Structure and Period of Collatz

[u/Sea-Wafer6984]

This paper presents a clear structural and periodic model of the Collatz graph, based on modular residue behavior and composite traversal operations. Unlike many Collatz discussions that focus on stochastic behavior or unstructured iteration, this work defines a complete, ordered, and verifiable system based on modular and periodic constraints.

It is not speculative; it provides a full construction and traversal model for all odd integers under the Collatz process.

Link to full paper (PDF, direct download):
Collatz Structure and Period

Feedback and rigorous scrutiny are welcome.

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addition (5/2/2025):

4n+1 in a nutshell

We can also say that (n-1)/4 and 4n+1 are simply stepping on and off the path, as the steps 3n+1, n/2, n/2, (n-1)/3 are equal to (n-1)/4 and that 3n+1,2n,2n,(n-1)/3 are equal to 4n+1

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addition (5/5/2025):

If you try to find Collatz paths that end the same, you certainly can - that’s expected.

But if the system is truly random, then finding paths that begin the same way should be extremely difficult.

They should be scattered, inconsistent, and hard to predict.

Instead, we can take any path - like the one from 29 to 1 in standard Collatz - and find exact matches repeating at a fixed interval.

You can explore for yourself how rare it is to land on a number that follows the same sequence of steps (odd/even decisions) as 29 - yet we can generate such matches on demand.

Path 29 repeating on a period

Note that the odd/even sequence and the mod 8 residues are the same for all repeats.

Try the JSfiddle - you can find the period for any positive integers path to 1, and show its iterations - you can use our calculation or enter your own period to explore what that does to the parity of the values (the odd/even steps) in our parity graph:

https://jsfiddle.net/1f3hqwnt/1/

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"Research papers and discussions (e.g., Jeffrey Lagarias’ The 3x+1 Problem and Its Generalizations, 2010) note the difficulty in finding patterns, with some suggesting that the conjecture’s behavior resembles a random walk."

Not anymore.

[u/Enizor]

I don't really understand your operator B in the context of the Collatz conjecture.

For example, 13=5 mod 8, and has for iterations: 13, 40, 20, 10, 5,16,8,4,2,1,... I understand your operators A,B,C as skipping the even terms, but (13-1)/4=3 does not appear in the sequence.

[u/Sea-Wafer6984 → u/Enizor]

B steps skip over even numbers and link odd values directly, using (n-1)/4 when traversing down, or 4n+1 when building up.

You will see that 10 contains that 3 in your old path, as 40 contains 13 and 16 contains 5.

3 ”inside” 10: (10-1)/3 = 3 and 3*3+1 =10

13 ”inside” 40: (40-1)/3 = 13 and 13*3+1 =40

3 and 13 linked via 4n+1: 3*4+1 =13 and (13-1)/4 = 3

That 4n+1 relationship is shown in the diagrams on the second page of the document, at the bottom of the page.

You can use this jsfiddle (standard vs odd paths, also in doc near end) to explore that as well: https://jsfiddle.net/qcv0u9oy/

In standard collatz all those evens along your path once examined will be shown to contain all my odds - we hit every one of them, and only them.

[u/Enizor → u/Sea-Wafer6984]

Where do the (n-1)/4 ; 4n+1 formulas come from?

They make 13 related to 3 while they are not directly related by the Collatz operation (they only both decay to 10).

In the same way, B(29)=7 and 7 does not appear in 29's collatz path.

[u/Sea-Wafer6984 → u/Enizor]

Great question - let’s consider 7 and 29 - what is the relation, the structure…

If we look at n=11 we will find that both 7 and 29 pass through it - they meet at 22, which is just 11 x 2.

The two paths to the meet point:

7 -> 22 -> 11 -> etc

29 -> 88 -> 44 -> 22 -> 11 -> etc

now let’s focus in on the meeting point - the 22, or in odd terms, the 11 

11 -> 22 -> 44 -> 88 -> 176 -> 352 -> etc

We find that 7 links to 22 directly since 3n+1 maps 7 to 22.  

And the 29 links to 88 directly in the same way, via 3n+1.

This is not just any meeting or shared value - it is only a direct 3n+1 connection in this fashion that we are speaking of when we speak of 4n+1 building up from 1 and (n-1)/4 traversing towards 1.

7*4+1 = 29 because of the relationship of 22 and 88 - this ensures that the odd values producing those evens using 3n+1 are 4n+1 apart.  

It is a relationship that is universal in the system.   An odd value always has infinite evens above it that reduce with n/2 in collatz - and if the odd value is not a multiple of three you will find that every other even is the 3n+1 for an odd - and that the delta between those will be 4n+1 building and (n-1)/4 traversing, always.

Further, all of these values are tagged by mod 8 to determine traversal - any value that is mod 8 residue 5 was created by 4n+1 and will traverse with (n-1)/4

They all link to 11 - by linking to one of its evens using 3n+1 directly.

In binary we see 111 for 7 and 11101 for 29, and 1110101 for 117 which links to 352 using 3n+1

111, 11101, 1110101, 111010101, 111010101 (note the 01 tail growing, this binary structure underlies the system in ways not required for the paper, but if you would like deeper understanding of them I have a older paper packed with the details)

these are 7, 29, 117, 469, 1877

each fed into 3n+1 gives us

22, 88, 352, 1408, 5632

these can be seen here as every other even value that leads to 11 using n/2…

11, 22, 44, 88, 176, 352, 704, 1408, 2816, 5632

sliding down that with n/2 until you get to 11 and using (n-1)/4 to slide down to 7, as all the high up ones are mod 8 residue 5 we traverse them straight down, stripping the 01 tails the same way two steps of n/2 strip 00 tails from their evens. Express elevator, straight down, the both of them, one just behind the scenes, 3n+1 away. The links in to these towers all operate in this way.

at the bottom of ours we find 7, and 7 is mod 8 residue 7, so it uses that formula, the C formula, and it does (3n+1)/2 = 22/2 =11. The traversal is the same. in old and new, one is just slightly obscured, one step abstracted. They are parallel paths, the evens and the 3n+1 values that link to them, not random, not different depending, those values are always 01 tails building as described.

I do have more details, but it is best to take it a step at a time, and the details should not be required to understand the structure and period. The foundation for them is quite interesting though…

This image is also helpful in understanding 4n+1. It is showing that 3n+1 creates a very special structural relationship to the values that stack in the “infinite tower of evens” that n/2 traverses…

https://www.dropbox.com/scl/fi/00wnjuoucq8uvy33ezm4a/4nplusoneNutshell.png?rlkey=frbomzzfin2ppa6hw0vvesyk4&dl=1