Temporal adjoints and injectivity lengths in stationary quantum processes STRONG
Taking local matrix adjoints along a forward random process does not reverse its temporal order. We give finite, ergodic qubit examples in which first and eventual Kraus-span injectivity lengths change under this operation. The failure persists for bistochastic, two-sided faithful local channels; a three-phase example also changes the onset distribution. These examples contradict the pointwise adjoint-length identity in Lemma A.1 of arXiv:2607.01446v1, without providing a counterexample to that manuscript's main parent-Hamiltonian or spectral-gap theorems. We state a corrected inverse-base identity. It preserves fixed-block distributions, and, under local and dual faithfulness, preserves first and eventual onset laws. We also replace an invalid common-kernel step in the source appendix by a Choi-rank convergence argument, recovering its eventual-strict-positivity/injectivity implication under the cited ergodic convergence theorem. The required normalization and endpoint shift are explicit. Fixed-block, first-occurrence, eventual, and pathwise assertions remain distinct.
Working paper. Citable, but not yet screened for the AIRR collection.