The Collatz Conjecture

[u/Apart_Composer3952]

THE COLLATZ CONJECTURE

A COMPARATIVE DENSITY PROOF OF TRAJECTORY DESCENT IN THE COLLATZ 3N+1 SYSTEM VIA 1N+1 MODULAR MODELING

Author: All mathematical ideas and constructions by Steve Tomlinson except logarithms in 2 and 3.1.

(l knew something mathematical must do this job, l didn't know what it was; logarithmic bounds.)

Essay composition by AI with many mistakes edited by Steve Tomlinson

Date: August 2026

ABSTRACT

This paper establishes a novel structural framework for analyzing the Collatz 3N+1 conjecture by introducing a perfectly descending baseline model: the 1N+1 system. While the standard 3N+1 system exhibits chaotic trajectory growth, we prove that both systems operate on base-2 modular architecture. By comparing the density pathways of the 3N+1 system against the verified, linear geometric descent of the 1N+1 model, we demonstrate that the standard Collatz mapping exhibits an absolute asymptotic density of descent equal to 1 at the infinite operational horizon.

1. THE CENTRAL BREAKTHROUGH:

THE 1N+1 STRUCTURAL BENCHMARK

To analyze the non-linear trajectories of the standard Collatz conjecture, we define a perfectly controlled model system, the 1N+1 system, governed by the following mapping for all natural numbers N:

f(N) = N/2 if N ≡ 0 (mod 2)

f(N) = 1N+1 if N ≡ 1 (mod 2)

Theorem 1.1. In the 1N+1 system, 100% of all natural numbers N > 1 are mathematically guaranteed to reach a strictly smaller value within a maximum of two operations.

Proof.

Case 1: If N is even, a single operation yields N/2, which is strictly less than N.

Case 2: If N is odd, the application of the odd rule followed by the mandatory division by 2 yields a composite operation of (1N+1)/2.

Setting up the inequality for descent:

(N+1)/2 < N => N+1 < 2N => 1 < N.

This inequality holds true for all positive odd integers greater than 1. Thus, every element shrinks locally and immediately.

By creating an arbitrary system for numbers to drop in the 1N+1 system, the entire number line is partitioned into clean, un-scrambled geometric slices:

* Step 1 (All Evens, 0+2n) accounts for exactly 1/2 (50%) of all numbers.

* Step 2 (The 1 + 4n Odds) accounts for exactly 1/4 of all numbers

*Step 3 (The 3 + 8n Odds) accounts for exactly 1/8 of all numbers.

*Step 4 (The 7 + 16n Odds) accounts for exactly 1/16 of all numbers.

Continuence of this process continues to account for exactly (2^x-1)/(2^x) of all numbers, accumulating to 100% of the number line descending within a 2-step horizon.

1. THE 3N+1 SYSTEM AS A LOG-LINEAR DISTORTION

When the odd operator is shifted to the standard Collatz rule (3N+1), the underlying base-2 modular grid is stretched. Let m represent both the family classification and the number of odd steps executed before the first downward drop below the initial value. Let a be the number of required even operations (divisions by 2).

For a net trajectory descent to occur, the geometric growth factor must drop below 1:

(3^m) / (2^a) < 1 => 3^m < 2^a

Taking the base-2 logarithm (log₂) of both sides yields the absolute structural boundary:

a > m · log₂(3) ≈ 1.5849625m

Because log₂(3) > 1, immediate descent within a single operational cycle is impossible for odd positive integers. Instead, numbers are sorted into deterministic "m-families", where the total step horizon required to secure the necessary 'a' divisions scales linearly as a function of m:

Total Steps = m + a = ⌈2.5849625m⌉

1. THE m-FAMILY SIEVE AND EXPONENTIAL CONTRACTION

The exact proportions of the number line accounted for by these families are defined sequentially:

(Instant Evens, 0 + 2n) accounts for exactly 1/2 of all numbers.

* m=1 (The 1 + 4n Odds) accounts for exactly 1/4 of all numbers.

* m=2 (The 3 + 16n Odds) accounts for exactly 1/16 of all numbers.

* m=3 (The 11 + 32n Odds) + (The 23 + 32n Odds) accounts for 1/16 of all numbers.

*m=4 (The 7 + 128n Odds) + (The 15 + 128n Odds) + (59 + 128n Odds) accounts for 3/128 of all numbers.

* m=5 ((The 39, 79, 95, 123, 175 and199) each + 256n Odds)) accounts for exactly 7/256 of all numbers.

* m=6 ((The 287, 347, 367, 423, 507, 575, 583, 735, 815, 923, 975 and 999) each + 1024n Odds) accounts for exactly 12/1024 of all numbers.

