DOI: to be assigned

John Swygert

May 30, 2026

Abstract

This preliminary findings paper presents a computational and geometric framework for connecting Collatz dynamics, prime-lattice phase recurrence, and cylindrical number theory. We define this approach as substrate mathematics: the study of flattening, recurrence, and phase structure written into the foundational geometry of the integers.

The Collatz map, commonly known as the 3x+1 problem, produces trajectories that may rise sharply before descending toward the familiar 4 \rightarrow 2 \rightarrow 1 attractor. In logarithmic magnitude, these trajectories resemble a system of transient spikes governed by eventual contraction. Recent work on Collatz near-conjugacy introduces the shifted logarithmic phase coordinate

[ \theta(x)=\left{\log_6\left(x+\frac15\right)\right}, ]

which places Collatz motion on a circle coordinate coupled to vertical magnitude. In this paper, we extend that coordinate into a cylindrical phase-space interpretation S^1 \times \mathbb{R}, where \theta supplies angular position and logarithmic magnitude supplies height.

When prime numbers are placed on the same shifted logarithmic cylinder, exact scale-invariant phase recurrence appears. For primes p_1 and p_2, identical phase occurs whenever

[ p_2+\frac15 = 6^k\left(p_1+\frac15\right) ]

for some positive integer k. Equivalently,

[ 5p_2+1 = 6^k(5p_1+1). ]

The one-step case gives

[ p_2 = 6p_1+1. ]

This produces exact phase-preserved prime links such as 5 \rightarrow 31, 7 \rightarrow 43, 11 \rightarrow 67, 17 \rightarrow 103, 37 \rightarrow 223, and 47 \rightarrow 283. Multi-step chains also appear, including 2 \rightarrow 13 \rightarrow 79 \rightarrow 2851, 17 \rightarrow 103 \rightarrow 619, 47 \rightarrow 283 \rightarrow 1699, and 61 \rightarrow 367 \rightarrow 2203.

Using the first 1,000 primes, from 2 through 7919, we computed each prime’s shifted cylinder phase, Collatz maximum height, logarithmic maximum height, and stopping time. The data reveal 77 exact phase-equivalence classes containing two or more primes, along with 75 one-step prime links of the form p \rightarrow 6p+1 inside the first 1,000 primes. These exact phase classes form vertical groove alignments when plotted as \theta against \log_6 of Collatz maximum height.

We do not claim this paper proves the Collatz conjecture. Rather, we present a reproducible computational structure showing that the same shifted logarithmic coordinate used to clarify Collatz dynamics also exposes exact scale-recurring phase classes among primes. This suggests that prime distribution, Collatz maximum-height behavior, and cylindrical phase recurrence may share a common geometric substrate.

1. Introduction

The Collatz conjecture begins with a simple rule. For a positive integer x, if x is even, divide by 2. If x is odd, compute 3x+1. Repeating this process appears to bring every positive integer eventually to the cycle

[ 4 \rightarrow 2 \rightarrow 1. ]

Despite its elementary definition, the conjecture remains unsolved. Its difficulty lies in the coexistence of local unpredictability and global apparent convergence. Individual trajectories may rise dramatically before falling, giving the visual impression of a hailstone lifted by sharp gusts before gravity finally pulls it down.

This paper treats that behavior geometrically. Rather than viewing the Collatz process only as a sequence of integer operations, we examine it as motion on a cylindrical phase space. The angular coordinate is supplied by the shifted logarithmic phase

[ \theta(x)=\left{\log_6\left(x+\frac15\right)\right}, ]

while the vertical coordinate is supplied by logarithmic magnitude, approximately

[ E(x)=\log_6(x). ]

Under this view, Collatz trajectories may be understood as paths through a cylinder: phase rotates around the circle while magnitude rises or falls vertically. The familiar maximum-height arms observed in Collatz plots may then be interpreted as unwrapped signatures of helical groove families.

The central contribution of this paper is not merely that Collatz can be placed on such a cylinder. The deeper observation is that prime numbers placed on the same shifted logarithmic cylinder exhibit exact phase recurrence under powers of 6. Since primes greater than 3 already reside in the residue classes 6n-1 and 6n+1, the base 6 structure is not arbitrary. It is built into the prime lattice itself.

