We present a new algebraic reformulation of the uniqueness problem for periodic orbits of the Collatz map $c(n) = n/2$ ($n$ even) or $c(n) = 3n+1$ ($n$ odd). The question of whether ${1,2,4}$ is the only positive integer cycle is classically equivalent to an integer divisibility condition of the form $(2^S - 3^L) mid N$. We recast this condition as the vanishing of an explicit integer polynomial --- the cycle polynomial $P_G(t)$ --- evaluated at an arithmetic point $t_0$ of multiplicative order $L$ modulo $D = 2^S - 3^L$. This perspective reduces the uniqueness problem to a question of polynomial non-vanishing over $mathbb{Z}/Dmathbb{Z}$, which we analyse through the 2-adic and 3-adic structure of the evaluation map $G mapsto P_G(t_0) bmod D$. Using this framework we establish two partial results. First, for every mixed valuation sequence $G$ --- one in which the accumulated deviations $varepsilon_i$ take both positive and negative values --- the cycle polynomial satisfies $P_G(t_0) otequiv 0 pmod{D}$ in the special case where exact integer vanishing $A(G) = B(G)$ would be required; this follows from a parity obstruction on 2-adic valuations together with the step-size constraint $G_i geq 1$. Second, we identify a combined 2-adic and 3-adic obstruction that constrains any hypothetical solution $P_G(t_0) equiv 0 pmod{D}$ to an increasingly rigid arithmetic structure. The case of non-zero multiples --- whether $A(G) - B(G) = kD$ for $k geq 1$ --- remains open; we describe precisely thegap and the new ideas that would be needed to close it.