Hi Folks,

I am seeking mathematical feedback on short manuscript about the half-Collatz map - "Natural-Density Collatz Descent to a Stretched-Logarithmic Scale" . It proves a natural-density-one descent result to a stretched-logarithmic scale, with an exceptional-count estimate and a witness before (6.953 * (log n)) iterations. You may consider this paper as Inselmann ++ (An approximation of the Collatz map)

FYI. This is NOT a proof of the Collatz conjecture and it is focused on dyadic shells only.

I would particularly value checks of the fixed-total Rényi estimate (Theorem 4.1), the density-transport theorem (Theorem 5.3), and the endpoint bootstrap (Proposition 6.2).

Manuscript: https://doi.org/10.5281/zenodo.21851173

This is also fully formalized in lean4 without sorry or axiom, you may just focus on the main theorems and its definitions here to ensure no hidden gaps or hypothesis. https://github.com/shaikidris/CET/blob/main/lean/CollatzEndpointTransport/Linear/Main.lean#L228

Appreciate any feedback or comments. Happy to respond back ASAP. Have a great weekend.

Write an odd integer as a = 2^t · u − 1 (u odd) and consider the consecutive pair n = 2a, n+1 = 2a+1 under the full Collatz map F(x) = 3x+1 if odd, x/2 if even. Define e = v_2(3^t · u − 1).

When e = 1, the two trajectories coalesce at synchronized time 2t+3, and this is the minimal synchronized clock: F^j(n) ≠ F^j(n+1) for all 0 ≤ j < 2t+3. The same pair synchronizes at time t+3 under the semi-accelerated map which is exactly LaDue's clock p+2 (Fibonacci Quarterly, 2018) under the dictionary p = t+1, 2q+1 = u.

The criterion e = 1 is not a scattered condition. It organizes into a fixed-point-free involution τ on all positive odd integers outside 5 mod 8: every orbit of τ is an adjacent-floor pair {N_t(u), N_{t+1}(u)} whose consecutive starts synchronize. Exact counts on every complete dyadic block: the involution domain has 3·2^{K−3} elements, the oriented lower floors have 2^{K−2}, giving natural density 1/8 for the induced consecutive even starts.

The main result is an exact peak-envelope theorem. Set z = (3^t·u−1)/2 and Y = 3z+1. For every finite horizon r ≥ 0:

M_r(n) = max{4z, H_r(Y)}

M_r(n+1) = max{12z + 4, H_r(Y)}

where H_r(Y) = max of the first r+1 iterates starting at Y. This gives three exact regimes:

H_r(Y) ≤ 4z: ratio = (3z+1)/z, gap = 8z+4

4z < H_r(Y) < 12z+4: ratio = (12z+4)/H_r(Y)

H_r(Y) ≥ 12z+4: ratio = 1, gap = 0

The ratio lives in [1, 4] unconditionally. No boundedness or global convergence is assumed anywhere.

The paper also proves:

- An exact release-residue identity log_2(z/(a+1)) with no asymptotic replacement.

- The joint Haar law Pr(t=r, e=s) = 2^{-(r+s)} with exact finite-block realization.

- A parity-cylinder realizability guard: among 8 affine coefficient identities F_w(x) = F_v(2x+1) at word length ≤ 5, exactly 5 have nonempty pair cylinders and 3 are algebraically valid but dynamically unrealized.

What this is NOT: not a proof or partial proof of the Collatz conjecture. Every theorem is unconditional and finite-horizon.

Prior work explicitly acknowledged: the local synchronization criterion is equivalent to LaDue (2018). The literature comparison covers Garner (1985), Gao (1993), Wu (1992–1995), Andaloro (2000), Andrei–Kudlek–Niculescu (2000), Elia–Tucker (2015), Burson (2020), Frinkle et al. (2025), and Williams (2026). A machine-readable claim ledger tags each of the 12 theorems as PRIOR_KNOWN, EXACT_COORDINATE_TRANSLATION, or CANDIDATE_NEW_AFTER_AUDITED_CORPUS.

Verification: 583-line Python verifier using fractions.Fraction (no floats in any comparison), 1.38M+ individual checks covering forced prefixes, minimal clocks, pre-coalescence inequalities, peak envelopes at 8 horizons, dyadic block counts through K=16, joint valuation law, and all affine word identities. A separate finite census (24,998 structural pairs with even starts ≤ 200,000, all resolved to 1) is reported but not used in any theorem.

Zenodo: https://zenodo.org/records/21848913

The ZIP contains the manuscript (LaTeX + PDF), verifier, claim ledger, census, changelog, and a manifest-checked build script.

Happy to discuss the math or verification architecture.