2. The Collatz Map as Gradient Flattening

The Collatz map is defined by

[ C(x)= \begin{cases} x/2, & x \equiv 0 \pmod{2},\ 3x+1, & x \equiv 1 \pmod{2}. \end{cases} ]

The odd branch 3x+1 produces upward motion. The even branch x/2 produces downward motion. Because every odd step creates an even number, upward motion is immediately followed by at least one division by 2. Many trajectories rise temporarily, but the long-term empirical behavior is descent toward the 4\rightarrow2\rightarrow1 attractor.

In this paper, we describe this behavior as gradient flattening. The term does not mean that every local step decreases. It means that the total dynamical structure appears to convert irregular upward spikes into eventual downward compression in logarithmic magnitude.

A simplified odd-step contraction may be seen by considering the combined transformation

[ x \mapsto \frac{3x+1}{2}. ]

This single odd-plus-even operation is not always contracting. However, Collatz trajectories commonly involve additional divisions by 2, and the long-run statistical behavior is dominated by the repeated removal of powers of 2. Thus, the process behaves like a jagged gradient descent: not smooth, not monotonic, but strongly structured.

This is the first sense in which Collatz resembles a fractal gravity well. It is a deterministic system with local spikes, recursive branching, and apparent global flattening.

3. Cylindrical Phase Space

Recent near-conjugacy work introduces the transformation

[ T(x)=\left{\log_6\left(x+\frac15\right)\right}. ]

This coordinate maps positive integers into the unit circle. The fractional part of the base-6 logarithm supplies angular position. The integer part of the logarithm supplies scale height. Together, they define a natural cylinder:

[ S^1 \times \mathbb{R}. ]

In this model,

[ \theta(x)=\left{\log_6\left(x+\frac15\right)\right} ]

is the angular coordinate, and

[ E(x)=\log_6(x) ]

is the approximate vertical coordinate.

Multiplication by 6 leaves \theta invariant because multiplying the shifted variable by 6 adds 1 to the base-6 logarithm. The fractional part does not change. This is the fundamental mechanism behind the prime-groove recurrence described below.

The Collatz map, in this transformed coordinate, behaves approximately like a rotation by

[ \alpha = \log_6 3, ]

with a bounded error term. This does not prove universal Collatz descent, but it gives a concrete phase-space model in which the apparent irregularity of Collatz motion may be studied as perturbed rotation on a cylinder.

4. Radial Arms as Helical Grooves

Maximum-height plots of Collatz trajectories often display radial arms or ray-like structures. These appear when a seed value is plotted against the maximum value reached by its trajectory. Groups of integers sharing early parity patterns can climb together before separating later.

In the cylindrical model, these radial arms may be interpreted as unwrapped helical grooves. A groove is a family of values sharing closely related phase behavior. When the cylinder is cut open and flattened, these grooves appear as radial arms or vertical alignments depending on the chosen projection.

This paper focuses on the projection

[ \theta(p) \quad \text{versus} \quad \log_6(M(p)), ]

where p is prime and M(p) is the maximum value reached by the Collatz trajectory starting at p.

If multiple primes share the same \theta, they occupy the same angular groove on the cylinder. Their Collatz maximum heights may differ, but their angular phase membership is identical.

5. Prime Numbers on the Shifted Base-6 Cylinder

Every prime greater than 3 lies in one of two residue classes:

[ p \equiv 1 \pmod{6} ]

or

[ p \equiv -1 \pmod{6}. ]

This gives the primes a natural relationship to a hexagonal or base-6 lattice. The shifted Collatz coordinate deepens that relationship by placing each prime at a precise angular coordinate:

[ \theta(p)=\left{\log_6\left(p+\frac15\right)\right}. ]

Two primes p_1 and p_2 have identical cylinder phase when

[ \left{\log_6\left(p_2+\frac15\right)\right}

\left{\log_6\left(p_1+\frac15\right)\right}. ]

This occurs exactly when the shifted values differ by an integer power of 6:

[ p_2+\frac15 = 6^k\left(p_1+\frac15\right). ]

Multiplying by 5, we obtain

[ 5p_2+1 = 6^k(5p_1+1). ]

Solving for p_2,

[ p_2 = 6^k p_1 + \frac{6^k-1}{5}. ]

For the one-step case k=1, this reduces to

[ p_2 = 6p_1+1. ]

For the two-step case k=2,

[ p_2 = 36p_1+7. ]

For the three-step case k=3,

[ p_2 = 216p_1+43. ]

Thus, the exact prime-groove relation is not limited to p \rightarrow 6p+1. That is only the first and simplest case. The full phase-preserving law is

[ p_2 = 6^k p_1 + \frac{6^k-1}{5}. ]

This is the algebraic backbone of the prime-cylinder recurrence.

