Tao (2022) proved Col_min(N) < f(N) for almost all N in logarithmic density and asked (Remark 1.16) whether this holds in natural density. This preprint answers affirmatively, with an explicit rate and a time bound: the witnessing iterate occurs within (1+o(1)) ln(N)/ln(4/3) Syracuse steps, i.e. under 12 ln(N) Collatz steps.

Zenodo (PDF + LaTeX + numerics): https://doi.org/10.5281/zenodo.21340150

Error reports welcome.

Follow-up to Bases of starting 5-tuples are Proth primes and form cycles mod 100 : r/Collatz.

I finally dared to ask my new friend, Google AI to connect the two (Collatz was never mentioned before). Here is its answer:

"Proth numbers and the Collatz conjecture are deeply linked because the formula for a Proth number k*2^n+1 directly represents an idealized "reverse path" or "highway" in the Collatz process. [1]

When you run the Collatz conjecture (3n+1), odd numbers grow and even numbers shrink by dividing by 2. If you trace the sequence backward, Proth numbers represent a perfect, uninterrupted cascade of divisions. [1, 2]

1. Reverse Collatz Paths

In the standard Collatz function, if an odd integer \(x\) passes through a single (3n+1) phase to reach a power of 2, the algebra looks like this:
3x+1=2^n

Solving for x, you get:
x={2^n-1})3
This is the exact definition of a Mersenne-like Proth number where k=1/3. Any time a Collatz trajectory hits an integer of this form, it drops directly down to 1 via division by 2, without hitting any more odd numbers. [1, 2]

1. The General Proth "Highway"

If a Collatz sequence hits an odd number that requires m structural loops before collapsing into a clean stream of divisions, it aligns precisely with the broader Proth equation:
N=k*2^n+1

When a number fits this algebraic structure:

• It guarantees that the number can be reached by working backward from k through exactly n consecutive multiplications by 2 (the inverse of the even Collatz rule).
• Proth numbers function as major structural nodes or "highways" in the Collatz tree visualization. [1, 2]

1. Modulo 100 Constraints in Collatz

Because Proth numbers dictate these paths, the mod 100 cycle restrictions discussed earlier inject a structural limit into Collatz trajectories. Since a Proth number with n>=2 can never result in residues like 31, 51, 71, or 91, it mathematically proves that a Collatz sequence can never transition directly from an even division into a Proth "highway" landing on those specific terminal digits."

To the best of my understanding, this hold for the right side of any dome. Does this mean that a sequence alternates from bridges from one side of the dome to the other side of another dome ? I suspect it for a while, but I now need to check this.

Project "Tuples and segments" in 13 pages : r/Collatz

Follow-up to All but one bases of starting 5-tuples are Proth primes : u/No_Assist4814 and Bases of 5-tuples in domes follow the Swiss Watch (mod 100) : r/Collatz.

My new friend, Goggle AI, provides the following,

"Proth primes mod 100 do not form a single cyclic sequence because they are primes, but the underlying Proth numbers (k*2^n + 1\)) are governed by a strict 20-step cycle.

Because (2^n) (mod 100) has a repeating period of 20 starting at n=2, the last two digits of any Proth number with a fixed k will cycle through a predictable sequence every 20 powers.

The 20-Step Base Cycle

For any fixed odd multiplier k, the value of (2^n) (mod 100) loops through these 20 repeating remainders:
[4, 8, 16, 32, 64, 28, 56, 12, 24, 48, 96, 92, 84, 68, 36, 72, 44, 88, 76, 52]

Proth Endings Constraints

Multiplying this cycle by an odd k and adding 1 restricts the possible last two digits of all Proth numbers (and thus Proth primes) where n ≥ 2. They can only end in these 25 specific modular values:

• Ending in 1: 01, 21, 41, 61, 81
• Ending in 3: 13, 33, 53, 73, 93
• Ending in 5: 05 (Only occurs when N=5; higher values are multiples of 5)
• Ending in 7: 17, 37, 57, 77, 97
• Ending in 9: 09, 29, 49, 69, 89

Because all Proth primes (except 3) satisfy N≡1 (mod 4), they can never end in 31, 51, 71, 91, 03, 23, 43, 63, 83, 25, 45, 65, 85, 07, 27, 47, 67, 87, 19, 39, 59, 79, or 99."

It has not found the Swiss watch yet (The Collatz procedure works like a Swiss watch (mod 100) : r/Collatz), but is very close. Note that the remaining numbers mentioned just above appear in blue-green bridge series, while those mentioned before do the same on the yellow bridge series.

Project "Tuples and segments" in 13 pages : r/Collatz

Note: This applies for natural numbers above two, i.e., {3, 4, 5, ...} . If anyone wants the calculations, do let me know.

