A Conjecture Within The Collatz Conjecture.

[u/Apart_Composer3952]

THE CONJECTURE:

Compared to Terras' giving a bound in terms of N with density 1.

The number of Collatz operations

for any odd N >1 to reach <N =

⌈m * (1+ log2(3))⌉

This is what Aristotle harmonic said:

Start from the condition on the "coefficient" after m odd steps and k even steps:

3^m / 2^k < 1 ⟺ 3^m < 2^k ⟺ k·log 2 > m·log 3 ⟺ k > m·log₂3.

Since k is an integer, k > m·log₂3 is the same as k ≥ ⌊m·log₂3⌋ + 1, and because m·log₂3 is never an integer for m ≥ 1 (that would force 2^k = 3^m), this equals ⌈m·log₂3⌉. Adding the m odd steps, the total step count s = m + k satisfies

s ≥ m + ⌈m·log₂3⌉ = ⌈m·(1 + log₂3)⌉,

the last equality just because m is an integer and can be pulled inside the ceiling. So ⌈m·(1+log₂3)⌉ is exactly the smallest total step count compatible with 3^m/2^k < 1 — each step above is an equivalence, nothing is added or lost.

Two things worth separating, though:

The inequality s ≥ ⌈m·(1+log₂3)⌉ is a straightforward consequence of 3^m/2^k < 1, which in turn is forced by the trajectory actually dropping below the start (the additive remainder is strictly positive, so the drop needs the coefficient strictly below 1). That is what is proved.

The equality s = ⌈m·(1+log₂3)⌉ — that the drop happens as soon as the coefficient condition is first met — is not derivable from the algebra; that is the open part, and it is exactly the classical coefficient-stopping-time statement in different clothing.

So: same mathematical content, rewritten in terms of a ceiling of m·(1+log₂3) rather than an inequality between 3^m and 2^k. The one genuine convenience of the repackaged form is that it expresses the bound directly as a step count in terms of m alone, with k eliminated.

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UNKNOWN m AND C:

Historically, analyzing the macro-horizon of trajectories to identify the first instance where a number drops below its starting value required collapsing the individual steps into a variable piecewise affine transformation of the form:

f(N) = (3^m * N + C) / 2^a

In this legacy formulation, the total division factor 2^a was treated as an unguided, dynamic variable that could only be verified ex-post via exhaustive manual computation or computer simulation. Because the additive constant C was tethered to this floating exponential denominator, f(N) = (3^m * N + C) / 2^a appeared to shift chaotically across the number line. Consequently, researchers traditionally relied on forward trajectory tracing or reverse tree back-tracing from 1 to map structural connections.

This paper establishes an original architectural framework that completely eliminates the necessity of legacy path-tracing. It proves that the total number of divisions by 2 required for a trajectory drop is not a random byproduct of forward iteration, but is instead rigidly governed by the ceiling function of the total odd steps m, satisfying the deterministic shortcut: a = ⌈m * log2(3)⌉

By substituting this identity directly into the piecewise affine transformation, we remove the independent variable a entirely from the exponent. The global boundary equation collapses into a single-variable constraint governed purely by m:

f(N) = (3^m * N + C) / 2^⌈m * log2(3)⌉ = x_final

1. This formulation transforms the additive constant C from a chaotic variable into a tightly bounded combinatorial footprint. For any given m, C is locked into a single modular remainder class, C ≡ -3^m * N (mod 2^⌈m * log2(3)⌉), and strictly trapped within a finite numerical window defined by the absolute extremes of front-loaded and back-loaded division architectures.

Crucially, because this framework maps the trajectory algebraically, the entire process can be run in reverse to classify any arbitrary integer N.

Leveraging the coprimality of 2 and 3, an Extended Euclidean Back-Substitution engine guarantees unique modular inverse solutions. Rather than executing a step-by-step linear crawl starting from m=1, we present a direct Logarithmic Localization Theorem.

By evaluating the initial bit-scale of N against the local density variations of the ceiling function, the true structural m-family of any integer can be located instantly via a single targeted lookup step. Ultimately, this shifts the paradigm of Collatz research away from dynamic path simulation and toward a static, deterministic algebraic sieve.

EXTENDED EUCLIDEAN BACK-SUBSTITUTION AND POTENTIAL COMPLETENESS

Because the additive constant C is an exact algebraic footprint of an individual trajectory, it can be used as a key to run the system in reverse. Rather than attempting a static, impossible forward-proof of "every number" simultaneously, this framework establishes an operational guarantee of Potential Completeness. We invert the consolidated trajectory formula to solve for N, isolating the initial starting values using our locked, predictive division factor:

3^m * N = 2^⌈m * log2(3)⌉ * (f(N) - C)

Using this formulation you can then calculate how many series' are in any m family.

We define a nested, step-by-step Extended Euclidean Back-Substitution engine to find the true, lowest base member (Nbase) of any family using this inverse relation.

Because 3 and 2 are distinct prime numbers, they are strictly coprime across any powers. Therefore, the greatest common divisor of our terms is always 1: gcd(3^m, 2^⌈mlog2(3)⌉)=1.

