The normalized odd Collatz map is analyzed as a discrete dynamical system in the ring of 2-adic integers. It is proven that the existence of periodic macro-cycles is equivalent to the solvability of a Diophantine system governed by an ensemble of projective 2 2 matrices. The finiteness of a macro-cycle is established to be consistent with the formation of invariant attractor hole windows. The topological isolation of these windows induces an overdetermined system of independent cyclic traversals, which strictly necessitates bitwise palindromic symmetry of the local 2-adic configurations (hole kernels). It is shown that for window widths L 9, the fundamental incommensurability of left-sided multiplicative expansion and right-sided additive truncation deterministically breaks this symmetry. Within the kinematic corridor of the orbit, this dynamical symmetry breaking algebraically forbids the existence of real periodic macro-cycles of length N > 7.
Eduard Dyachenko (Thu,) studied this question.