[
  {
    "id": "ARR-2026-5BWNCWFTY79W79RF-v2",
    "type": "article",
    "title": "Temporal adjoints and injectivity lengths in stationary quantum processes",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          10,
          29
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-5BWNCWFTY79W79RF",
    "version": "v2",
    "genre": "Technical note",
    "URL": "https://airr.science/notes/ARR-2026-5BWNCWFTY79W79RF/versions/v2/",
    "note": "Technical note; ARR-2026-5BWNCWFTY79W79RF v2. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.",
    "abstract": "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."
  }
]