Parity-Controlled Collatz Dynamics and a Probabilistic Phase Transition

We study a class of Collatz-type dynamical systems in which the parity sequence (odd/even steps) is treated as an external control, either taken from the classical Collatz map or modeled probabilistically. By separating parity control from numerical update rules, we show that the qualitative behavior of such systems is governed by an entropy balance between multiplicative growth and halving. Using an empirical even-step frequency p≈log⁡2/log⁡3≈0.631p \approx \log 2 / \log 3 \approx 0.631p≈log2/log3≈0.631, motivated by heuristic and rigorous work of Tao, we identify a critical multiplier mc≈1.85m_c \approx 1.85mc≈1.85 separating average contraction from average expansion. This framework explains why the classical 3x+13x+13x+1 problem is difficult, why 2x2x2x is not dynamically special under statistical parity, and why Collatz-type problems exhibit a phase-transition phenomenon.