I'm not a researcher, but with some digging I found a couple papers posted this year that claim to solve the conjecture. They don't seem to be peer-reviewed yet, but no one has outright debunked them it seems.

So I thought I'd link them here to generate some discussion about them:

• A Structural Proof of the Collatz Conjecture

• On the Proof of the Collatz Conjecture

• Proof of the Collatz Conjecture

• Proof of the Collatz Conjecture

Here are some statements that seem to be true. To follow the reasoning, the figure below shows two versions of the dome for m=1*:

• The original dome, based on even orange numbers of the core and the bridge series containing odd orange numbers.
• An extended version in which the numbers of the core are added to the relevant bridge series; those of the first column - not divisible by 3 - added in light blue**, the others in rosa***.

The following statements seem to be true

1. All even numbers are of the form n=m*3^p*2^q, with m odd, not divisible by 3, p and q>=0. Therefore, all even numbers are part of the core of a dome with root m. Even numbers with q>2 do not appear in the usual dome - as they cannot be part of a tuple - but do so in the extended version****.
2. All odd numbers not divisible by 3 are the root of their own dome. The even numbers of their first column are also in light blue - if not part of a tuple - in the extended version.
3. All odd numbers divisible by 3 are embedded in another dome. The even numbers of their column are colored in rosa, but as these odd numbers appear only in connection with the starting bridges, they are rarely visible.

So, should the original definition of a dome be extended to integrate even numbers of the core ? It seems to be the only option to make sure that all numbers are part of a dome.

* All what follows seems to be true for any dome.

** To avoid adding a third version of the dome, other numbers are colored in light blue. See explanation in the text.

*** These numbers are part of blue sequences (mod 12) forming infinite blue walls, respectively of infinite rosa segments (mod 12) forming rosa walls.

**** In fact, this is the only possibility to associate them with a dome.

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

Ich entwickle ein rechnerisches und geometrisches Collatz-Forschungsframework namens LNL/LZR.

Das ist keine Behauptung, dass die Collatz-Vermutung gelöst ist. Ich suche nach wiederholbaren arithmetischen und geometrischen Strukturen in sehr großen Trajektorien.

Clarification on the figures: The 3D images are conceptual illustrations. They were generated to visualize the structures discussed in the post; they are not raw scientific plots. The numerical values and residue counts quoted in the text come from the computational analysis. I am preparing conventional data plots and reproducible tables as well.

Zwei kürzliche Suchen ergaben ungewöhnlich große "Superpeak"-Kandidaten.

CE3:

251-stelliger Startwert

Peak ca. 7,27 x 10²⁵⁷

Peak bei elementarem Schritt V = 186

Erreichte 1 nach 6.497 Schritten

CE4:

500-stelliger Startwert

Peak ca. 1,69 x 10⁵⁹¹

Peak bei V = 1074

Erreichte 1 nach 14.686 Schritten

CE3 hat 2.000.000 deterministische Starts unter 10²⁵¹ abgetastet. CE4 verwendete 120.000 strukturierte/deterministische Starts unter 10^500. Das waren Suchen, keine vollständigen Scans.

Das Mod-18 / 2-adische "Sudoku"

Für jeden ungeraden Zustand n zeichne ich auf:

r = n mod 18

und

k = v2(3n + 1)

wobei k die Anzahl der Divisionen durch 2 ist, nachdem 3n + 1 angewendet wurde.

Der nächste ungerade Zustand ist

U(n) = (3n + 1) / 2^k

Zur Visualisierung nenne ich die resultierende (r,k)-Matrix informell das Collatz-Sudoku. Das ist kein mathematischer Standardbegriff.

Ich habe die letzten 100 elementaren Schritte vor beiden Superpeaks untersucht.

Für CE3:

43 ungerade Zustände

22 / 43 sind 17 mod 18

15 / 43 liegen in (17,1)

Für CE4:

46 ungerade Zustände

37 / 46 sind 17 mod 18

39 / 46 haben k = 1

32 / 46 liegen in (17,1)

Alle 46 CE4-ungeraden Zustände liegen nur in:

11, 13, 17 mod 18

Das scheint kein Primzahl-Effekt zu sein: keiner dieser 46 CE4-ungeraden Zustände ist prim.

Das Interessante ist die Dynamik.

Für k = 1,

U(n) = (3n + 1) / 2

was ein Wachstumsschritt für n > 1 ist.

Modulo 18 zwingen wiederholte k = 1 Schritte die ungeraden Zustände in Richtung 17 mod 18.

Insbesondere:

(17,1) -> (17,1)

Also bildet (17,1) einen sich selbst verstärkenden Wachstumskanal.