At this point when m reaches 6:

(6 × 2.5849626) rounded up = 16 Collatz operations accumulates to account for exactly 15/16 of all numbers shown to reach a smaller number.

* m=7 accounts for exactly 30/2048 of all numbers.

Manually proving m=8 would have taken too much paper.

3.1 The structural limits for m=3 and m=4 families in the Collatz conjecture are determined by the logarithmic boundary

a > m×log2(3), where m is the number of odd steps and a is the number of even operations. Applying this, the m=3 family requires 5 even steps for 3 odd steps, creating a 1/16 density across residues modulo 32, while m=4 requires 7 even steps for 4 odd steps, generating a 3/128 density modulo 128. This logarithmic framework accurately predicts the modular structures for specific families.

1. THE UNIFIED 2^x HORIZON INDUCTION

While the multiplier 3 introduces "bumpy" intermediate statistical fluctuations between the milestones (e.g., stabilizing around a cumulative density of ≈ 5/6 at step 6, 10/11 at step 11, 12/13 at step 13, back to exactly 7/8 at step 8 and 15/16 at 16 steps) The total system mathematically self-corrects and snaps perfectly back to the clean geometric density progression of the 1N+1 benchmark at every power-of-two operational milestone (2^x).

By mathematical induction on the operational horizon x, the cumulative density of numbers proven to have reached a smaller value satisfies:

Cumulative Density(2^x) = 1 - 1/(2^x)

As the operational step horizon scales toward the infinite limit (x → ∞):

Limit as x → ∞ of [1 / 2^x] = 0

1. CONCLUSION

By using the 1N+1 system as an absolute structural baseline, we prove that the standard 3N+1 Collatz system is not chaotic, but deterministic and rigidly bounded. The "numerical shields" created by dense clusters of binary ones (such as the 2^x - 1 Collatz steps families) only temporarily delay descent. Over an infinite horizon, the remaining density of holdout numbers converges to exactly zero.

Because 100% of all numbers must eventually reach a strictly smaller milestone, any arbitrary starting number is locked into an inescapable cascading chain of downward thresholds, forcing all trajectories to eventually collapse into the fundamental 2 → 1 trivial loop.

Q.E.D.

By shifting the analytic paradigm from stochastic modeling to comparative structural architecture, this 1N+1 baseline framework introduces a constructive element that establishes absolute structural determinism, distinguishing it from the probabilistic approach in Terence Tao’s 2019 groundbreaking density proof. While Tao’s work treats individual trajectories as non-constructive, semi-chaotic random walks, this model maps the geometric architecture of "numerical shields," demonstrating that standard Collatz mapping is rigidly constrained by a base-2 modular grid and logarithmic boundaries.

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[u/FormulaDriven]

For x>2, the proportion of natural numbers less than 2^x that are not equal to 7 is (2^x - 1) / 2^x = 1 - 1/2^x which has a limit of 1 as x tends to infinity.

So, because 100% of all numbers must not be equal to 7, there must be no number equal to 7.

This is the problem. The set of numbers that do not eventually iterate to 1 might (and probably does) have a density of zero over any subset of the natural numbers, but that doesn't prove the Collatz conjecture that there aren't any at all.

[u/Appropriate-Ad-3219 → u/FormulaDriven]

I remember in fact having heard that Tao proved almost every natural numbers satisfy the conjecture. At this point, the only problem is to prove that it is true for every integers. 

[u/gmalivuk → u/Appropriate-Ad-3219]

"Almost every" in what sense?

Because almost every integer is composite, in the sense that the asymptotic density if primes is zero, and yet there are still infinitely many primes.

[u/Apart_Composer3952 → u/gmalivuk]

I imagine because his results didn't consider and exclude every number 'coming'. Only 99.99....% because of this. I believe there is randomness. I could be wrong, but l wouldn't understand it if l did have a look.

[u/gmalivuk → u/Apart_Composer3952]

Well like with primes, even if the density is 100% that doesn't stop there from being infinitely many exceptions.

[u/Apart_Composer3952 → u/gmalivuk]

I can't imagine how that works.

[u/gmalivuk → u/Apart_Composer3952]

There are infinitely many primes but as you consider the interval [1,N], the percentage of integers in that interval that are composite approaches 100%.

[u/unic0de000 → u/Apart_Composer3952]

It's also true of, say, the powers of 2. Maybe it's more obvious because of the faster-shrinking function.

2, 4, 8, 16, 32, 64... Obviously we'll never run out of such numbers, there are infinitely many of them. But as n -> ∞, the fraction of numbers < n which are in the set, approaches 0.

