{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRIC_GROUP_THEORY-0134",
  "title": "Exact Heisenberg Determinant Limits and Tower-Dependent Homological Torsion",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We compute the exact finite determinant limit for Kammeyer's element f=1-2a+2b in the integral group ring of the discrete Heisenberg group. Along the congruence tower modulo powers of any fixed odd prime p, the normalized Fuglede-Kadison determinant converges to 2^(1/(p+1)), whereas the infinite determinant is 2. An exact central-character factorization and an integral-lattice tail bound justify the ordinary limit without passing an unbounded logarithm through weak spectral convergence. Attaching a two-sphere and a three-cell to the Heisenberg nilmanifold realizes the same discrepancy in integral homology. The resulting finite connected three-dimensional CW complex has determinant-L2-acyclic universal cover and L2-torsion log 2, but its normalized alternating integral homology torsion tends to log 2/(p+1). This limit therefore depends on the residual tower of a single space and refutes the unrestricted finite-CW formulation of modified homological torsion approximation in Hughes and Luck, arXiv:2510.20959v2, Conjecture 1.2. The space is neither aspherical nor a closed manifold; no aspherical-manifold conjecture or original AIM 11.1 closure is claimed. Kammeyer's underlying counterexample and infinite determinant, Deninger's determinant formula, Luck's nilmanifold vanishing theorem, and Boschheidgen's representation-theoretic ingredients are explicitly credited. Two unchanged standard-library exact checkers support the displayed finite calculations; the general conclusions rest on the analytic proof, not finite sampling. This English preprint is AI-assisted, originating-researcher self-audited, and unrefereed. Novelty remains undetermined. No independent human review, proof-assistant verification, guaranteed indexing, or absolute-priority claim is made.",
  "result_type": "COMPLETE_SCOPED_COUNTEREXAMPLE_AND_EXACT_LIMIT",
  "categories": [
    "math.AT",
    "math.GR",
    "math.OA"
  ],
  "keywords": [
    "Heisenberg group",
    "Fuglede-Kadison determinant",
    "homological torsion",
    "residual towers",
    "counterexample"
  ],
  "manuscript_version_date": "2026-10-02",
  "publication_date": "2026-10-02",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-02",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometric-group-theory-0134/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometric-group-theory-0134/paper.pdf?v=1722a5873be5",
  "doi": "10.5281/zenodo.23103819",
  "zenodo_record_url": "https://zenodo.org/records/23103819",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For H=H_3(Z), f=1-2a+2b and every fixed odd prime p, the normalized determinants on H_3(Z/p^i Z) converge to 2^(1/(p+1)). An explicitly constructed finite connected three-dimensional CW complex Y has pi_1(Y)=H, determinant-L2-acyclic universal cover and L2-torsion log2, while its normalized alternating integral homology torsion along that p-tower converges to log2/(p+1). This refutes the finite-CW formulation of Hughes--Luck arXiv:2510.20959v2 Conjecture1.2. Y is neither aspherical nor a closed manifold, so neither AIM11.1 nor the frozen record is counted resolved. Kammeyer's example/infinite determinant and Boschheidgen's representation ingredients are credited. AI-assisted, self-audited, unrefereed preprint. No independent human review, proof-assistant formalization or absolute-priority certification is claimed.",
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      "sha256": "1722a5873be585b127d9a8031016612a64cf33fc4b30ae2de1f0ed073c4b8b8e"
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    "source.zip": {
      "sha256": "2b24d39ed2aa55bcfab06b997eb7292328e60f3c842f837cac958ff07ccd7683"
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    "verification_report.md": {
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text.",
  "original_problem_resolved": false,
  "theorem_scope_resolved": true,
  "whole_source_record_resolved": false,
  "retained_public_get_count": 4,
  "retained_metadata_stable_before_after": false,
  "adjacent_conjecture_refuted": true,
  "adjacent_source_in_frozen_corpus": false,
  "frozen_record_closure_increment": 0
}