2-Adic Obstructions to Presburger-Definable Characterizations of Collatz Cycles
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909995996545024 |
|---|---|
| author | Dhiman, Madhav Pandey, Rohan |
| author_facet | Dhiman, Madhav Pandey, Rohan |
| contents | I investigate structural limitations of Presburger-arithmetic-based approaches to the Collatz problem. I show that the Collatz cycle equation admits a unique solution in the $2$-adic integers, which I term a \emph{ghost cycle}. These ghost cycles are shown to be genuine periodic orbits of the $2$-adic Collatz map, satisfying all local parity constraints.
I prove unconditionally that the divisibility predicate $\mathcal{D}_y = \{(x, C) \in \mathbb{N}^2: (2^x - 3^y) \mid C\}$, which acts as the algebraic necessary condition for integrality, is not semilinear for any fixed number of odd steps $y \ge 1$. This result is established by demonstrating that the fibers of $\mathcal{D}_y$ exhibit unbounded periods, an obstruction to Presburger definability. Consequently, strategies relying solely on Presburger arithmetic or finite automata to define the integrality constraint cannot capture the distinction between ghost cycles and genuine integer cycles. I conclude with a heuristic argument suggesting that because ghost cycles satisfy the algebraic cycle equation, the non-existence of integer cycles cannot be proven solely through algebraic manipulation of the cycle equation itself. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12772 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | 2-Adic Obstructions to Presburger-Definable Characterizations of Collatz Cycles Dhiman, Madhav Pandey, Rohan Number Theory Dynamical Systems I investigate structural limitations of Presburger-arithmetic-based approaches to the Collatz problem. I show that the Collatz cycle equation admits a unique solution in the $2$-adic integers, which I term a \emph{ghost cycle}. These ghost cycles are shown to be genuine periodic orbits of the $2$-adic Collatz map, satisfying all local parity constraints. I prove unconditionally that the divisibility predicate $\mathcal{D}_y = \{(x, C) \in \mathbb{N}^2: (2^x - 3^y) \mid C\}$, which acts as the algebraic necessary condition for integrality, is not semilinear for any fixed number of odd steps $y \ge 1$. This result is established by demonstrating that the fibers of $\mathcal{D}_y$ exhibit unbounded periods, an obstruction to Presburger definability. Consequently, strategies relying solely on Presburger arithmetic or finite automata to define the integrality constraint cannot capture the distinction between ghost cycles and genuine integer cycles. I conclude with a heuristic argument suggesting that because ghost cycles satisfy the algebraic cycle equation, the non-existence of integer cycles cannot be proven solely through algebraic manipulation of the cycle equation itself. |
| title | 2-Adic Obstructions to Presburger-Definable Characterizations of Collatz Cycles |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2601.12772 |