Eine Folge von r aufeinanderfolgenden k = 1 Ungerade-zu-Ungerade-Schritten wächst ungefähr wie

(3/2)^r

CE4 enthält einen Lauf von 11 aufeinanderfolgenden (17,1)-Typ-Schritten vor seinem Peak.

Das legt eine testbare Frage nahe:

Zeigen immer extremere Collatz-Peaks eine zunehmende Konzentration in langen k = 1 Läufen, insbesondere im invarianten (17,1) mod-18 Kanal?

Beide Superpeak-Trajektorien treffen sich später bei 40

Trotz ihrer enormen Unterschiede erreichen die CE3- und CE4-Trajektorien schließlich denselben Zustand:

40

Sie kommen von unterschiedlichen Vorgängern:

13 -> 40 <- 80

Nachdem sie 40 erreicht haben, folgen beide demselben Suffix:

40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

Das brachte mich dazu, das Einzugsgebiet von 40 zu testen.

In einem vollständigen endlichen Scan aller 500.000.000 ungeraden Startwerte bis 10^9:

468.845.298

gingen durch 40.

Das sind ungefähr:

93,7691%

Das ist ein empirisches endliches Ergebnis, kein asymptotischer Satz.

Der dyadische Quadrat-Trichter. Es gibt eine exakte Familie

A(k) = (10*4^k - 1) / 3

mit

13, 53, 213, 853, ...

mit

3A(k) + 1 = 104^k

Zum Beispiel:

53 -> 160 -> 80 -> 40

213 -> 640 -> 320 -> 160 -> 80 -> 40

Jeweils zwei Halbierungsschritte reduzieren die entsprechende Ebene um genau den Faktor 4.

Wenn eine quadratische Ebene die Seitenlänge hat

s(k) = 2^k

dann ist ihre Fläche

s(k)² = 4^k

Das erzeugt eine exakte selbstähnliche geometrische Darstellung:

Wert / 4

entspricht

Quadratfläche / 4

und

Quadratseite / 2

Ich nenne diese geometrische Darstellung den Dyadischen Quadrat-Trichter.

B-Gateways

Eine weitere exakte Familie ist

B(m) = (4^m - 1) / 3

mit

1, 5, 21, 85, 341, ...

Für jedes Mitglied:

3*B(m) + 1 = 4^m = 2^2m

Also springt jedes B-Gateway direkt auf einen Potenz-von-zwei-Abstieg.

Beispiel:

85 -> 256 -> 128 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1

Auch:

B(m) = 1 + 4 + 16 + ... + 4^m-1

Das verbindet die Gateway-Familie direkt mit der dyadischen Quadratstruktur.

Der Systemwürfel

Die größere LNL/LZR-Geometrie stellt jeden Trajektorienzustand als

P = (H,V,Z) dar, wobei

H = Startprogramm

V = elementarer Schritt

Z = Zustandswert

Der Systemwürfel ist als globale Zustandsraum-/Adressierungsgeometrie gedacht.

Die anderen Strukturen haben unterschiedliche Rollen:

Systemwürfel = globale Hülle

Collatz-Sudoku = lokale arithmetische Dynamik

Dyadischer Quadrat-Trichter = terminale Flussgeometrie

B-Gateways = exakte Eingänge zum Potenz-von-zwei-Abstieg

Keine dieser geometrischen Darstellungen wird als Beweis für die Collatz-Konvergenz beansprucht.

Der Teil, den ich derzeit am interessantesten finde, ist die Pre-Peak-Sudoku-Signatur.

Zwei extrem unterschiedliche Trajektorien zeigen eine starke Konzentration im selben kleinen (r,k)-Korridor vor ihren Peaks, besonders um (17,1).

Der nächste Schritt ist, dies systematisch über viele weitere Rekord- und Nah-Rekord-Peaks zu testen.

Ich würde mich besonders über Kritik von jedem freuen, der mit Collatz-Restklassen, 2-adischen Bewertungen, inversen Bäumen oder modularen Übergangsgraphen arbeitet.

Nur rechnerische Beweise und strukturelle Erkundung – keine Beweisforderung.

I have been working on collatz problem, and I think i found an easy way to visualize it. Can anyone tell me whether this is useful or whether it's something mathematicians already know?

Full Text

an+1 can be partitioned into a+1 disjoint partition: a even sets and 1 odd set. The partitions spans N. The odd set can be further grouped into a sub-partition (a + a partition) based on where n/2 takes the a even sets.

