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Abstract

We characterize convergence in trace norm for positive operator sequences whose cocycle subadditivity is imposed only on nearly equal block lengths. For a positive trace-preserving map on trace class, every normalized sequence converges precisely when the map is strongly mean ergodic and the scalar corridor has divergent square-width integral. We also obtain an exact robustness criterion: for a nonincreasing normalized trace-error profile, universal convergence holds precisely when the same two conditions hold and the error profile is integrable on logarithmic scale. A positive stationary oscillation proves necessity, including profiles with jumps. The scalar width and summability principles have classical or previously established origins; the operator transfer uses recurrent-support absorption, compact positive orbits and a regularized root-fidelity energy that pays order slack linearly by trace. The criterion includes absorbing channels with nonfaithful stationary states. Finite-dimensional selfadjoint sequences satisfy a complementary trace criterion, and unitary dynamics admit an exact classification by pure-point spectrum and corridor width. All corridor inequalities are required in both orders, with measurable nonincreasing width and eventual access to adjacent integer pairs.