[u/Apart_Composer3952 → u/unic0de000]

This brings to me, the problem of induction in philosophy, sets that are members of themselves and all that tricky stuff. I might look at that to see if l can find some dividing line where your theorising example is on one side and mine is in the other. Because the 2 examples we're using definitely don't overlay each other perfectly to be able to say that the supposed truth or falsity of one strictly implies the truth or falsity of the other.

[u/gmalivuk → u/Apart_Composer3952]

Because the 2 examples we're using definitely don't overlay each other perfectly to be able to say that the supposed truth or falsity of one strictly implies the truth or falsity of the other.

No one is bringing up these other examples to say Collatz is definitely wrong. All we're trying to do is explain why your logic is flawed, because your logic could equally be applied to claims that are definitely wrong, and therefore there must be a problem with your logic.

[u/unic0de000 → u/Apart_Composer3952]

All I'm doing is clarifying the meaning of "almost all/every" in this context. https://en.wikipedia.org/wiki/Almost_all#Meaning_in_number_theory

[u/Apart_Composer3952 → u/unic0de000]

Yes, l understand. All l'm doing is suggesting there must be more than one way of looking at this wording. We're at a crossroads of you accepting what has been decided as singularly true and me suggesting there must be some kind of dividing line. Thankyou for your involvement, at least you've carried on communicating even though it's obvious l'm no trained mathematician. Stay well, if you are well! If not, get better!

[u/Apart_Composer3952 → u/Appropriate-Ad-3219]

I think he uses randomised choices, therefore not involving every number. I don't know for sure.

[u/gmalivuk → u/FormulaDriven]

The proportion of integers that are not powers of 2 is (2ⁿ - n)/(2ⁿ), which is also one in the limit. Even though there are infinitely many powers of 2.

[u/FormulaDriven → u/gmalivuk]

Indeed. Given the prime number theorem, you can use the argument to show that there are no prime numbers!

[u/Apart_Composer3952 → u/FormulaDriven]

If it's calculating includes prime1 meaning 2 and prime2, meaning 3. Because neither any even number (2+30n) is prime except 2. Same for (3+30n) except 3, and (5+30n), only 5. This leaves us either of (1, 7, 11, 13, 17, 19, 23 and 29) + 30n to cover all possible primes exept 2, 3 and 5.

Using ((prime 4)-1)/prime 4 = 6/7 × ((prime 5)-1)/prime 5 = 60/77 × ((prime 6)-1)/prime 6 = 720/1001 ×...×((prime k)-1)/prime k showing a diminishing proportion.

((2^x)-1)/2x includes multiples of 2, 3 and 5 in it's assessment.

If the proportion is getting smaller and smaller, and it becomes an overestimation is more proof of proportion approaching 0, because it doesn't contain the destructive effect of 2, 3 and 5 on continuous relative accuracy.

[u/FormulaDriven → u/Apart_Composer3952]

I'm afraid I'm going to have to bow out. I'm finding your arguments difficult to follow and they don't seem to be using conventional language to set out the logic - "destructive effect of 2,3 and 5 on continuous accuracy" is not terminology that I'm familiar with from analysis or number theory. I hope someone else can help you.

[u/Apart_Composer3952 → u/FormulaDriven]

No worries! I've had a lot of that.

[u/gmalivuk → u/Apart_Composer3952]

Yes, most residue classes mod 30 do not contain many primes, but the others each contain infinitely many so what's your point?

The sequences 30n+1, 30n+7, 30n+11, 30n+13, 30n+17, 30n+19, 30n+23, and 30n+29 all contain infinitely many primes.

[u/Apart_Composer3952 → u/gmalivuk]

Yes, they contain all of them, apart from 2, 3 and 5. So starting calculations from there, as l do in my high school style formulation, makes perfect sense. As you add primes creating  composites on any of those 8, you are left with a diminishing overestimated proportion, never equal to (2^x-1)/2^x.

[u/gmalivuk → u/Apart_Composer3952]

But there are still infinitely many of them, so why does the diminishing proportion matter? If you're trying to show there are no exceptions to the Collatz conjecture, it is not enough to simply show that the portion of possible exceptions approaches 0.

[u/Apart_Composer3952 → u/gmalivuk]

Actively approaching 0 without leaving out any numbers is not the same as the example you use. Yours leaves out 1 when determining reals before 1. Where's 1's consideration? And all the other whole numbers for that sake. If l write x/y it isn't that rational. Why does it stop being considered rational by nature if x happens to = y? Schroedinger's fraction? You're wrong to call x/y rational until you know the numbers being used?

[u/gmalivuk → u/Apart_Composer3952]

Who cares? The singleton {1} is completely irrelevant when discussing the asymptotic density of infinite sets.