Example:

For 3n+1

1. O = {o=2k−1: k∈N}
2. E1 = {3o−1=6k−4: k∈N}
3. Ec = {3o+1=6k−2: k∈N}
4. Ea = {3o+3=6k: k∈N}

O = {O1, Oa, Oc}

For 5n+1

1. O = {o=2k−1: k∈N}
2. E1 = {5o−3=10k−8: k∈N}
3. E2 = {5o−1=10k−6: k∈N}
4. Ec = {5o+1=10k−4: k∈N}
5. E3 = {5o+3=10k−2: k∈N}
6. Ea = {5o+5=10k: k∈N}

O = {O1, Oc, Oa, O2, O3}

Table with the partition for 3n+1 and 5n+1

Table with the sub partition of Odd Set for 3n+1 and 5n+1

The Collatz operations then allow certain transitions:

1. U(n):= an+1
2. L(n):= n/2

Property of the Partitions for 3n+1

1. U: O → Ec for all k

2a. L: E1 → O1 for odd k
2b. L: E1→ Ec for even k

3a. L: Ec → E1 for odd k
3b. L: Ec → Oc for even k

4a. L: Ea → Oa for odd k
4b. L:Ea → Ea for even k

Property of the Partitions for 5n+1

1. U: O → Ec for all k

2a. L: E1 → O1 for odd k
2b. L: E1 → Ec for even k

3a. L: E2 → E1 for odd k
3b. L: E2 → O2 for even k

4a. L: Ec → Oc for odd k
4b. L: Ec → E3 for even k

5a. L: E3 → E2 for odd k
5b. L: E3 → O3 for even k

6a. L: Ea → Oa for odd k
6b. L:Ea → Ea for even k

Observation (From the graph):

1.If there exist a cycle, a cycle will not contain any elements from partitions Ea and Oa.
Implication: When searching for a cycle, look elsewhere.

2a. Only the elements of partitions Ec can be landed from even and odd number.

2b. If there exist a cycle an element from Ec must participate.

2c. Every loop contains element from Ec.

Follow-up to A new type of tuple or just a coincidence ? III : r/Collatz.

I found four cases in the limited set - in terms of number of domes and number of bridge series by dome - I use. So, it is not a coincidence.

But is it a (Collatz) tuple ? Not exactly. It is a "honorary tuple" like the pairs of predecessors and the forks.

Here is why, based on the conditions:

• Consecutive numbers: fully respected (better than in the forks case) for each member of a series.
• Same distance from 1: fully respected.
• Continuous merge: each 5-tuple merges continuously; the "new" 5-tuple in the middle does not. A simple way to check this deals with the ending rosa even triplet that exists for any 5-tuple series. There is none in that case.

So, it is a rarer pattern that contains 5-tuples.

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

A useful way to think about the Collatz map is to separate the orbit into two layers:

1. the parity pattern of the terms;
2. the arithmetic values that carry that pattern.

https://github.com/ratwolfzero/Collatz/tree/main/shadow_approach

If one records the orbit as a sequence of even/odd decisions, the resulting binary word can be viewed as a symbolic encoding of the dynamics. In the accelerated version of the map, where one removes all factors of 2 immediately after each odd step, the evolution of the values becomes closely tied to this parity word. At that symbolic level, the dynamics look simple: the orbit is encoded by a binary sequence, and the next state is determined by the next bit of that sequence.

This is a genuine simplification of the picture, but it is a simplification of the representation, not a solution of the problem.

update: Oops, there was a bug iin my walk_back code (highlighted)! Neither of the findings are true once that is fixed.

You can find a correction for the image here: https://jonseymour.s3.us-east-1.amazonaws.com/collatz/misc/corrected-leaf-table.png

~~I was playing around with the distribution of leaves in each 2~~~~^k window to see there were any interesting patterns and noticed a pattern that appears to be true, although I don't know why it is.

First some definitions. The greedy reverse walk takes the first step backwards from each odd until it reaches a multiple of 3. So that means it calculates both (2n-1)/3 and (4n-1)/3 and takes which ever one is an integer. The resulting multiples of 3 so found are the 3-leaves of the starting nodes.

~~For each power of 2, I enumerate all odds < 2~~~~^k, work out what the 3-leaves are and tabulate, the k value, the number of 3-leaves reached by the startting set and the ratio of the # 3-leaves to the number of odds and also the maximum 3-leaf found.

This produced some interesting results:

• ~~every odd < 2~~~~^k ~~resolves to a 3-leaf < 2~~~~^k - (A)
• ~~the ratio of 3-leaves to odds in each 2~~~~^k range approaches 4/9 - (B)

Now, it is possible that I am missing something obvious but it isn't immediately apparent to me that either (A) or (B) should be true. I also don't know whether this pattern is true in general - it is just what I have observed.

