{
  "abstract": "Let G be a torsion-free group whose real group algebra R[G] has no zero divisors, and let u -> u^{*theta} be an l2-isometric anti-involution of R[G] (a composition of the inversion involution with a ring automorphism and a sign character). We prove a compactness theorem: for every finite *theta-closed support window W the constant mu*(W) = min{ ||w^{*theta}w||_2 : ||w||_2 = 1, supp(w) in W } is strictly positive, and every real theta-unitary element (u^{*theta}u = 1) supported in W satisfies ||u||_2 <= mu*(W)^{-1/2}. In particular the integer theta-unitary elements supported in W form a finite, effectively enumerable set: the a priori infinite \"height\" direction of the unit search collapses to a single real constant. For the Promislow (Hantzsche-Wendt) group P and the window B(4) that hosts Gardam's counterexample to the unit conjecture over F_2, we combine this with exact SAT-certified height ladders (heights <= 31 per stratum, DRAT-certified master cell at larger heights), a radius-free depth-tail theorem, and a mirror symmetry between the +- strata, reducing the vanishing of all nontrivial theta-unitary units u = +-1 (mod 2) with supp(u) in B(4) to a single certified lower bound on mu*(B(4)). We report numerical estimates mu*(B(4)) ~ 1.4e-3, well above the required threshold, and prove three structural results about the remaining certification problem: no linear (Cauchy-Schwarz) dual certificate exists, because P carries theta-anti-unitary trivial elements; and, numerically, the level-2 sum-of-squares relaxation is boundary-pinned with an explicit slope, both in the direct and in the ideal-multiplier formulation -- the obstruction being a spurious pseudo-moment that no measure can realize. We also contrast the mechanism with Z[D_infinity], where torsion produces zero divisors, mu* = 0, and genuinely unbounded unipotent families of twisted unitaries.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25015",
  "author_total": 1,
  "authors": [
    {
      "name": "Moe Tabei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bd8k58d2d4exh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25015",
  "fulltext_url": null,
  "id": "bd8k58d2d4exh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25015",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GR"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7214066956",
  "pdf_url": "https://arxiv.org/pdf/2609.25015",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bd8k58d2d4exh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25015",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25015"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A compactness theorem for twisted-unitary elements of integral group rings, with a certified route to the theta-unitary Case A at window B(4) of the Promislow group",
  "title_text": "A compactness theorem for twisted-unitary elements of integral group rings, with a certified route to the theta-unitary Case A at window B(4) of the Promislow group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/bd8k58d2d4exh",
  "venue": "arXiv",
  "versions": []
}