Prime-Groove Chains in the Co-Rotating Collatz Frame: A Supplemental Note on Substrate Mathematics

Prime-Groove Chains in the Co-Rotating Collatz Frame: A Supplemental Note on Substrate Mathematics

DOI: to be assigned

John Swygert

May 30, 2026

Abstract

This supplemental paper extends the earlier paper, “Substrate Mathematics: Collatz Dynamics as Fractal Gradient Flattening on the Cylindrical Prime Lattice,” by introducing a co-rotating coordinate for viewing prime-groove chains inside the shifted Collatz cylinder.

The earlier paper established that the shifted logarithmic phase coordinate θ(p) = fractional part of log₆(p + 1/5) produces exact phase recurrence among primes whenever p₂ + 1/5 = 6ᵏ(p₁ + 1/5). Equivalently, 5p₂ + 1 = 6ᵏ(5p₁ + 1). The one-step case gives the especially simple prime-link relation p₂ = 6p₁ + 1.

Using the first 1,000 primes, from 2 through 7,919, we found 77 exact phase-equivalence classes containing two or more primes, 75 one-step links of the form p → 6p + 1, and 100 exact pairwise same-groove relationships when higher powers of 6 are included.

This supplemental note examines what happens when the Collatz rotational component is conceptually removed. The near-conjugacy coordinate places Collatz motion close to a circle rotation by α = log₆3, with a bounded error term. By moving into a co-rotating frame, the apparent helical twist of the cylinder is reduced, and prime-groove chains can be viewed as spoke-like radial structures rather than only as wrapped helical structures.

The result strengthens the geometric interpretation: prime phase classes are not merely scattered coincidences in a logarithmic plot. They are exact scale-invariant groove families on the shifted base-6 cylinder. The co-rotating frame makes those families easier to see.

This paper does not claim to prove the Collatz conjecture. It presents a reproducible geometric and computational observation: prime chains generated by the shifted base-6 law occupy stable phase grooves, and the same coordinate system also describes the rotational structure of Collatz dynamics.

1. Purpose of This Supplemental Note

The first paper introduced the main result: prime numbers placed into the shifted logarithmic coordinate θ(p) = fractional part of log₆(p + 1/5) form exact phase-equivalence classes whenever their shifted values differ by powers of 6.

That result is algebraic, not speculative.

If p₂ + 1/5 = 6ᵏ(p₁ + 1/5), then p₁ and p₂ have identical θ phase because multiplying by 6ᵏ adds k to the base-6 logarithm. The fractional part remains unchanged.

This supplemental note adds a second layer: the co-rotating frame.

The Collatz near-conjugacy shows that the Collatz map behaves approximately like rotation by α = log₆3 in the shifted coordinate, with a bounded perturbation. If the ordinary θ coordinate shows the prime-groove lattice on the cylinder, then the co-rotating coordinate asks a different question:

What does the structure look like when the dominant rotational motion is removed?

The answer is visually and conceptually important. The helical twist becomes easier to read as a spoke system. The wrapped groove becomes a radial grammar.

2. The Shifted Cylinder Coordinate

The primary coordinate is:

θ(x) = fractional part of log₆(x + 1/5).

This maps each positive integer onto a circular phase coordinate. The full logarithmic value log₆(x + 1/5) can be split into two parts:

1. an integer shell index, representing scale level;
2. a fractional phase θ, representing angular position.

Thus, the integers may be placed on a cylinder:

angular position = θ(x)

vertical scale = approximately log₆(x)

This is the basic cylindrical substrate used in the first paper.

3. Exact Prime-Groove Equivalence

Two primes p₁ and p₂ occupy the same shifted phase groove when:

p₂ + 1/5 = 6ᵏ(p₁ + 1/5).

Multiplying by 5 gives:

5p₂ + 1 = 6ᵏ(5p₁ + 1).

Solving for p₂ gives:

p₂ = 6ᵏp₁ + (6ᵏ − 1)/5.

The one-step case, k = 1, gives:

p₂ = 6p₁ + 1.

The two-step case, k = 2, gives:

p₂ = 36p₁ + 7.

The three-step case, k = 3, gives:

p₂ = 216p₁ + 43.

These are not approximate formulas. They are exact phase-preserving transformations in the shifted base-6 coordinate.

4. First 1,000 Prime Verification

We computed the first 1,000 primes, from 2 through 7,919. For each prime p, we computed:

θ(p) = fractional part of log₆(p + 1/5)

the Collatz maximum height M(p)

the logarithmic maximum height log₆(M(p))

the number of Collatz steps required to reach 1

the exact reduced phase key obtained from 5p + 1 after removing powers of 6

The exact reduced phase key is obtained as follows:

Start with S = 5p + 1.

