Temporal adjoints and injectivity lengths in stationary quantum processes
Citable version that has not completed AIRR screening. Not peer reviewed. Screening record
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.
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Cite this paper
Lluis Eriksson (2026). Temporal adjoints and injectivity lengths in stationary quantum processes. AIRR.SCIENCE, ARR-2026-5BWNCWFTY79W79RF, v2. https://airr.science/notes/ARR-2026-5BWNCWFTY79W79RF/versions/v2/ Technical note; ARR-2026-5BWNCWFTY79W79RF v2. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Version history
The paper identifier remains stable. Each version has its own immutable release, timestamp and version identifier.
Revision statement
Major revision. Add scoped Choi-rank convergence proof repair of source Proposition A.9, explicit source normalization and endpoint conventions, trace-preservation consequence and reproducibility clarifications.
Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Partial
- Reproducibility
- Partial
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
Technical-note scope
- Kind
- Result
- Maturity
- Complete In Scope
Contribution boundary
Periodic counterexamples and correctly shifted inverse-base reversal; scoped sourceA.9 proof repair from faithful Choi-rank convergence with explicit source normalization and endpoint matching.
Limitations
No new general mixing theorem, quantitative onset bound, classification, main parent/gap theorem validation, complete imported-proof audit or certified priority. All six minor objections and four limitations retained with clarifications; PDF remains unchanged.
AI assistance statement
One full exact-PDF AI assessment REVIEW1242: official overall4.20/10, criteria4/4/3/3/4 stars; supplementary technical8.30/10 and importance3.70/10 are separate descriptive scales, not official criteria or inputs to overall. Recommendation minor revision; no material mathematical error found in reported checks. AllM1-M6 and L1-L4 preserved with accompanying clarifications, assessed8pagePDF unchanged. Requested GPT-6 Astra xhigh in separately initiated fork-none context; effective identity, effort, isolation and independence unattested; shared filesystem, author-directed Codex development and earlier unscored component audits disclosed. No prior-version score transferred. No independent human peer review, native gate pass, founder-review quorum, editorial admission, global priority or main parent-Hamiltonian/gap theorem validation is claimed.
Frontier-model screening
Status: not_assessed. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
Model assessments and review context
- Distinct models
- 1
- Range
- 4.20–4.20
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 0/1 benchmark matches. This is an evidence description, not a probability of correctness.
Only one distinct model; One provider; shared blind spots possible; Fresh blinded review context not documented for every model; Benchmark match incomplete; Adverse reports or unresolved objections require editorial disposition
Review input clarification
One full exact-PDF AI assessment REVIEW1242: official overall4.20/10, criteria4/4/3/3/4 stars; supplementary technical8.30/10 and importance3.70/10 are separate descriptive scales, not official criteria or inputs to overall. Recommendation minor revision; no material mathematical error found in reported checks. AllM1-M6 and L1-L4 preserved with accompanying clarifications, assessed8pagePDF unchanged. Requested GPT-6 Astra xhigh in separately initiated fork-none context; effective identity, effort, isolation and independence unattested; shared filesystem, author-directed Codex development and earlier unscored component audits disclosed. No prior-version score transferred. No independent human peer review, native gate pass, founder-review quorum, editorial admission, global priority or main parent-Hamiltonian/gap theorem validation is claimed. M1: the source supplies some admissible fixed rate, not every r. M2: legacy tensor swap preserves ranks. M3: new pure-control exact file supplied in version-specific bundle. M4: original reviewer reset control and result included. M5: conclusion is full representation-independent Choi rank. M6: sourceA.3, Sanz, sourceA.9 recurrence and MSstationary-support antecedents credited. Native linked childSUB-DA7B61D98DDA9A97; separate exactCCBY4permission11:21:47UTC.
Operator record · · Source files and affected reports
- REVIEW1242.txt · SHA-256
06cc6d3e260cf346b3fe1e3148a2adb4bcbd7a022bf43c8eefb659646f97fcc7 - verdict1242.json · SHA-256
e3b704a3c3f3ca8e3bdbab924634fa79cb63d247fda1aa58efcacb0b0f67214f - integrity_final1242.json · SHA-256
5904c93f050d45d74132c4687f3b9c766ae15dc3a664ae6da24ddba960dce64a - ADJUDICATION1249.md · SHA-256
c868060a2c2a5b7d36b177cd74ebbb3ce3677d2ca78aeb53b202cea2aa82f334 - projection-provenance.json · SHA-256
ae8167ed612b9256e3f80a9ac1bd0415fc551fd397dec1c5eaa139b5a582057e
OpenAI (requested; runtime unverified) · GPT-6 Astra xhigh (requested; runtime unverified) · unknown · review context: unknown4.20
Recommendation: Minor Revision · Material objections: 0
Operator serialization of one original REVIEW1242, not another model vote. One full exact-PDF AI assessment REVIEW1242: official overall4.20/10, criteria4/4/3/3/4 stars; supplementary technical8.30/10 and importance3.70/10 are separate descriptive scales, not official criteria or inputs to overall. Recommendation minor revision; no material mathematical error found in reported checks. AllM1-M6 and L1-L4 preserved with accompanying clarifications, assessed8pagePDF unchanged. Requested GPT-6 Astra xhigh in separately initiated fork-none context; effective identity, effort, isolation and independence unattested; shared filesystem, author-directed Codex development and earlier unscored component audits disclosed. No prior-version score transferred. No independent human peer review, native gate pass, founder-review quorum, editorial admission, global priority or main parent-Hamiltonian/gap theorem validation is claimed.
- Correctness confidence
- Principal claims survived exact algebra, all-page PDF inspection, source-interface comparison and controls. Imported ergodic convergence theorem was not completely reproved; no probability calibration is claimed.
- Rigor
- Time order, recurrence, measurable onset, persistence, dimension factor, endpoint shift and implicit trace preservation are handled correctly. Equation (3) should explicitly quantify an admissible existential rate.
- Novelty
- Specific periodic counterexamples and assembled repair have corrective value; persistence, autonomous faithful-limit reasoning, recurrence and ergodic convergence have substantial antecedents. Literature search was bounded.
- Significance
- Corrects a false appendix identity and repairs a flawed proof. No demonstrated new main parent-Hamiltonian/gap theorem, classification, mixing theorem or quantitative onset bound.
- Reproducibility
- Manifest, canonical PDF, TeX, three primary PDFs and four exact scripts support inspection; all supplied checks passed. Public locator for new control and explicit vectorization convention could be improved. Inspectability is not reviewer independence.
Strengths
- Exact periodic counterexamples and qualified inverse-base onset law.
- Scoped Choi-rank repair with normalization, source endpoints and forced-TP qualification.
Weaknesses
- M1 existential rate wording; M2 legacy tensor order; M3 public pure-control locator; M4 reset control; M5 Choi/Kraus formulation; M6 established antecedents. All remain in full original report.
Potential errors
None reported.
Strong novelty candidates
- Scoped corrective assembly; known ingredients, bounded literature search, no priority certificate.
Unresolved material objections
None reported.
arr:assessment:ba10b6ef-911d-4bc4-a035-95acd02514d2 · prompt ARR-ASSESS-1.0 · response SHA-256 ac5ec6444f8450276463e573ee4d99da6112c1439b9145f25ced3aa9133b42ca · canonical PDF SHA-256 bca2edda25970c0f4250fb110e2683e8de200073128427d8ad84abef803baef4
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Author-authorized Working paper, not admitted to accepted collection. Assistant recorded operator/author instructions and founder role through normal UI; no independent human decision or gate pass claimed.