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Exploring Collatz Dynamics with Human-LLM Collaboration

Edward Y. Chang

Abstract

We investigate structural properties of the Collatz iteration through two phenomena observed in large computational exploration: modular scrambling of residue classes and a burst--gap decomposition of trajectories. We prove several structural results, including a modular scrambling lemma showing that the gap-return map acts as an exact bijection on high bits, a persistent exit lemma characterizing gap structure after persistent states, and a decay property for known portions of binary representations under gap-return dynamics. We further prove that, in the modular model, gap lengths and $2$-adic valuations follow geometric distributions, while persistent run lengths are geometric with expected burst length $E[B]=2$; together these predict strict orbit contraction. These results suggest a conditional framework in which convergence would follow from suitable orbitwise hypotheses on burst and gap lengths, which in turn are suggested by an orbit equidistribution conjecture. However, the key hypotheses remain open, and the framework is exploratory rather than a complete reduction. The paper also documents the human-LLM collaboration through which these observations were developed.

Exploring Collatz Dynamics with Human-LLM Collaboration

Abstract

We investigate structural properties of the Collatz iteration through two phenomena observed in large computational exploration: modular scrambling of residue classes and a burst--gap decomposition of trajectories. We prove several structural results, including a modular scrambling lemma showing that the gap-return map acts as an exact bijection on high bits, a persistent exit lemma characterizing gap structure after persistent states, and a decay property for known portions of binary representations under gap-return dynamics. We further prove that, in the modular model, gap lengths and -adic valuations follow geometric distributions, while persistent run lengths are geometric with expected burst length ; together these predict strict orbit contraction. These results suggest a conditional framework in which convergence would follow from suitable orbitwise hypotheses on burst and gap lengths, which in turn are suggested by an orbit equidistribution conjecture. However, the key hypotheses remain open, and the framework is exploratory rather than a complete reduction. The paper also documents the human-LLM collaboration through which these observations were developed.
Paper Structure (25 sections, 15 theorems, 35 equations, 3 figures, 1 table)

This paper contains 25 sections, 15 theorems, 35 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. The 1/4 Persistent-Transition Law
  4. The convergence chain
  5. Entry--Occupancy Equivalence
  6. Entry bound implies convergence
  7. The Persistent Exit Lemma
  8. The Burst-Gap Criterion
  9. The Scrambling Lemma
  10. Statement and proof
  11. The pattern-determination bound
  12. Known-Zone Decay
  13. Conditional convergence and reduction to orbit equidistribution
  14. The conditional convergence theorem
  15. The Orbit Equidistribution Conjecture
  16. ...and 10 more sections

Key Result

Lemma 2.8

Let $a$ be an odd integer and $K \ge 1$. Then the map $\delta \mapsto a \cdot \delta \bmod 2^K$ is a bijection on $\{0, 1, \ldots, 2^K - 1\}$.

Figures (3)

  • Figure 1: The burst-gap decomposition of a Collatz orbit. Bursts (red) are maximal runs of iterates with $k \ge 2$; gaps (green) are runs of safe iterates ($k = 1$). The Persistent Exit Lemma (Lemma \ref{['lem:gap']}) shows that when a burst ends at a persistent state, the subsequent gap has length exactly $1$.
  • Figure 2: Architecture of the conditional framework. Green boxes denote results proved in this paper. Red boxes denote open components. Solid arrows represent proved implications; dashed arrows denote steps depending on open inputs. The conditional framework is intended to reduce the Collatz conjecture to the Orbit Equidistribution Conjecture together with the orbitwise regularity needed to supply the mean burst and mean gap bounds.
  • Figure 3: Known-Zone Decay for $M = 12$. The known zone $Z_k$ decreases by at least $3$ per gap-return. The worst-case bound $Z_k \le M - 3k$ reaches zero after $\lceil M/3 \rceil = 4$ gap-returns. Empirically, typical orbits reach $Z_k = 0$ in $1$--$3$ steps.

Theorems & Definitions (47)

  • Definition 2.1: Standard Collatz map
  • Conjecture 2.2: Collatz conjecture
  • Definition 2.3: Syracuse map
  • Definition 2.4: Persistent and safe states
  • Definition 2.5: Burst-gap decomposition
  • Definition 2.6: Gap-return map
  • Definition 2.7: Critical persistent frequency
  • Lemma 2.8: Bijection lemma
  • proof
  • Theorem 3.1: $1/4$ Persistent-Transition Law
  • ...and 37 more