A Collatz-type dynamical system on Eisenstein integers: cycle classification by double resonance and exhaustive itinerary enumeration

We introduce an accelerated Collatz-type map T* on the Eisenstein integers Z[ω], using the prime π = 1 − ω (of norm 3) as divisor. We prove that the only nonzero fixed points are ±2ω (Theorem 1), establish an explicit norm formula for the cycle denominator π^K − 2^m revealing a phase obstruction: small denominators require both size resonance (K/m ≈ log4/log3) and phase alignment K ≡ 0 mod 12 (Theorem 2). For the first non-trivial resonant pair (m,K) = (19,24), we classify all cycles exhaustively by meet-in-the-middle enumeration: exactly 12 orbits exist, forming 6 opposite pairs (Theorem 3, computer-assisted). Two additional cycles for (m,K) = (28,36) are certified (Theorem 4). Convergence to a periodic orbit is observed on 10^6 points with zero divergence. Supplementary material includes reproducible Python code and a complete cycle catalog.