2-Adic Obstructions to Presburger-Definable Characterizations of Collatz Cycles

TL;DR

This paper explores the limitations of Presburger arithmetic in characterizing Collatz cycles, introducing the concept of ghost cycles in 2-adic integers and demonstrating algebraic obstructions to their definability.

Contribution

It proves that certain divisibility predicates related to Collatz are not semilinear, revealing fundamental barriers to algebraic and automata-based approaches for the problem.

Findings

Ghost cycles are genuine 2-adic periodic orbits.

Divisibility predicates are not semilinear for fixed odd steps.

Presburger-based methods cannot distinguish ghost cycles from true cycles.

Abstract

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.