[
  {
    "id": "ARR-2026-5K8R1DT82Q95JSHY-v1",
    "type": "article",
    "title": "Error-space geometry and positivity orders of encoded reduction maps",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          10,
          25
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-5K8R1DT82Q95JSHY",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-5K8R1DT82Q95JSHY/versions/v1/",
    "note": "Research preprint; ARR-2026-5K8R1DT82Q95JSHY v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.",
    "abstract": "We study tensor products of reduction maps after restriction to perfect-tensor encodings. An error-support expansion of the unnormalized Choi matrix accommodates multiple logical reference parties and arbitrary complex absolutely maximally entangled states. For one logical party, odd physical length n and 0<t<=1, averaging orthogonal error families over fixed-size site subsets gives a completely positive output complement under the sufficient condition d>=2n/t. The encoded map then has an exact Schmidt-rank sign criterion and optimal dimension-uniform positivity order floor(t^(-n)). At t=1/2, a separate estimate covers d>=2n. For two logical parties and four physical outputs, the half-parameter map is completely positive exactly in the admissible dimensions 2<=d<=12. In dimensions d>=13 it is not completely positive, but is k-positive at least through floor(2d(d-2)/(d-6)). Explicit polynomial-evaluation codes give failure at rank9d for every prime d>=43, establishing linear growth up to unmatched constants for those examples. Code existence, the Schmidt-overlap criterion and the reduction-map family are prior ingredients. The results retain substantive code-support restrictions and belong to the completely copositive class."
  }
]