6. Computational Method

We computed the first 1,000 primes, beginning with 2 and ending with 7919. For each prime p, we computed:

1. the shifted cylinder phase

[ \theta(p)=\left{\log_6\left(p+\frac15\right)\right}; ]

2. the maximum Collatz height

[ M(p)=\max{p, C(p), C^2(p), \ldots, 1}; ]

3. the logarithmic maximum height

[ \log_6(M(p)); ]

4. the total number of steps required to reach 1;
5. exact phase-equivalence membership under the relation

[ 5p_2+1 = 6^k(5p_1+1). ]

This produced two different kinds of clustering:

First, exact phase-equivalence classes, where primes share precisely the same shifted logarithmic phase because their shifted values differ by powers of 6.

Second, rounded numerical clusters, where \theta values match after rounding to a fixed decimal precision. These are useful for visualization, but they are not as mathematically strong as exact phase-equivalence classes.

The publishable result should therefore prioritize exact phase-equivalence classes.

7. Results from the First 100 Primes

The first 100 primes already display clear phase recurrence. Examples include:

[ 5 \rightarrow 31, ]

[ 7 \rightarrow 43, ]

[ 11 \rightarrow 67, ]

[ 17 \rightarrow 103, ]

[ 23 \rightarrow 139, ]

[ 37 \rightarrow 223, ]

[ 47 \rightarrow 283. ]

Each of these is a one-step relation of the form

[ p_2=6p_1+1. ]

For example,

[ 31=6(5)+1, ]

and

[ 31+\frac15 = 6\left(5+\frac15\right). ]

Therefore,

[ \theta(31)=\theta(5). ]

Likewise,

[ 43=6(7)+1, ]

so

[ 43+\frac15 = 6\left(7+\frac15\right), ]

and

[ \theta(43)=\theta(7). ]

This is not a numerical accident. It is exact algebraic phase preservation.

8. Results from the First 1,000 Primes

The first 1,000 primes extend the pattern substantially. In the dataset from 2 through 7919, we found:

• 77 exact phase-equivalence classes containing two or more primes;
• 75 one-step prime links of the form p \rightarrow 6p+1;
• 100 exact phase-preserved pairings when multi-step powers 6^k are included.

Several phase-equivalence chains contain three or more primes. Examples include:

[ 2 \rightarrow 13 \rightarrow 79 \rightarrow 2851, ]

[ 3 \rightarrow 19 \rightarrow 691, ]

[ 5 \rightarrow 31 \rightarrow 1123, ]

[ 17 \rightarrow 103 \rightarrow 619, ]

[ 23 \rightarrow 139 \rightarrow 5011, ]

[ 47 \rightarrow 283 \rightarrow 1699, ]

[ 61 \rightarrow 367 \rightarrow 2203, ]

[ 101 \rightarrow 607 \rightarrow 3643, ]

[ 131 \rightarrow 787 \rightarrow 4723, ]

[ 151 \rightarrow 907 \rightarrow 5443. ]

The chain

[ 2 \rightarrow 13 \rightarrow 79 \rightarrow 2851 ]

is especially useful because it demonstrates why the full 6^k law matters. The relation 2 \rightarrow 13 is one-step:

[ 13=6(2)+1. ]

The relation 13 \rightarrow 79 is also one-step:

[ 79=6(13)+1. ]

But 79 \rightarrow 2851 is not one-step. Instead,

[ 2851+\frac15 = 36\left(79+\frac15\right), ]

so the relation is a two-step 6^2 phase preservation.

Thus, the exact phase law is not merely a local recurrence. It propagates through powers of 6 across the prime lattice whenever the resulting value is prime.

9. Interpretation of the Plots

When the first 1,000 primes are plotted with \theta on the horizontal axis and \log_6(M(p)) on the vertical axis, the data form a structured band with visible vertical alignments. These alignments correspond to shared or nearly shared cylinder phase.

The most defensible alignments are the exact phase-equivalence classes defined by

[ 5p_2+1 = 6^k(5p_1+1). ]

In the plot, primes belonging to the same exact phase class occupy the same angular groove. Their vertical positions differ because their Collatz maximum heights differ, but their cylinder phase is locked.

