{
  "abstract": "We present a localized rooted-operator argument for Weil positivity in the real odd logarithmic channel, retaining the polar rank-one term throughout. The three gates are structural levels rather than a serial dependency: Gate I is an independent strict cellular/shell positivity theorem and diagnostic; Gate II carries the load-bearing operator argument; Gate III uses compression, closure, and the restricted odd Weil criterion. In Gate II, the inherited parent response is placed in the same post-old-core/common-cut primal form in which the arithmetic mismatch is charged. The audited MASTER-P2 same-block splice combines the aligned reference contribution with the actual Schur response before the target-ground coefficient is used, while the literal-slot/true-ground mismatch is paid explicitly by a leakage term. The source elimination is written as one exact common-source Schur completion on the same harmonic vector, and the endpoint fold is used only as unitary coordinate transport, so it creates no second source debit. An outward-rounded finite computation and analytic tail give a strictly positive MASTER margin for k >= 7. Starting from the rigorously certified physical coefficient-2 endpoint Y=7, Schur induction yields positivity at the integer endpoints; zero-extension gives non-negativity at every finite support radius. Closure and the restricted odd Weil criterion then give the conclusion stated in the main theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20367",
  "author_total": 1,
  "authors": [
    {
      "name": "Marco Desogus"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2526fy8aqtv4g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20367",
  "fulltext_url": null,
  "id": "2526fy8aqtv4g",
  "issns": [],
  "landing_url": "https://hal.science/hal-05760729v1",
  "language": "en",
  "licence": "LicenseRef-publisher-terms",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
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      "openalex",
      "2602|Analytic Number Theory Research"
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  "openalex": "W7213608306",
  "pdf_url": "https://hal.science/hal-05760729/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2526fy8aqtv4g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:39Z",
      "native_id": "hal-05760729",
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      "url": "https://hal.science/hal-05760729v1"
    },
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2609.20367",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20367"
    }
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Three Gates: A Rooted-Operator Approach to Weil Positivity",
  "title_text": "The Three Gates: A Rooted-Operator Approach to Weil Positivity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/2526fy8aqtv4g",
  "venue": "HAL",
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}