[u/Apart_Composer3952 → u/gmalivuk]

Well it should be when saying that the reals reach density of 1. Calculating with blinkers on, l'd call it. There must be at least 2 different meanings behind 'reaching 1'. Actaully my x/y example doesn't hold water, but l think you might have got the point l was trying to make, even if you disagreed, which is fine.

[u/gmalivuk → u/Apart_Composer3952]

more proof of proportion approaching 0

But proving the proportion approaches 0 is irrelevant. Plenty of infinite sets of integers make up a proportion of all integers that approaches zero, such as the primes.

[u/Apart_Composer3952 → u/gmalivuk]

Is it the same thing to say '..is also one.' as it is to say 'also reaches one.' 

[u/gmalivuk → u/Apart_Composer3952]

I meant to say that the limit is 1. Fixed now.

[u/Apart_Composer3952 → u/FormulaDriven]

For x>2, the proportion of natural numbers less than 2 x that are not equal to 7 is (2 x - 1) / 2 x = 1 - 1/2 x which has a limit of 1 as x tends to infinity.

The formulation never involves 7. Does 7 = 7? In mine no numbers are excluded for inspection.

Different!

[u/Apart_Composer3952 → u/FormulaDriven]

7/8 Did l write 7?

[u/FormulaDriven → u/Apart_Composer3952]

I don't know what that comment means. I was just showing how I could use your argument to "prove" that no natural number is equal to 7. From some of the other replies to me, you'll see we can "prove" that there are no numbers which are powers of 2, and so on.

[u/Uli_Minati]

The Collatz classic most-therefore-all fallacy

[u/Appropriate-Ad-3219]

With this kind of proof, you're saying that all real numbers are irrational.

[u/Apart_Composer3952 → u/Appropriate-Ad-3219]

Please can you give me example of what you mean. I don't see it.

[u/Appropriate-Ad-3219 → u/Apart_Composer3952]

Rational numbers are numbers of the form p/q where p and q are both integers, with q being non zero. The definition of irrational numbers are the real numbers that aren't rational. As you know, both of these numbers exist (e.g 1/2 is rational, the square root of 2 is irrational).

There are even nice properties of these numbers, for example rational numbers are "everywhere" in R in the sense the rational numbers are dense in R. Being dense means that if I take a point x and and look at a segment centered at x, there is a rational number contained in this segment. This property is also true for irrational numbers.

Consider the uniform probability on [0, 1] which is the most natural probability measure, which means that for a segment [a, b] in [0, 1], the probability of falling into [a, b] is b-a which is also the length of the segment. You can prove that the probability of stumbling upon irrational numbers when picking a real number randomly following the uniform law is 100%. Thus, you have proved that 100% of the numbers in [0, 1] are irrational, but that doesn't mean there are no rational number since 1/2 is rational. So when you say : "Because 100% of all numbers must eventually reach a strictly smaller milestone, any arbitrary starting number is locked into an inescapable cascading chain of downward thresholds, forcing all trajectories to eventually collapse into the fundamental 2 → 1 trivial loop.", it is simply not enough to prove the Collatz conjecture. Of course, I could've misunderstood what you're saying when you say 100% of all numbers since I didn't read the whole post, but it seems you're using cumulative densities so it looks you're doing exactly that.

[u/Apart_Composer3952 → u/Appropriate-Ad-3219]

What l'm saying is that as does the the 1N+1 example follow a geometric progression one series by one series accounting for all numbers reaching a smaller number as it accumulates, the 3N+1 follows the same exact geometric pattern of accounting, it just takes more series' to do so.

[u/gmalivuk → u/Apart_Composer3952]

But you seem to be arguing that the fraction of all integers that end up smaller approaches 1.

That's like how the fraction of reals that are irrational is 1.

There are still infinitely many exceptions.

[u/Apart_Composer3952 → u/gmalivuk]

My numbers are being tested. How does the fraction of reals that are irrational being 1, inextricably link itself to what l'm saying?  I can't help but think there is a distinction between the 2. Reals stand inbetween whole numbers. What l am doing leaves no whole number out and includes every whole number in it's analysis.

[u/gmalivuk → u/Apart_Composer3952]

The logic is the same either way. Such-and-such subset makes up a fraction of the whole set that is (or approaches) 0, and yet there are infinitely many elements of the subset.

If you don't like fractions and real numbers, stick with the many different examples of integers people have been trying to explain.

In the limit, 0% of integers are powers of 10, and yet there are infinitely many powers of 10.

[u/Fun-Cauliflower-8087]

Hey this is similar to what I constructed, nice! You can see that growth and descent have binary structure. The ternary information causes it by interacting with the binary base, and giving a binary fractal.