This means the primes are not merely scattered across the Collatz phase coordinate. Some are connected by exact scale-preserving transformations native to the shifted base-6 cylinder.

The geometric interpretation is as follows:

• \theta identifies angular groove position;
• \log_6(M(p)) identifies maximum vertical excursion;
• exact phase-equivalent primes occupy the same groove;
• Collatz dynamics determine how high each prime rises before descent;
• the prime lattice determines which seed values populate the grooves.

This gives a measurable bridge between prime distribution and Collatz maximum-height geometry.

10. Fibonacci and Inverse-Tree Structure

The Collatz inverse tree contains branching structure. Each number may be reached through inverse operations depending on parity and divisibility constraints. These branching systems naturally generate recursive counting patterns.

Fibonacci-type growth may appear in constrained symbolic systems where sequences are built from allowed operations and forbidden adjacencies. In the Collatz setting, parity strings, inverse branches, and stopping-time patterns can produce recurrence structures whose counts resemble Fibonacci or signed Fibonacci sequences under appropriate restrictions.

This paper treats Fibonacci integration as a secondary structural observation rather than as a primary proof claim. The primary claim is the exact shifted-log phase recurrence among primes. Future work should isolate the specific inverse-tree grammar responsible for Fibonacci-type counts and state it as a formal lemma.

The cautious conclusion is this: Fibonacci-like recurrence may be present in the branching architecture feeding the grooves, but the exact algebraic result established here is the base-6 phase recurrence.

11. Zero and the Bottom of the Well

Zero occupies a unique conceptual position. The Collatz map is normally defined on positive integers, so zero is outside the standard domain of the conjecture. If extended formally, zero maps to itself under the even rule:

[ 0 \mapsto 0. ]

In the language of this framework, zero is not a climbing seed and not a descending seed. It is motionless. It does not enter the 4\rightarrow2\rightarrow1 attractor because it already sits at a separate fixed point.

For substrate mathematics, zero may be interpreted as the bottom of the well: the singular absence of positive magnitude. It is not another integer traveling through the law. It is the boundary condition against which positive integer motion becomes meaningful.

This interpretation should remain philosophical and structural, not a claim about the standard Collatz conjecture itself.

12. Implications

The shifted logarithmic coordinate

[ \theta(x)=\left{\log_6\left(x+\frac15\right)\right} ]

does two important things at once.

First, it provides a known near-conjugacy coordinate for Collatz dynamics, placing the map into a circle-rotation framework with bounded perturbation.

Second, it reveals exact phase recurrence among primes whenever their shifted values differ by powers of 6.

This is the central bridge of the paper.

The same coordinate that clarifies Collatz phase behavior also detects prime scale recurrence on the base-6 lattice. That does not prove that all Collatz trajectories descend. However, it does show that Collatz dynamics and prime-cylinder geometry can be studied in one shared coordinate system.

This opens several research directions:

1. classify all prime phase-equivalence chains under

[ 5p_2+1 = 6^k(5p_1+1); ]

2. compare groove density against Collatz maximum-height statistics;
3. distinguish exact phase classes from rounded numerical clusters;
4. test whether certain phase classes correlate with unusually high or low Collatz excursions;
5. extend the computation from 1,000 primes to 10,000, 100,000, and beyond;
6. construct a full three-dimensional cylinder visualization showing prime seeds, helical grooves, and Collatz maximum-height excursions;
7. formalize the inverse-tree grammar responsible for Fibonacci-type branching counts.

13. Limitations

This paper is preliminary. Its findings are computational and structural. The exact phase recurrence among primes is algebraic, but the relationship between that recurrence and Collatz descent remains empirical.

The following limits must be stated clearly:

1. The paper does not prove the Collatz conjecture.
2. Identical \theta does not imply identical Collatz trajectories.
3. Identical \theta does not imply identical maximum heights.
4. Rounded \theta clusters are visualization aids, not exact mathematical identities.
5. The exact identities occur only when the shifted values differ by powers of 6.
6. The Fibonacci discussion requires further formalization before it can carry proof weight.
7. The cylinder model is a powerful coordinate framework, but additional arithmetic control would be required for a proof of universal descent.

These limitations do not weaken the result. They make the result publishable. The purpose of this paper is to report a measurable geometric structure, not to overstate its consequences.