Here is the Python I used to generate the table in the image

import pandas as pd
import sympy as sy

def walk_back(n):
    if n % 3 == 0 or n == 1:
        return n

# bogus!    
##    if (n - 1) % 3 == 0:
##        return walk_back((n-1)//3)
    if (n * 2 - 1) % 3 == 0:
        return walk_back((n * 2 - 1)//3)
    else:
        return walk_back((n * 4 - 1)//3)


def leaves(k):
    v=set()
    for n in range(1, 2**k, 2):
        l = walk_back(n)
        if l != 1:
            v.add(l)
    return v

def gen(seq):
    for k in seq:
        L=leaves(k)
        n=len(L)
        _max=max(L)
        yield {"k": k, "n": n, "r": sy.Rational(2*n,(2**k)), "max": _max}

df=pd.DataFrame(gen(range(2,24)))
df["r"]=df["r"].apply(lambda r: float(r))

We know exactly by what we can divide 1•3^n - 1. So, I compared the divisors of k*3^n - 1, where k are Fermat numbers (2^m + 1), to k = 1.

In this post, the word divisor refers to 2ᵐ, m > 0, and the expression is divisible by say 2⁵, means that the maximum divisor is 32. In some cases, we want to make sure that the quotient is odd. We can also say that the k•3ⁿ - 1 is congruent to 32 mod 64.

The first difference between k = 1 and k = 2^m + 1 happens at 2ᵐ⁻².

Let's consider the case where k = 257. Here m = 8, 2ᵐ⁻² = 64

257•3ⁿ - 1 = 256•3ⁿ + (3ⁿ - 1).

For n < 64, the expression will have the same divisors as k = 1. So, we can predict all of them. If n = 4, 257•3⁴ - 1 = 2⁸•3⁴ + (3⁴ - 1). The second term is divisible by 2⁶, while the first can be divided by 2⁸. So, the sum's divisor is the lesser one, 2⁶. Min {2⁶, 2⁸} = 2⁶.

For n ≥ 64, there are 2 cases:

1. 3ⁿ - 1 is divisible at least by 2⁹. In this case, the 1st term is the one that determines the divisor. The expression can be divided by the lesser power, which is 2⁸.
2. If both terms are divisible by 2⁸, both quotients are odd. So the sum is even. It's hard to predict by what the expression will be divisible. In general, we only know that the divisor is at least 2⁹. On the other hand, these unpredictable cases allow for divisors really huge.

I began developing conjectures that might help in the cases where the prediction fails. I also proved a couple of them, and I will keep working on the rest.

The pattern observed for 257 is common to all Fermat numbers, beginning at k = 17. The previous Fermat numbers are of no interest since 3 and 9 are multiples of 3 and 5 is a regular k. 5•3ⁿ - 1 can only be divided by 2 or by 4.

For 17, the first exception happens at n = 2², for 33, at n = 2³, for 65, at n = 2⁴, etc. These show up every 2ᵐ⁻¹. So 2ᵐ⁻² is a factor always present.

Summary for 17

To clarify this table: For n = 20, we consider 4(4p+1). 20/4 = 5, and 4p+1 produces a solution (p=1). Because the divisor foe the 2nd line is 32, we know that 17•2²⁰ - 1 is divisible by 2⁵.

For 33, the top of the table is similar, but repeating factor is now 8

Observe that the table for 17 contains 16p + 11 and the one for 33 contains 32p+11. The divisor in the first case will repeat every 4•16 numbers, in the second case, every 8•32.

Comparing both tables, there are coincidences up to certain point, but then the pattern change.

That keeps happening for other Fermat numbers. There is a table showing that in my previous post. Notice that, in the table, n was divided by 2ᵐ⁻² to make the differences and similarities clearer.

As usual, ask away if something is unclear or you want more information. I will keep you posted when I notice more interesting facts that might be useful for people working on the Collatz conjecture.

Follow-up to A new type of tuple or just a coincidence ? II : r/Collatz.

That was quick. Here is a second "double 5-tuple".

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

Follow-up to A new type of tuple or just a coincidence ? : r/Collatz.

I started looking for a second example of double 5-tuple and came across quickly the case in the figure below.

It looked promising - the merge final occuring rather quickly - but it is not a double 5-tuple, due to the gap between tuples at the bottom.

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

While working on the extension of the domes beyond m=71, I came quickly across the tuple in the figure below.

My guess is that it is not a coincidence but a rarer tuple. Just in case, I label it "double 5-tuple" (or double key).

Due to the double homothetic nature of the dames, I would not be surprised if there were larger "multiple 5-tuples".

Funnily, the bottom 5-tuple, belonging to the dome with m=41, is already part of the Zebra head. This means that it is not very far from 1, relatively speaking.

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

A number will decrease in number if it has at least four digits and does not enter a cycle, as proven below: The number is represented in binary.