While S is divisible by 6, divide S by 6.

The final remaining integer is the exact groove key.

Primes with the same final groove key occupy the same shifted phase groove.

This integer method avoids floating-point ambiguity. It proves exact phase membership without relying on rounded decimal θ values.

The first-1,000-prime computation found:

77 exact phase-equivalence classes containing two or more primes.

75 one-step prime links of the form p → 6p + 1.

100 exact pairwise same-groove relationships when higher powers of 6 are included.

165 of the first 1,000 primes belong to a repeated phase class.

These numbers show that the groove structure is not rare. It appears repeatedly even in the first small sample of the prime sequence.

5. Representative Prime-Groove Chains

The strongest examples are the exact chains.

Examples include:

2 → 13 → 79 → 2851

3 → 19 → 691

5 → 31 → 1123

17 → 103 → 619

23 → 139 → 5011

47 → 283 → 1699

61 → 367 → 2203

101 → 607 → 3643

131 → 787 → 4723

151 → 907 → 5443

Each chain preserves the same shifted cylinder phase θ.

The chain 2 → 13 → 79 → 2851 is especially useful because it shows both one-step and higher-power phase preservation.

13 = 6(2) + 1.

79 = 6(13) + 1.

But 2851 is not 6(79) + 1. Instead:

2851 + 1/5 = 36(79 + 1/5).

So 79 and 2851 share phase through a 6² relation.

This proves that the groove law is not limited to the simple p → 6p + 1 case. The deeper law is:

p₂ + 1/5 = 6ᵏ(p₁ + 1/5).

6. The Co-Rotating Collatz Frame

The near-conjugacy coordinate shows that Collatz motion is closely related to rotation by:

α = log₆3.

In ordinary θ-space, Collatz dynamics carry a rotational component. That means the visual structure can appear twisted or helical when plotted on the cylinder.

A co-rotating frame removes the dominant rotation. Conceptually, the co-rotating coordinate is:

φ = θ − nα mod 1.

Here, θ is the shifted phase coordinate, α = log₆3 is the Collatz rotation angle, and n is the relevant rotation count or scale index used for the projection being studied.

The purpose of φ is not to create a new prime identity. The exact prime identity already exists in θ. The purpose of φ is visual and dynamical: it lets the observer subtract the dominant rotational drift and inspect the groove structure in a frame moving with the Collatz rotation.

In ordinary language:

θ shows the grooves wrapped around the cylinder.

φ helps show what those grooves look like when the rotational twist is removed.

The helical groove becomes a radial spoke.

7. Why This Matters

A helical groove and a radial spoke may be the same structure viewed in different frames.

This is common in geometry and physics. A path that looks twisted in one coordinate system may become straight in another. A rotating system may look complicated from the outside but simple from the moving frame.

That is the key reason the co-rotating frame matters.

The original θ coordinate reveals exact phase recurrence.

The co-rotating φ coordinate clarifies the underlying spoke-like organization.

Together, they suggest that the prime-groove structure is not merely a plot artifact. It is a coordinate-stable feature of the shifted base-6 cylinder.

8. Collatz Maximum Height and Groove Excursion

For each prime seed p, the Collatz trajectory has a maximum height M(p). Plotting θ(p) against log₆(M(p)) shows where the prime begins angularly and how high its Collatz trajectory rises vertically.

Primes in the same exact groove do not necessarily have the same maximum height. They share angular phase, not identical trajectory.

This distinction is essential.

Identical θ does not mean identical Collatz behavior.

Identical θ means identical shifted cylinder groove membership.

The maximum height then records how far each seed climbs before descending toward the 4 → 2 → 1 attractor.

Thus, a groove is not a prediction that all members behave identically. It is a geometric family. Different members of the family may rise to different heights, but they begin from the same angular grammar.

9. The Grammar Interpretation

The earlier paper described substrate mathematics as the study of flattening laws written into the foundational geometry of the integers.

This supplemental note sharpens that language.

If numbers are the symbols, then the shifted cylinder is a grammar-space.

The primes are not merely isolated marks. They occupy angular positions.

The Collatz map is not merely a recursive trick. It is a dynamical syntax acting on those positions.

The exact relation p₂ + 1/5 = 6ᵏ(p₁ + 1/5) is a grammar rule.

The co-rotating frame reveals how that grammar looks when the dominant rotational motion is subtracted.

This suggests a careful philosophical statement:

Mathematics may be not only the language of the universe, but also the grammar by which structure is permitted to transform.

In this paper, that claim remains mathematical and modest. The grammar under study is the exact shifted base-6 grammar of prime phase recurrence.

