume the distribution of probabilities are:
100% / 2^K = probability of a even number having K factors

If K = 1 then: (3N + 1) / 2 = 1.5N + 0.5, N will grow

if you apply this formula in almost 50% of cases, N will grow by a factor of 1.5N + 0.5

Since this formula is applied in 1 out of every 2 cases, we will have: 1.5N + 0.5

So what would the worst-case scenario have to be for it to decrease?

If K = 2 then:
(3N + 1) / 2 = 1.5N + 0.5
[(1.5N + 0 .5) * 3 + 1] / 4 = 1.125N + 0.625, N stills grows

If K = 3 then:
[(1.125N + 0.625) * 3 + 1] / 8 = 0.421875N + 0,359375, N will decrease

The chance of K being at least 3 is 25% because:
(12.5% of K = 3 + 12.5% of K > 3) = 25% of K >= 3

The average case to get K >= 3 will be 4 so we will get the following average:
2 cases of K = 1 (50%)
1 case of K = 2 (25%)
1 case of K = 3 (worst case because if K > 3 then will decrease even more)

The result will be:
(3N + 1) / 2 = 1.5N + 0.5
[(1.5N + 0.5) * 3 + 1] / 2 = 2.25N + 1.25

[(2.25N + 1.25) * 3 + 1] / 4 = 1,6875N + 1,1875

[(1,6875N + 1,1875) * 3 + 1] / 8 = 0,6328125N + 0,5703125, N will decrease

This proof for every 4 cases the probability of worst case decreasing is higher than worst case of 2 cases and for an average of 4 cases the worst case decrease is more than the worst case of 2 cases, proofing the more you repeats this formula the higher will be the probability of N decreasing.

Another fact that need to be mentioned is that:

K is limited to the intervals of 2^E to 2^E + 1
So the max number K can be in the interval of 2^E to 2^E + 1 is E

If N is present in 2^E to 2^E + 1 then getting K = 1:
(3N + 1) / 2 = 1.5N + 0.5, still on this interval

Getting K = 1 again:
[(1.5N + 0.5) * 3 + 1] / 2 = 2.25N + 1.25, goes to the next interval

For example:
2^4 to 2^5 = 16 to 32
the max number K can be is limited to 4

If we get K = 1 two times the interval will increase, consulting the probability distribution we have:

50% of evens have 1 factor of 2
25% of evens have 2 factors of 2
12.5% of evens have 3 factors of 2
6.25 % of evens have 4 factors of 2
...

Meaning the chance of decreasing will also become higher if N grows.

Finally, we know that up to 1,000,000,000,000 it is true, and once N reaches a number less than that, it is guaranteed to become 1

And also it's impossible for a number bigger than this get directly turned into 1 because it's N / 2 every time N is even ;

plus the more you repeat this formula the higher the chances of N decreasing, proving this conjecture is true.

Guys I need help. I was wondering if J.simons and de.werger's application of baker's theory on linear forms of logarithms could be applied to a rather different equation describing collatz map in one equation, using similar logic used by C.Bohm and G.sontacchi's criterion for loop in collatz. As I do not have a clear grasp at transcendental number theory, Can anyone tell me if this line of inquiry is even appropriate?

I received the following objection:

>“A finite state identity cannot capture the data needed for the transitions; there is no ‘faithfulness’ in infinite variety. That is pretty much the point of infinite variety.”

I want to ask a narrow technical question about this.

I am not claiming that a finite quotient encodes the whole infinite orbit, and I am not asking whether this proves the Collatz conjecture. I am only asking about the local next-transition claim:

>Can a finite quotient carry enough information to determine the next local transition?

I attached two standalone Lean 4 files.

https://www.wow1.com/CenteredFramework.lean checks the finite centered transition algebra. It proves that:

• C mod 8 uniquely determines the survivor residue;
• C determines the next centered state C';
• the admissibility lock is preserved;
• lifted representatives of the form D*u + C reduce to the same quotient representative modulo D = 3^17;
• lifted next representatives also reduce to the same finite next state modulo D.

https://www.wow1.com/NextStepFaithfulness.lean checks the same issue at the lifted/raw branch level. It keeps full lifted branch data and proves that the normalized next centered coordinate is uniquely determined, and that affine-family parameters do not create different next finite transition data modulo 3^17.

So the specific question is:

>Do these Lean files adequately answer the local version of the objection, namely that the finite quotient cannot contain enough information for the next transition?

Equivalently, if this local claim still fails, what is the missing datum?

A concrete failure would have the shape:

>two lifted branches with the same finite quotient, but different next finite transition data.

That is the kind of objection I am trying to isolate. I am not asking here about global orbit completeness or the full proof route, only whether the quotient contains enough information to determine its local transition.

I’ve been exploring a custom Collatz-type dynamical system for the past couple months and wanted feedback from people interested in number theory / dynamical systems.

Instead of using a fixed odd rule like (3n+1), my variant uses increasing odd multipliers:

(1,3,5,7,\dots)

Definition:

• If (x) is even: [ x \to x/2 ]
• If (x) is odd: [ x \to mx+1 ] and then the multiplier updates: [ m \to m+2 ]

starting from (m=1).

So the odd multipliers used successively are:
[
1,3,5,7,\dots
]

I ran computational tests for starting values up to 10,000 with a 200,000-step cutoff.

Observations:

• A small minority quickly reach small numbers like ({1,2,3,4,5})
• Most trajectories instead grow extremely large and never returned within the computational limit
• I also proved there are no cycles in the full ((x,m)) state space because the multiplier strictly increases after odd steps

I wrote a short experimental paper about the system and I’d appreciate feedback, criticism, or suggestions for further directions.

I’m not claiming a breakthrough — just sharing an interesting dynamical-system experiment inspired by Collatz.

For those who would like to take a break from the main problem or look at it from a different perspective, I propose a similar hypothesis:

Conjecture (Division by the smallest divisor): If we take any prime number p > 2, multiply it by 3, add 2, and continue this process until we get a composite number, and when we get a composite number, we divide it by the smallest prime divisor until we get a prime number again, then we will eventually get into a cycle of length 19: 5 → 17 → 53 → 23 → 71 → 43 → 131 → 79 → 239 → 719 → 127 → 383 → 1151 → 691 → 83 → 251 → 151 → 13 → 41 → 5

I was analyzing the Goldbach conjecture yesterday and today, and a while back I tried to solve the Collatz conjecture, and I concluded that the Goldbach conjecture is even harder than the Collatz conjecture.

I would appreciate feedback on the logical structure of the following finite-state reduction idea for the Collatz problem.

The idea is to encode the remaining “live” part of the dynamics by a finite state

K = (C, t mod 2, p),

where p is a finite label and

0 <= C < 3^17.

The update has the form

C' = (C + 3^17 m)/8,

where m is determined by the current state, 0 <= m <= 7, and the numerator is divisible by 8.

Then

0 <= C + 3^17 m < 8*3^17,

so

0 <= C' < 3^17.

So, if this encoding is faithful, the live part moves inside a finite deterministic graph. Any infinite live path would eventually repeat.

My question is:

If every repeating component and every exit leaf from the finite graph can be shown to either reach 1, reach a smaller odd number, or enter a family that was already closed earlier, is that enough in principle for a global descent proof?

Or is there some standard trap in Collatz arguments where this kind of finite-state reduction still fails?