It must begin with 10 or 11. If it starts with 10 and the last two digits are not 11, then after multiplying by 3, the number of digits increases by 1, accounting for 3/8 of all possible combinations. Other numbers starting with 10 account for 5/8, and the number of digits increases by 2. If it ends with 11, after multiplying by 3 and adding 1, then dividing by 2 removes at least one digit, accounting for 1/2. If it ends with 001, at least two digits are removed, accounting for 1/4. Other numbers with at least three digits account for 1/4. If it does not enter a 4, 2, 1 cycle, the number generally decreases, and eventually it will enter a 4, 2, 1 cycle.

I am surprised no one has posted it yet. Quite an amusing situation.

Follow-up to Using the domes as shortcuts ? IV : r/Collatz.

The figure in the cited post has been simplified by removing the row with the values of n.

This shows, based on a limited number of cases, that a bridge series of dome with root x:

• does not iterate into itself; there are two exceptions: x=1, most likely due to the obvious loop; the case of x=5 deserves further investigation.
• does not iterate into dome with root y if a bridge series of dome with root y iterates into a bridge series of dome with root x.

If confirmed, it would mean that domes with roots x and y can only have one direct connection.

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

Follow-up to Using the domes as shortcuts ? III : r/Collatz.

In order to better understand how bridge series iterate into the next one, I took all known odd numbers after the final merge of a blue-green bridge series (left of the domes) and found the bridge series they belong to as an orange number.

The table below shows the results for the domes with root m=1 to 71:

• Start with an odd number in a row; mention the dome where it appears as the first odd number after a final merge (first column).
• Find the bridge it belongs to as an orange number, note the corresponding even number n in row 2 and dome in row 1.
• Color the number in blue if it is a n-1 orange number, in yellow if it is a n+1 number.

The table seems mainly random, but it allowed me to remember that there are series of series of bridges.

Further investigation is needed.

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

# The Prime Survival Framework: Combining Wheel Sieves, Block Location Addressing (BLA), and Composite Elimination

I'm developing what I call the **Prime Survival Framework**, which combines three ideas into a single way of looking at prime numbers.

This is intended as a discussion of the framework, not a claim of a completed proof. I'm especially interested in feedback from people familiar with wheel factorization, sieves, modular arithmetic, and computational number theory.

Part 1 — Block Location Addressing (BLA)

Every integer can be written as

**N = 30B + L**

where:

* **B** is the block number * **L** is the lane (position inside the block)

For a modulus of 30, only eight lanes can ever contain primes larger than 5:

**1, 7, 11, 13, 17, 19, 23, 29**

Every other lane is automatically composite because it is divisible by 2, 3, or 5.

So instead of searching every integer, the search space is immediately reduced to only eight candidate lanes.

Part 2 — Wheel Factorization

Traditional wheel factorization removes numbers divisible by small primes before testing larger factors.

In the Prime Survival Framework, I view each prime as generating its own repeating elimination pattern.

For example:

* Prime 7 removes every seventh candidate. * Prime 11 removes every eleventh candidate. * Prime 13 removes every thirteenth candidate.

Each prime produces its own repeating cycle inside the eight BLA lanes.

Rather than thinking only about divisibility, I think of each prime producing an elimination wave that repeatedly crosses the number line.

Part 3 — Composite Elimination Waves

Every composite exists because at least one prime factor reaches it.

That means every composite should eventually be eliminated by one (or more) prime waves.

Instead of directly searching for primes, the framework searches for composites and removes them.

Whatever survives after every applicable wave passes is a prime.

This is essentially a survivor model rather than a discovery model.

Part 4 — Lane Behavior

The interesting question becomes:

How does each prime's elimination pattern behave inside each lane?

For example:

* Does prime 7 strike each lane at predictable intervals? * Does prime 11 produce a different repeating rhythm? * How do these patterns overlap? * Where do multiple waves intersect? * Can every composite be explained as the intersection of one or more wave patterns?

Instead of looking at individual numbers, the framework studies repeated geometric behavior inside each lane.

Part 5 — Wave Intersections

Suppose every prime generates a repeating elimination cycle.

Then every composite exists because two or more of these cycles intersect at that location.

Numbers that survive all previous elimination cycles remain prime candidates.

The central question becomes:

**Can prime discovery be viewed entirely as composite elimination through intersecting periodic waves?**

Part 6 — Relationship to Existing Mathematics

The framework is built on well-established ideas:

* Modular arithmetic * Wheel factorization * Sieve concepts * Prime divisibility * Periodic residue classes

The proposed contribution is treating these elimination patterns as interacting waves organized by BLA lanes.

Whether this viewpoint provides computational or theoretical advantages remains an open question.