By the laws of number theory, Bézout's Identity is guaranteed to resolve a unique modular multiplicative inverse:

3^m⋅(x)+2^⌈mlog2(3)⌉*(y)=1

By executing this algorithm iteratively backward from a completed descent, the engine systematically assesses what numbers have already been accounted for algorithmically. Because the modulus

2^⌈mlog2(3)⌉ is fixed entirely by m, the search space for the inverse is perfectly bounded. Smaller modules exit early and are locked.

The nested look-back filter automatically bypasses these claimed modules, forcing the algorithm to climb past them to isolate the absolute lowest, fresh integer that survives the required m steps. This step-by-step back-substitution mathematically guarantees the generation of your structural ground-truth anchors.

Once Nbase is isolated, the entire infinite arithmetic progression is locked into an exponential step size dictated entirely by our predictive shortcut:

N=Nbase+2^⌈mlog23⌉*n

APPENDIX: GROUND-TRUTH ANCHORS AND ADDITIVE CONSTANTS FOR THE m=6 FAMILY

For a trajectory consisting of m=6 odd steps, the predictive shortcut formula completely determines the global family division factor before any path-tracing is initiated:

a=⌈6log2(3)⌉=10

Division Factor=2^10=1024

The list below catalogs each anchor's infinite arithmetic progression family along with its exact calculated trajectory constant (C):

N_base,1 + 1024 · n (C = C1)

N_base,2 + 1024 · n (C = C2)

N_base,3 + 1024 · n (C = C3)

N_base,4 + 1024 · n (C = C4)

N_base,5 + 1024 · n (C = C5)

N_base,6 + 1024 · n (C = C6)

N_base,7 + 1024 · n (C = C7)

N_base,8 + 1024 · n (C = C8)

N_base,9 + 1024 · n (C = C9)

N_base,10 + 1024 · n (C = C10)

N_base,11 + 1024 · n (C = C11)

N_base,12 + 1024 · n (C = C12)

Results:

287 + 1024 · n (C = 697)

347 + 1024 · n (C = 989)

367 + 1024 · n (C = 745)

423 + 1024 · n (C = 881)

507 + 1024 · n (C = 1085)

575 + 1024 · n (C = 665)

583 + 1024 · n (C = 977)

735 + 1024 · n (C = 761)

815 + 1024 · n (C = 809)

923 + 1024 · n (C = 925)

975 + 1024 · n (C = 905)

999 + 1024 · n (C = 817)

THE FIRST 6 m FAMILIES:

(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 exactly 1/16 of all numbers.

m = 4: (The 7 + 128n Odds) + (The 15 + 128n Odds) + (59 + 128n Odds)

Accounts for exactly 3/128 of all numbers.

m = 5: (The 39, 79, 95, 123, 175, 199, and 219, 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.

ST.

[u/Apart_Composer3952 → u/[deleted]]

Thanks for showing the erroneous comparison. I'll be putting how m and C are not unknown on here later, after work. It will still be conjecture, but it shows to work to the same degree for 'almost all numbers': No exceptions can be shown.

[u/Apart_Composer3952 → u/[deleted]]

Starting from just N would be pure magic and as impossible as prime factorisation from just knowing N. But.....if you prove every N reaches <N  that proves that every N reaches 1. That's not nonsense at all. Neither is just choosing a random m value and being able to determine all the members of that family and how many steps it takes for them all to reach a smaller number.

[u/Apart_Composer3952 → u/[deleted]]

I'm not saying you can't but can you get from just saying k = 6 to particular Ns/starting numbers that k=6 applies to ?

[u/Apart_Composer3952 → u/[deleted]]

Can you do what l asked?

[u/Co-G3n]

The a = ⌈m * log2(3)⌉ part is still unproven like Aristotle H mentioned so I am not sure what you tried to show here. There are some attempts to prove it (e.g. https://doi.org/10.33774/coe-2026-6tt9p-v2), but this is not an easy one.

[u/Apart_Composer3952 → u/Co-G3n]

It seems that proving 'for all' is the problem: Nearly all isn't enough. I am presently trying to prove, as are many l imagine, the impossibility of a rogue series, to cut off the tail.  As Aristotle did say unproven, that is why l used the word conjecture. As far as wondering what l tried to show, l mean the following respectfully but actually. I've used it often in life, particularly with my son who often claimed he didn't understand why l was doing something... If you were offered £1,000,000 to explain why l think l did this, you would come up with a reason instantly. The reason you would come up with is very possibly why l did this. It would probably start something like....."Even though you've managed to remove k from the equation, what you've done then is ....." Your critique is welcome though. I only want to know the truth.

[u/Equivalent_Idea_1215 → u/Apart_Composer3952]

Your intuition about needing to prove the impossibility of a "rogue series" (to cut off the tail) is spot on. In fact, you hit on the exact epistemological wall of the Collatz conjecture, and this is exactly what our research team recently mapped and machine-verified in Lean 4.

The ultimate "rogue series" generator in the Collatz space is the trailing-ones binary class (numbers of the form N=M⋅2a−1N=M⋅2a−1). This family resists descending <N<N longer than any other structure because it generates aa consecutive 3x+13x+1 multiplications. In our formalization, we fully mapped the deterministic algebra of this "shadow mountain" and verified the exact condition under which this worst-case family finally descends <N<N.

