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arxiv: 2607.01718 · v1 · pith:QUB75N36new · submitted 2026-07-02 · 🧮 math.NT

A Coordinate System for Collatz Dynamics

Jennifer Williams This is my paper

Pith reviewed 2026-07-03 07:16 UTC · model grok-4.3

classification 🧮 math.NT
keywords Collatz dynamics3-smooth factorizationcoordinate systemtriangular partitionprime obstructionskeletonsresidue classes
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The pith

A coordinate system organizes Collatz chains into triangles and proves that rows congruent to 2 mod 4 with k at least 6 in the principal skeleton contain no primes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a partition of the nonnegative integers into countably many infinite triangles based on the unique 3-smooth factorization of odd positive integers. In this setup, each row within a triangle forms a Collatz chain of alternating parity, and the coordinates (a, b) define a skeleton that permits a deterministic diagonal movement. This framework makes it possible to demonstrate algebraically that the principal skeleton has no primes in rows whose length k satisfies k congruent to 2 modulo 4 and k at least 6, and that this is the only residue class with such a complete obstruction. A reader might care because the results hold regardless of whether the Collatz conjecture is true, offering a structural view of these sequences.

Core claim

Building from the unique representation of every odd positive integer as n equals lambda times 2 to the a times 3 to the b minus 1 with gcd of lambda and 6 equal to 1 and a at least 1, the nonnegative integers are partitioned into countably many infinite triangles. Each row k equals a plus b forms a Collatz chain of alternating parity within the skeleton L_lambda. The coordinate system supports a deterministic flow from (a, b) to (a-1, b+1), with exit at a equals 1 to another skeleton. As an application, rows k congruent to 2 modulo 4 with k at least 6 in the principal skeleton L_1 contain no primes, and this residue class is the unique one admitting complete algebraic obstruction. All resul

What carries the argument

The skeleton L_lambda coordinate system, which assigns to each odd positive integer the pair (a, b) from its 3-smooth factorization and indexes rows by k equals a plus b, enabling deterministic diagonal flows and boundary transitions between skeletons.

If this is right

  • Rows k congruent to 2 modulo 4 and at least 6 in the principal skeleton L_1 contain no prime numbers due to algebraic obstruction.
  • This residue class is the only one that admits a complete algebraic obstruction to containing primes.
  • The triangular partition and coordinate system apply to every nonnegative integer.
  • The framework and its consequences do not depend on the truth of the Collatz conjecture.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The coordinate system could connect different skeletons through boundary exits, potentially forming larger structures for analyzing Collatz trajectories globally.
  • Similar algebraic obstructions might be identifiable in other skeletons L_lambda for lambda not equal to 1 using the same coordinate approach.
  • Checking small values of k in L_1 provides a direct way to verify the absence of primes in the obstructed rows.

Load-bearing premise

Every odd positive integer can be written uniquely in the form lambda times a power of 2 times a power of 3 minus one, where lambda is coprime to 6 and the power of 2 is at least 2 to the first.

What would settle it

Observing a prime number in any row k where k is congruent to 2 modulo 4 and k is greater than or equal to 6 in the principal skeleton L_1 would show that the algebraic obstruction does not hold for that row.

Figures

Figures reproduced from arXiv: 2607.01718 by Jennifer Williams.

Figure 1
Figure 1. Figure 1: Distribution of Collatz chain lengths for n ∈ {0, 1, . . . , N} with N = 107 . Chains are categorized as {ci} ⊆ [0, N] (all elements ≤ N) or {ci} ̸⊆ [0, N] (at least one element > N). (a) Linear scale showing chains of length 1, 3, 5, . . . , 23. (b) Logarithmic scale showing all observed chain lengths from 1 to 47. (c) Total count of chains intersecting [0, N] compared with theoretical prediction N/(6 · 2… view at source ↗
Figure 2
Figure 2. Figure 2: Crown triangles from the two crown families. Left: T0 with crown ≡ 0 mod 12. Right: T8 with crown ≡ 8 mod 12. Crowns are marked with dashed borders and grey fill. Orange edges (right side of each triangle) show recurrence 2n + 2; Blue edges (left side) show recurrence 3n + 4. Odd elements are shown in bold. Gray arrows indicate Collatz function flow along each row. Triangles are infinite and extend indefin… view at source ↗
Figure 3
Figure 3. Figure 3: Prime counts in rows k ≡ 2 (mod 4). Left: In skeleton L1, rows with k ≡ 2 (mod 4) contain zero primes for k ≥ 6; the row k = 2 is an exception, containing the primes 3 and 5. Right: Across 11 skeletons (λ ∈ {1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31}) and 10 rows (110 pairs total), only L1 exhibits this zero￾prime property; all other skeletons contain primes in these rows. The proof uses only elementary divi… view at source ↗
Figure 4
Figure 4. Figure 4: Top left: Mersenne numbers 2k −1 (position 1, i.e., b = 0, in L1), with primes starred. Top right: Thabit numbers 3 · 2 k−1 − 1 (position 3, i.e., b = 1, in L1). Bottom left: All odd positions in L1, showing the Pierpont family 2a · 3 b − 1. Position 2b + 1 corresponds to exponent b. Blue indicates b = 0 (Mersenne), orange indicates b = 1 (Thabit), and gray indicates b ≥ 2. Bottom right: Prime density comp… view at source ↗
Figure 5
Figure 5. Figure 5: Left: Skeleton λ value at each Collatz step for selected starting values. The orbit of n = 27 (red) reaches much higher λ values compared with other starting n values. Right: Maximum λ visited by orbits starting from n = 3, 4, . . . , 200. The value n = 27 (red bar) is a local anomaly, reaching λ = 2309. This observation does not explain why the orbit of n = 27 behaves anomalously compared to other values … view at source ↗
read the original abstract