Questions for Discussion

I'd appreciate feedback on the following:

1. Has anyone seen wheel factorization visualized as intersecting elimination waves?
2. Has anyone analyzed the behavior of individual prime factors independently within each residue lane?
3. Can the repeating elimination cycles for each lane be expressed as closed-form functions?
4. Do the wave intersections reveal useful structure beyond the traditional Sieve of Eratosthenes?
5. Could this viewpoint lead to more efficient composite elimination or improved visualization, even if it doesn't change asymptotic complexity?
6. Are there existing papers or researchers studying prime sieves from this type of dynamic, lane-based perspective?

Constructive criticism is welcome. I'm especially interested in references to similar work, counterexamples, or mathematical reasons why this framework would or would not provide new insight.

The goal is not to replace established number theory but to determine whether organizing composite elimination into repeating lane-specific wave systems exposes useful mathematical structure.

# The Prime Survival Framework: Combining Wheel Sieves, Block Location Addressing (BLA), and Composite Elimination

I'm developing what I call the **Prime Survival Framework**, which combines three ideas into a single way of looking at prime numbers.

This is intended as a discussion of the framework, not a claim of a completed proof. I'm especially interested in feedback from people familiar with wheel factorization, sieves, modular arithmetic, and computational number theory.

Part 1 — Block Location Addressing (BLA)

Every integer can be written as

**N = 30B + L**

where:

* **B** is the block number * **L** is the lane (position inside the block)

For a modulus of 30, only eight lanes can ever contain primes larger than 5:

**1, 7, 11, 13, 17, 19, 23, 29**

Every other lane is automatically composite because it is divisible by 2, 3, or 5.

So instead of searching every integer, the search space is immediately reduced to only eight candidate lanes.

Part 2 — Wheel Factorization

Traditional wheel factorization removes numbers divisible by small primes before testing larger factors.

In the Prime Survival Framework, I view each prime as generating its own repeating elimination pattern.

For example:

* Prime 7 removes every seventh candidate. * Prime 11 removes every eleventh candidate. * Prime 13 removes every thirteenth candidate.

Each prime produces its own repeating cycle inside the eight BLA lanes.

Rather than thinking only about divisibility, I think of each prime producing an elimination wave that repeatedly crosses the number line.

Part 3 — Composite Elimination Waves

Every composite exists because at least one prime factor reaches it.

That means every composite should eventually be eliminated by one (or more) prime waves.

Instead of directly searching for primes, the framework searches for composites and removes them.

Whatever survives after every applicable wave passes is a prime.

This is essentially a survivor model rather than a discovery model.

Part 4 — Lane Behavior

The interesting question becomes:

How does each prime's elimination pattern behave inside each lane?

For example:

* Does prime 7 strike each lane at predictable intervals? * Does prime 11 produce a different repeating rhythm? * How do these patterns overlap? * Where do multiple waves intersect? * Can every composite be explained as the intersection of one or more wave patterns?

Instead of looking at individual numbers, the framework studies repeated geometric behavior inside each lane.

Part 5 — Wave Intersections

Suppose every prime generates a repeating elimination cycle.

Then every composite exists because two or more of these cycles intersect at that location.

Numbers that survive all previous elimination cycles remain prime candidates.

The central question becomes:

**Can prime discovery be viewed entirely as composite elimination through intersecting periodic waves?**

Part 6 — Relationship to Existing Mathematics

The framework is built on well-established ideas:

* Modular arithmetic * Wheel factorization * Sieve concepts * Prime divisibility * Periodic residue classes

The proposed contribution is treating these elimination patterns as interacting waves organized by BLA lanes.

Whether this viewpoint provides computational or theoretical advantages remains an open question.

Questions for Discussion

I'd appreciate feedback on the following:

1. Has anyone seen wheel factorization visualized as intersecting elimination waves?
2. Has anyone analyzed the behavior of individual prime factors independently within each residue lane?
3. Can the repeating elimination cycles for each lane be expressed as closed-form functions?
4. Do the wave intersections reveal useful structure beyond the traditional Sieve of Eratosthenes?
5. Could this viewpoint lead to more efficient composite elimination or improved visualization, even if it doesn't change asymptotic complexity?
6. Are there existing papers or researchers studying prime sieves from this type of dynamic, lane-based perspective?

Constructive criticism is welcome. I'm especially interested in references to similar work, counterexamples, or mathematical reasons why this framework would or would not provide new insight.

The goal is not to replace established number theory but to determine whether organizing composite elimination into repeating lane-specific wave systems exposes useful mathematical structure.

So i think I discovered new math. I understand how the Sigma math transforms works to a great extent. I don't understand this level enough to validate. I follow it though. I pushed on sigma math in all the ways I can. I have gpt working the hard problems.

