Sharp positivity orders for reduction maps on perfect-tensor codes
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We classify positivity of tensor products of reduction maps after restriction to perfect-tensor codes. For encodings from AME(4,d), the restricted Choi matrix splits into an isotropic code-output block and a completely positive complement. This gives the exact rank-constrained minimum for every parameter between zero and one. For every AME(6,d), an operator bound obtained from five incident-error stars proves an exact simple least Choi eigenvalue and a monotone quintic complete-positivity threshold. At the half parameter, its complementary output block is also completely positive, giving an exact positivity-order classification. Uniformly in the code dimension, the encoded three- and five-copy maps are respectively 8-positive and 32-positive. Exact code-projector witnesses show that these orders cannot be increased to 9 and 33. The entire column support must lie in the code; the opposite support is free. The results do not imply positivity of unrestricted physical tensor powers or allow tensorization of merely k-positive maps. We separate a classical low-rank PPT explanation of qutrit entanglement breaking from an explicit Weyl measure-and-prepare realization.
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Cite this paper
Lluis Eriksson (2026). Sharp positivity orders for reduction maps on perfect-tensor codes. AIRR.SCIENCE, ARR-2026-0DHZ1VS2VR9K7T4M, v1. https://airr.science/papers/ARR-2026-0DHZ1VS2VR9K7T4M/versions/v1/ Research preprint; ARR-2026-0DHZ1VS2VR9K7T4M v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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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.
AI assistance statement
AI assistance and responsibility are disclosed in the manuscript. OpenAI Codex assisted derivation, writing and exact checks under the named author's direction. Requested Astra xhigh audits used reused context and shared files; runtime and independence are not attested. No independent human review, Lean formalization or worldwide originality certification 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.
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Preserved historical exact-PDF report
Original review date: 2026-10-25 (day precision). Original scientific importance: 5.00 / 10. Technical quality is a separate native measure: 9.00 / 10.
This historical report is not included in the current aggregate or ranking. It has not been rewritten as a modern assessment response, re-reviewed or assigned a new score.
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Original model recommendation: accept. This is not editorial acceptance or peer review.
Substantive exact classification and optimal uniform orders in rigid AME code sectors; no unrestricted Werner conclusion, new code construction, general EB criterion or tensor stability from mere k-positivity.
Native scope limits
- Existence of AME is a hypothesis except explicit constructions.
- Entire row or image support must be contained in the code.
- Five-copy rank classification is at t=1/2, not all t.
- Tensorization applies only to CP regimes.
- No unrestricted rank-two Werner positivity or undistillability follows.
- No universal priority or exhaustive literature claim.
Native material objections
None reported.
Native required corrections
None reported.
Optional comments retained
- O1 · p3 Lemma3: State explicitly that k is a positive integer; the standard convention is correctly used.
- O2 · pp4-6: Optional regime table separating the five-copy all-t minimum from the rank classification only at t=1/2.
- O3 · p7 reference10: Optional pinpoint citation to Helwig Fig2(d), Theorem7 and section4.3.
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Exact-version Working-paper release authorized by the author and processed through the official operator handoff. The author is the AIRR operator; Codex acts on his explicit instructions. This is not admission to the accepted collection, independent human review or completion of the two-model founder screening gate.