10. What Is Proven Here

This paper establishes the following limited but real claims:

1. The shifted phase coordinate θ(p) = fractional part of log₆(p + 1/5) assigns every prime a cylinder phase.

2. Two primes share exact phase when p₂ + 1/5 = 6ᵏ(p₁ + 1/5).

3. This is equivalent to 5p₂ + 1 = 6ᵏ(5p₁ + 1).

4. The one-step case is p₂ = 6p₁ + 1.

5. In the first 1,000 primes, there are 77 repeated exact phase classes.

6. In the first 1,000 primes, there are 75 one-step p → 6p + 1 prime links.

7. In the first 1,000 primes, there are 100 exact pairwise same-groove relationships when all powers 6ᵏ are included.

8. These exact phase classes can be plotted against Collatz maximum-height data.

9. The co-rotating coordinate φ = θ − nα mod 1 is a natural way to inspect the structure after subtracting the dominant Collatz rotation α = log₆3.

10. The resulting visualization supports the interpretation of prime-groove chains as spoke-like structures in the co-rotating frame.

11. What Is Not Proven Here

This paper does not prove the Collatz conjecture.

It does not prove that every positive integer reaches 1.

It does not prove that prime groove membership determines Collatz maximum height.

It does not prove that identical phase produces identical Collatz trajectories.

It does not prove that the co-rotating frame removes all complexity.

It does not claim that visual spoke structure is itself a theorem of universal descent.

These cautions are not weaknesses. They protect the result.

The exact algebra is strong enough without overclaiming.

12. Implications for Substrate Mathematics

This supplemental result strengthens the larger substrate mathematics program in three ways.

First, it shows that the shifted base-6 cylinder is not merely a Collatz visualization. It also organizes prime phase recurrence.

Second, it shows that exact prime-groove classes can be found by integer arithmetic alone, using 5p + 1 and powers of 6.

Third, it shows that the co-rotating Collatz frame may reveal a simpler spoke-like geometry beneath the apparent helical twist.

This suggests several next steps:

Extend the computation from 1,000 primes to 10,000 and 100,000 primes.

Measure the density of repeated exact phase classes.

Compare groove membership against unusually high Collatz maximum heights.

Track whether certain exact phase classes produce systematically different stopping-time behavior.

Build a three-dimensional cylinder visualization with θ as angle, log₆(p) as height, and log₆(M(p)) as excursion.

Study whether the co-rotating frame reduces visual complexity in larger datasets.

Develop a formal classification of prime-groove chains under the law p₂ = 6ᵏp₁ + (6ᵏ − 1)/5.

13. Conclusion

The first 1,000 primes reveal a repeated exact phase structure in the shifted base-6 cylinder. The governing law is:

p₂ + 1/5 = 6ᵏ(p₁ + 1/5).

Equivalently:

5p₂ + 1 = 6ᵏ(5p₁ + 1).

The one-step case is:

p₂ = 6p₁ + 1.

This law generates exact prime-groove chains such as:

2 → 13 → 79 → 2851

5 → 31 → 1123

17 → 103 → 619

47 → 283 → 1699

61 → 367 → 2203

These chains share the same shifted cylinder phase. When viewed in relation to Collatz maximum-height data, they form stable groove families. When the Collatz rotation α = log₆3 is conceptually removed through a co-rotating frame, the helical structure can be read as a spoke-like grammar.

The finding is preliminary, but it is reproducible.

The primes are not random marks on the cylinder.

They occupy exact phase grooves.

The Collatz map does not merely move numbers.

It rotates, perturbs, lifts, drops, and flattens them through a measurable phase grammar.

This is the second step in substrate mathematics: not the proof of Collatz, but the discovery of a coordinate system in which prime recurrence and Collatz motion begin to speak the same geometric language.

References

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.

Computational Appendix

The first 1,000 primes were computed from 2 through 7,919.

For each prime p:

θ(p) = fractional part of log₆(p + 1/5).

M(p) = maximum value reached by the Collatz trajectory starting at p.

H(p) = log₆(M(p)).

Exact phase membership was determined by reducing S = 5p + 1 through repeated division by 6.

If two primes reduce to the same final S value, they belong to the same exact shifted phase groove.

Results:

77 exact repeated phase classes.

75 one-step p → 6p + 1 prime links.

100 exact pairwise same-groove relationships when higher powers 6ᵏ are included.

165 of the first 1,000 primes belong to a repeated exact phase class.

Representative exact chains:

2 → 13 → 79 → 2851

3 → 19 → 691

5 → 31 → 1123

17 → 103 → 619

23 → 139 → 5011

47 → 283 → 1699

61 → 367 → 2203

101 → 607 → 3643

131 → 787 → 4723

151 → 907 → 5443

References 

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.