{
  "schema_version": 1,
  "problem_number": "AIM-COMPUTATION-0090",
  "title": "Exact Primary Kernels for Groups Generated by Linear Polynomials",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let p be prime, let S be a nonempty subset of F_p with d elements, and let h be a nonconstant polynomial with no zero in S. Let G_h be the subgroup of (F_p[X]/(h))^* generated by the classes of X-s for s in S. We determine the order, invariant factors, and integer relation lattice of the kernel K_h of reduction to G_rad(h). If h is the product of distinct monic irreducible factors f_i to powers e_i, put L_a = sum_i deg(f_i)(ceil(e_i/p^a)-1), tau = min{a: L_a < d}, and lambda = L_tau. Then K_h is isomorphic to (Z/p^(tau+1)Z)^lambda times (Z/p^tau Z)^(d-lambda), with trivial factors omitted. Its order is p^(sum_a min(d,L_a)). The proof uses logarithmic differentiation, exact Frobenius descent, and one final linear relation space. Consequently log|G_h|/d diverges uniformly as d grows whenever deg(h)-deg(rad(h)) is at least a fixed positive fraction of d. This is a complete primary-kernel classification, not a solution of the reduced, squarefree, or irreducible AKS-related growth question. Standard methods and prior bounds are credited; this unrefereed preprint makes no absolute novelty or priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.NT",
    "math.GR"
  ],
  "keywords": [
    "AIM-COMPUTATION-0090",
    "finite rings",
    "principal units",
    "primary kernel",
    "Frobenius",
    "invariant factors",
    "relation lattice"
  ],
  "manuscript_version_date": "2026-10-24",
  "publication_date": "2026-10-24",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-24",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-computation-0090/",
  "pdf_url": "https://eulersolve.org/papers/aim-computation-0090/paper.pdf?v=2f89daa48614",
  "doi": "10.5281/zenodo.23607000",
  "zenodo_record_url": "https://zenodo.org/records/23607000",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every prime p, nonempty S contained in F_p, and nonconstant h with no zero in S, the kernel of G_h to G_rad(h) has the stated exact order, two-adjacent-exponent invariant factors, and explicit integer relation lattice. The theorem covers all irreducible degrees and multiplicities under these hypotheses. The broader AIM growth question remains partially solved. This theorem determines the primary kernel, not the reduced subgroup, and gives no new information for squarefree moduli. Standard methods and prior bounds are credited; novelty and priority are not certified. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
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    "source.zip": {
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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."
}