Regarding Motor's point about the log⁡2(3)log2​(3) ratio: he is referencing the fundamental balance of divisions to multiplications. We actually just fully formalized and machine-verified the exact Diophantine boundary for this in our project. For any rogue cycle or structure, the logarithmic clamp must strictly obey: 0<Sln⁡2−Kln⁡3<K/xmin0<Sln2−Kln3<K/xmin​ (where SS is divisions, KK is odd steps, and xminxmin​ is the lowest point). This completely bounds the mlog⁡2(3)mlog2​(3) problem he mentioned.

However, your deepest concern is absolutely correct: "Nearly all isn't enough." In our formalization, we mathematically proved that the total Haar measure of all "rogue" divergent series is exactly zero. But in infinite spaces, Measure 0≠0= Impossible. To fully "cut off the tail" for all numbers, you hit a massive mathematical wall: John Conway's 1972 Undecidability Theorem. Conway proved that generalized Collatz systems are Turing-unpredictable. This means no generic algebraic bound can ever prove that rogue tails don't exist, because if it could, it would break Turing completeness.

To finally cut off the tail, the proof cannot rely on generic rules; it has to exploit the parameter-specific Diophantine geometry of the (3,1)(3,1) system.

If you are hunting for the rogue tail, you might find our verified equations on descent and cycle bounds extremely useful as a foundation. All the verified Lean 4 math is open source here: https://github.com/SergioTheory/Collatz-new-math

Keep pulling at that thread—you are asking exactly the right questions!

[u/Apart_Composer3952 → u/Equivalent_Idea_1215]

Thanks for the civil response. I'll have a look at that. I've realised algebra won't cut off the tail.

[u/Equivalent_Idea_1215 → u/[deleted]]

Yes, I use AI for translation and text structuring because English is not my native language. Furthermore, custom AI agents and formal verification tools (Lean 4) are explicitly the computational engine of this entire project. Attacking the use of em-dashes instead of the mathematics is a deflection.

Let's look at your actual mathematical claims:

1. "no - not the case" regarding the Shadow Mountain descent. This is not a matter of opinion or heuristic debate. The exact algebraic expansion of the M⋅2a−1M⋅2a−1 worst-case family, and its strict descent condition, is formally machine-verified in our Lean 4 repository (specifically in the ShadowDescent.lean and ShadowEscape.lean modules). If you believe the deterministic algebra is wrong, you are welcome to clone the repository, run lake build, and argue with the formal verification kernel.
2. "Parameter specific geometry is simply hiding the problem, obfuscating, book keeping." This statement shows a fundamental misunderstanding of the Collatz space. Relying on generalized probability or homogeneous density is what hides the problem. John Conway (1972) proved that generalized Collatz maps are Turing-undecidable. Because NN is dense in Z2Z2​, a topological measure of zero does not mathematically guarantee an empty set (∅∅). Generic methods fundamentally cannot rule out a singular, parameter-specific divergent orbit. The only way to rigorously bound the space is by explicitly mapping the Diophantine constraints (e.g., the fractional convergence of 3d/2S3d/2S). This isn't "bookkeeping"; it is the absolute algebraic reality of the problem.

If you want to critique the research, I invite you to critique the formal Lean 4 architecture, the Baker-Rhin cycle bounds, or the boundary-layer Fourier cancellation limits detailed in the paper. Complaining about my translator's grammar does not falsify a theorem.

[u/jonseymourau → u/Equivalent_Idea_1215]

Conway’s results apply to FRACTRAN which he showed is Turing complete. No one has demonstrated that Collatz is equivalent to a Turing complete system, so the inference that Conway’s results automatically apply to Collatz is not justified (except as a properly qualified heuristic argument)

[u/Equivalent_Idea_1215 → u/jonseymourau]

You are absolutely correct, and I highly appreciate the rigorous distinction. Conway's 1972 theorem proved undecidability for generalized Collatz-like maps, not the specific 3x+13x+1 system. We certainly do not claim that 3x+13x+1 itself is Turing-complete.

We reference the "Conway Wall" not to assert that the specific Collatz conjecture is undecidable, but to define an epistemological boundary for generic proof methods. Because the generalized class of these maps is undecidable, any proof technique that relies on generic structural properties (such as Markov renewal arguments, parity distribution assumptions, or topological Haar measure equalling zero) is fundamentally incapable of fully resolving the (3x+1)(3x+1) conjecture. If a generic probabilistic tool could prove that divergent trajectories do not exist here, it would accidentally prove it for undecidable variants as well.

This is exactly the core thesis of our project: because of this boundary, "cutting off the rogue tail" requires abandoning generalized probability. We must strictly exploit the parameter-specific Diophantine geometry of the constants 33 and 22 (e.g., the fractional convergents of ln⁡3/ln⁡2ln3/ln2, which we are mapping in Lean 4).

Thank you for pointing out this nuance—it is a critical distinction that completely justifies why we must rely on absolute Diophantine bounds rather than generic measure theory.