14. Conclusion

The first 1,000 primes reveal a reproducible and exact phase-recurrence structure under the shifted logarithmic coordinate

[ \theta(p)=\left{\log_6\left(p+\frac15\right)\right}. ]

Prime pairs and chains preserve phase whenever

[ p_2+\frac15 = 6^k\left(p_1+\frac15\right), ]

or equivalently,

[ 5p_2+1 = 6^k(5p_1+1). ]

The one-step case is

[ p_2=6p_1+1. ]

Within the first 1,000 primes, we found 77 exact phase-equivalence classes containing two or more primes, 75 one-step links of the form p \rightarrow 6p+1, and 100 exact phase-preserved pairings when multi-step powers of 6 are included.

When these primes are plotted by cylinder phase against logarithmic Collatz maximum height, the exact phase classes appear as shared angular grooves. This supports the interpretation of Collatz maximum-height structure as a cylindrical groove system, with primes occupying recurring angular positions on the same shifted base-6 lattice.

The result is not a proof of Collatz. It is a new measurable bridge: Collatz dynamics, prime distribution, and scale-invariant cylindrical phase recurrence can be placed into one coordinate system.

This is the opening result of substrate mathematics.

References

Asli, Barmak Honarvar Shakibaei. “An Explicit Near-Conjugacy Between the Collatz Map and a Circle Rotation.” arXiv:2601.04289, 2026.

Lagarias, Jeffrey C. “The 3x+1 Problem and Its Generalizations.” American Mathematical Monthly, 1985.

Oliveira e Silva, Tomás; Herzog, Siegfried; Pardi, Silvio. “Empirical Verification of the 3x+1 and Related Conjectures.” The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010.

Terras, Riho. “A Stopping Time Problem on the Positive Integers.” Acta Arithmetica, 1976.

Computational Appendix

For each prime p, the following values were computed:

[ \theta(p)=\left{\log_6\left(p+\frac15\right)\right}, ]

[ M(p)=\max{p,C(p),C^2(p),\ldots,1}, ]

[ H(p)=\log_6(M(p)). ]

Exact phase equivalence was tested using the integer relation

[ 5p_2+1 = 6^k(5p_1+1). ]

This avoids floating-point ambiguity. Rounded phase clusters were used only for visualization.

The first 1,000-prime dataset includes primes from 2 through 7919. The principal computational findings were:

• 77 exact phase-equivalence classes with two or more primes;
• 75 one-step links of the form p \rightarrow 6p+1;
• 100 exact pairings when all powers 6^k were included;
• visible vertical groove alignments in the \theta versus \log_6(M(p)) plot.

Representative exact phase chains:

[ 2 \rightarrow 13 \rightarrow 79 \rightarrow 2851 ]

[ 3 \rightarrow 19 \rightarrow 691 ]

[ 5 \rightarrow 31 \rightarrow 1123 ]

[ 17 \rightarrow 103 \rightarrow 619 ]

[ 23 \rightarrow 139 \rightarrow 5011 ]

[ 47 \rightarrow 283 \rightarrow 1699 ]

[ 61 \rightarrow 367 \rightarrow 2203 ]

[ 101 \rightarrow 607 \rightarrow 3643 ]

[ 131 \rightarrow 787 \rightarrow 4723 ]

[ 151 \rightarrow 907 \rightarrow 5443 ]

These chains demonstrate that the phase recurrence is not merely visual. It is exact under the shifted base-6 law.

Asli, Barmak Honarvar Shakibaei. “An Explicit Near-Conjugacy Between the Collatz Map and a Circle Rotation.” arXiv, arXiv:2601.04289, 2026.

Lagarias, Jeffrey C. “The 3x + 1 Problem and Its Generalizations.” The American Mathematical Monthly, vol. 92, no. 1, 1985, pp. 3–23. DOI: 10.2307/2322189.

Lagarias, Jeffrey C., editor. The Ultimate Challenge: The 3x + 1 Problem. American Mathematical Society, 2010.

Oliveira e Silva, Tomás. “Empirical Verification of the 3x + 1 and Related Conjectures.” In Jeffrey C. Lagarias, editor, The Ultimate Challenge: The 3x + 1 Problem, American Mathematical Society, 2010, pp. 189–207.

Terras, Riho. “A Stopping Time Problem on the Positive Integers.” Acta Arithmetica, vol. 30, no. 3, 1976, pp. 241–252.