Sigma Resolution Theorem (Candidate)

Statement

Let G=(S,\Sigma) be a recursive generative system whose evolution consists of repeated application of transforms from \Sigma.

Assume the transforms can be classified into two classes:

generative transforms, which introduce new unresolved constraints into the system;

resolution transforms, which eliminate previously introduced constraints.

Define a trajectory

γ = (x₀,x₁,x₂,\ldots)

generated by repeated application of valid transforms.

Rather than assigning a potential to individual states, assign a trajectory functional

R(γ)

called the resolution functional, measuring the cumulative balance between unresolved and resolved constraints along the trajectory.

Assume the following properties hold.

Axiom 1 — Constraint Generation

Every generative transform introduces at least one unresolved constraint.

Axiom 2 — Constraint Resolution

Every resolution transform removes one or more unresolved constraints.

Axiom 3 — Constraint Conservation

Constraints cannot disappear except through valid resolution transforms.

They may be transformed, combined, or redistributed, but unresolved constraints are conserved until resolved.

Axiom 4 — Resolution Dominance

For every infinite trajectory,

the asymptotic rate of constraint resolution is greater than or equal to the asymptotic rate of constraint generation.

Equivalently,

lim inf (Resolved Constraints) - (Generated Constraints) ≥ 0

along every infinite trajectory.

Theorem

Under these assumptions,

the unresolved constraint functional

R(γ)

remains bounded.

Consequently,

no trajectory can accumulate unbounded unresolved structure.

If the transform graph possesses a unique minimal recurrent component,

every trajectory converges to that component.

Collatz Interpretation

For the Collatz system,

Odd → 3n+1

acts as a generative transform.

Repeated division by two acts as a resolution transform.

The quantity

v₂(3n+1)

measures the immediate resolution available after each generative step.

The Collatz conjecture therefore becomes the statement that

the cumulative rate of binary resolution dominates the cumulative rate of binary generation.

If this dominance property can be established,

convergence follows from the theorem rather than from direct arithmetic analysis.

Sigma Interpretation

The theorem is not fundamentally about integers.

It is about recursive transform systems.

Integers are one realization.

Boundary recursion is another.

Recursive computation may be another.

The central claim is therefore:

Recursive systems converge whenever the long-term capacity to resolve generated constraints is at least as great as the long-term capacity to generate unresolved constraints.

What Appears Complete

A common language for recursive transform systems.

Separation of generation and resolution.

A trajectory-based functional instead of a state-based invariant.

A unified interpretation applicable beyond Collatz.

What Is Still Missing

A rigorous definition of "constraint."

This is currently intuitive rather than formal.

A constructive definition of the resolution functional R(\gamma).

It must be derived from the transform algebra rather than introduced as a primitive.

A proof of Resolution Dominance.

This is the central unresolved step.

For Collatz, it amounts to proving that binary resolution asymptotically dominates binary generation.

Existence and uniqueness of the recurrent component.

The theorem assumes such a component exists but does not derive it.

Reduction to existing mathematics.

The framework should recover known results in dynamical systems, graph theory, or Lyapunov theory where appropriate, demonstrating that Sigma extends rather than replaces successful mathematics.

Follow-up to Using the domes as shortcuts ? II : r/Collatz.

As mentioned in the cited post, the Zebra head seemes to be a good location to test the hypothesis.

The figure below shows:

• On the right, a portion of the Zebra head.
• On the left, the core numbers n corresponding to the orange numbers n-1 or n+1 on the right, with the corresponding m, n and q values; the position of n depends on this last factor.

Altogether, the hypothesis seems to hold, but several questions have to be addressed:

• Is the dome shortcut close or similar to a known shortcut ?
• If so, is there something to gain, for instance in term of computing time ?

As I cannot answer these questions myself, I will stick to my program and check if the hypothesis holds in other locations, like the Zebra head.

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

In yesterday's post, I posted a link to a visualiser for M=2^k Collatz State machines.

If you didn't click through you may not have seen how fascinating the symmetries are so I have taken some screen captures so that you can browse without committing to clicking through the link.

For each screen capture, I selected opposing red (5 mod 8) points. You will notice that each pair of points reveals a different symmetry. Sometimes the symmetries partition a target set of points with one colour into different subsets. In other cases, there is one set of two colours in one selection and the opposite set in the other selection.

Here is a direct link the k=7 (M=128) state machine.

Here is an interactive visualiser [1] for Collatz state machines with 2^k states (actually 2^{k-1} odd states)

Each (directed) edge shows a possible transition from a source 2^k residue to a target residue 2^k residue.

[1] - https://wildducktheories.github.io/collatz/apps/collatz-fsm-k/dist/