It is well-established that every odd positive integer $n$ can be written uniquely as $n = \lambda \cdot 2^a \cdot 3^b - 1$ where $\gcd(\lambda, 6) = 1$ and $a \geq 1$. Building from this 3-smooth factorization, we introduce a partition of the nonnegative integers into countably many infinite triangles where each row $k$ forms a Collatz chain of alternating parity. The partition admits a coordinate system as a skeleton $\mathcal{L}_\lambda$ using the pair $(a, b)$ for odd positive integers within a geometric structure where row $k$ corresponds to $k = a + b$. Each position $(a, b)$ maps to $(a-1, b+1)$, a deterministic diagonal flow requiring no number-theoretic input. At the boundary $a = 1$, the trajectory exits to another skeleton depending on the factorization of $\lambda \cdot 3^{b+1} - 1$. The coordinate system is new. As a concrete application, we prove that rows $k \equiv 2 \pmod 4$ with $k \geq 6$ in the principal skeleton $\mathcal{L}_1$ contain no primes, and show this is the unique residue class admitting complete algebraic obstruction. Our contribution is the framework that makes visible which nonnegative integers these arguments apply to, with all results independent of the Collatz conjecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces a coordinate system for Collatz dynamics grounded in the unique 3-smooth factorization of every odd positive integer n = λ · 2^a · 3^b − 1 (gcd(λ,6)=1, a≥1). This induces a partition of the nonnegative integers into countably many infinite triangles whose rows k = a + b form Collatz chains of alternating parity. Skeletons L_λ are defined via the pair (a,b), with a deterministic diagonal flow (a−1,b+1) at interior points and boundary exit rules depending on the factorization of λ · 3^{b+1} − 1. The principal application is an algebraic proof that rows k ≡ 2 (mod 4) with k ≥ 6 in the principal skeleton L_1 contain no primes, together with a demonstration that this is the unique residue class admitting complete algebraic obstruction; all results are stated to be independent of the Collatz conjecture.

Significance. If the derivations hold, the coordinate system supplies a new algebraic framework that renders visible which nonnegative integers are subject to standard factorization arguments, yielding an explicit, conjecture-independent result on the absence of primes in a specific arithmetic family of rows. The use of the well-established unique 3-smooth factorization together with the algebraic factorization of 2^a 3^b − 1 constitutes a concrete, falsifiable contribution; small explicit verifications (e.g., k=6,10) are consistent with the claim while other residue classes contain known primes (e.g., 53 for k=4).

minor comments (3)
  1. [Abstract] Abstract, first paragraph: the mapping from position (a,b) to the Collatz step (a−1,b+1) is described as deterministic and requiring no number-theoretic input; an explicit one-line verification that this step preserves the 3-smooth form of the next odd integer would improve clarity.
  2. The manuscript should include a short table or explicit list of the first few rows of L_1 for k=2,4,6,8,10 showing the explicit integers and confirming the algebraic factorization when k≡2 (mod 4).
  3. Notation: the boundary-exit rule at a=1 is stated to depend on the factorization of λ · 3^{b+1} − 1; a single worked example for a small λ would remove any ambiguity about how the trajectory continues across skeletons.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were enumerated in the report, so we have no individual points requiring point-by-point rebuttal or revision at this stage. We remain available to address any additional feedback.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper starts from the standard, externally verifiable fact that every odd positive integer n admits a unique factorization n = λ · 2^a · 3^b − 1 with gcd(λ,6)=1 and a≥1. From this it defines a coordinate system on countably many triangles whose rows are indexed by k=a+b and whose internal flow is the deterministic map (a,b)→(a−1,b+1). The central claim—that rows k≡2 (mod 4), k≥6 in L_1 contain only composites—is obtained by an algebraic factorization of 2^a 3^b −1 that is forced precisely when a+b≡2 (mod 4) and a+b≥6; this is a direct number-theoretic identity, not a fitted parameter or self-referential definition. No self-citations, ansatzes, or renamings of known results are load-bearing. The entire argument is stated to be independent of the Collatz conjecture.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The framework rests on the stated 3-smooth factorization (treated as background) and introduces new geometric objects (triangles and skeletons) whose only support is the definition given in the paper.

axioms (1)
  • domain assumption Every odd positive integer n can be written uniquely as n = λ · 2^a · 3^b - 1 where gcd(λ, 6) = 1 and a ≥ 1.
    Explicitly invoked in the first sentence of the abstract as well-established.
invented entities (2)
  • Infinite triangles partition no independent evidence
    purpose: Partition of nonnegative integers into Collatz chains indexed by k = a + b
    New structure introduced to support the coordinate system; no independent evidence supplied beyond definition.
  • Skeletons L_λ no independent evidence
    purpose: Coordinate system using pairs (a, b) with deterministic diagonal flow
    Core new object of the paper; defined via the factorization and flow rule.

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discussion (0)

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Reference graph

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