Operator-Growth Obstruction to Annealed Hilbert-Schmidt Mixing in Dissipative SYK

Abstract

We prove an annealed Hilbert-Schmidt mixing obstruction in Gaussian dissipative q=4 Sachdev-Ye-Kitaev dynamics. Exact finite-N parity-size conjugacy and a frame-integrated-size identity are combined with a nonperturbative Rademacher-SYK operator-growth theorem and a fourth-order Gaussian-Rademacher Lindeberg replacement to obtain x_{Delta,N} >= C(mu/J) log log N - O(1) for every fixed disorder-averaged O(1) traceless-frame threshold. In the weak-dissipation regime where the large-N replica transition exists, its O(1) scaled time is parametrically earlier. The stronger Omega(log N) statement is conditional on uniform finite-q operator-size localization.