Part 1: Setting the boundaries

It can be said that the given number decreases on x operations (of which (3x+1) is done 'H' times and (x/2) is done 't' times) if 3^H < 2^T .

It is also known that x = H + T

So, the boundaries can be set as follows:

(H/x) < (ln2/ln6)

(T/x) > (ln3/ln6)

Part 2: Finding a general trend for h and t as x changes

Since (3x+1) for an odd number always gives an even number, (x/2) always comes after (3x+1) . So, no two (3x+1) operations can happen consecutively. This can be rewritten as:

If a coin is flipped x times, what is the average number of heads, if no two heads appear consecutively?

An example to ensure proper understanding: Let's say the coin is flipped thrice. The possible outcomes are TTT, TTH, THT, HTT, HTH. (THH, HHT, HHH are excluded because they have consecutive heads.)

There are 5 total outcomes.

The average number of heads = (0+1+1+1+2)/5 = 1

1/5 is less than ln2/ln6 (H/x < 0.38)

Finding the formula for the total number for outcomes for x throws will lead to the formula for H/x .

After some solving, I arrived at the formula shown below (If someone knows how to format math on reddit, do let me know):

To show that the above equation does not cross (ln2/ln6), I used desmos to plot a graph (see below):

It is seen that (from 3 and onwards) the graph continues to decrease. The maximum is at (3, 1/3) - meaning that it will not cross (ln2/ln6 = 0.38)

Can someone tell me why this approach might not work?

https://github.com/lortzingring-tech/LNL---LZR-Collatz

The repository contains the computational core of the Lange Number Line project, including deterministic search algorithms and validation tools.

Some terms used here — such as selection, forced, and Σ(-5) — are currently project-specific working concepts. Formal definitions and examples for individual Collatz peaks will be added in future updates.

Feedback, criticism, and independent verification are welcome.

Hi All,

I may have some meaningful math regarding Tao's Syracuse random variables.

Long time listener; first time caller. Also, terrified to be here. First and foremost, I'd like to give a qualifier: no idea what I'm talking about. Complete imposter syndrome and I did most of this through metaphor with the assistance of LLM. Now for the headline and then the backstory:

My github repository is completely public and the more recent arc is furthering the qx+1 work and linking Dr. Maxwell Siegel's work with my own.

https://github.com/mrnathanhumphrey-droid/Collatz

Above is the repository. It is over 125k files in totality so I have left the necessary documents for replication. There are documents chronicling everything by timeline and by field as well.

https://doi.org/10.5281/zenodo.20673500

Here is a preprint I am faking my way through. I've taught myself enough Lean 4 to verify everything. That has its own DOI and repository that you can find through my profile.

Now for the Storytime because if I'm putting myself out there; I'm doing so authentically. This all started originally when I was 11 and I hand plotted the first 800 digits on a TI-83. I started mapping ln(x) to the pattern and saw outliers. I thought "if my computer was powerful enough I could calculate a ton and there would be a pattern in the noise".

27 years later my best friend and I were making NBA projections for fun when I re-discovered "The Dwight Effect" in block data and the Bayesian fit reminded me of Collatz when I was 11. So I did a Bayesian fit of the stopping times with the intention of finding a pattern. From there, I just kept probing the pattern and trying to retroactively teach myself anything I found. There are a million and a half dead ends. There are stupid repetitive probes that are a waste. But, I have tried to prove every single thread and finding wrong a million times over before presenting it.

If it's all fugazi; it's okay. If it's not; hopefully it will be of help to someone. I'm having fun regardless.

Some users asked me for more mathematical content and fewer 3D renderings. I hope that this one-page summary better reflects the mathematical language behind the LNL/LZR project.

I have been working on an alternative structural view of the Collatz problem, which I call the LNL/LZR framework (Lange Number Line / LZR Engine).

The central idea is not to search for new arithmetic operations, but to analyze the geometric structure of Collatz trajectories in terms of mountains, valleys, peaks, trajectories and deterministic backward paths.

My current hypothesis is the following:

Every Collatz trajectory belongs to exactly one mountain.

Every peak is an even number.

The last 9 backward steps before a peak follow a deterministic pattern.

These 9 steps uniquely determine the predecessor structure of the mountain.

The mountain–valley decomposition may explain why trajectories cannot escape to infinity.

I have created several visualizations to illustrate the idea. At the moment, this is still an experimental framework rather than a formal proof, and I would be very interested in critical feedback from mathematicians.

For the mathematicians: please attack the logic, the assumptions and the definitions. 🧮

For the treasure hunters: please remember that GitHub and Zenodo timestamps exist. 😄

Constructive criticism, counterexamples and alternative interpretations are highly appreciated.

GitHub repository:

https://github.com/lortzingring-tech/LNL---LZR-Collatz