{
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  "problem_id": "vibemathed:complete-rational-classification-of-fifth-order-autocorrelation-ambiguities-on-u",
  "title": "Complete rational classification of fifth-order autocorrelation ambiguities on $U_{30}$",
  "canonical_statement": "For rational-valued signals $f,g:C_{30}\\to\\mathbb Q$ with exact Fourier support $U_{30}$, equality of autocorrelations through order five is completely classified. After translating $g$, there are $\\alpha\\in\\mathbb Q(\\zeta_{30})^\\times$ and $z\\in\\mathbb Q(\\zeta_6)^\\times$, with $z\\bar z=1$, such that\n$$\n\\widehat f(u)=\\sigma_u(\\alpha),\\qquad\n\\widehat g(u)=\\sigma_u(z\\alpha)\n$$\nfor every $u\\in U_{30}$. Conversely, every such pair, extended by zero off $U_{30}$, is rational-valued and agrees through order five. Normalized parameters are translation-equivalent exactly modulo $\\mu_6$, and the sixth-order data agree exactly when $z^6=1$.",
  "plain_summary": "VibeMathed reports this item as candidate. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "candidate",
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      "id": "vibemathed:complete-rational-classification-of-fifth-order-autocorrelation-ambiguities-on-u:event",
      "problem_id": "vibemathed:complete-rational-classification-of-fifth-order-autocorrelation-ambiguities-on-u",
      "type": "proved",
      "status": "candidate",
      "occurred_at": "2026-08-07T00:00:00.000Z",
      "title": "Autocorrelation Phase Lattices on Cyclic Groups: Unit Supports and the 2pq Orbit-Stable Classification",
      "summary": "Agulnick and Busick-Warner exhibited a family of fifth-order ambiguities on the exact unit support $U_{30}$ and conjectured that it was not a complete classification because it did not use the full field $\\mathbb Q(\\zeta_{30})$. This work proves the complete classification. The larger field enlarges the common amplitude $\\alpha$, while every relative ambiguity remains a norm-one parameter in $\\mathbb Q(\\zeta_6)$. The result is a specialization of a theorem for every exact unit support $U_{6m}$.\n\nThe entry does not claim a complete parametrization for arbitrary supports: on the 255 automorphism-stable supports treated elsewhere in the paper, the broader result is a closing-degree classification. It does not treat noisy data or noncyclic groups, and it makes no novelty, priority, or firstness claim.",
      "ai_contribution": "ai_discovered",
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      "model": "GPT-5.6 Sol (Codex)",
      "provider": "OpenAI",
      "attempted_at": "2026-08-07T00:00:00.000Z",
      "human_collaborators": [],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "Following the self created protocol PAPP, that you can find here:https://github.com/aconsciousfractal/Gate-Disciplined-Computational-Mathematics\n GPT-5.6 Sol, operating through Codex under my direction, developed the central relation-lattice reduction, the phase-ratio normal form, and the Galois-descent argument that confines the relative phase to the norm-one torus in $\\mathbb Q(\\zeta_6)$. It also generated exact symbolic replays and the verification package. I selected the problem and scope, controlled the literature and claim boundary, directed repeated adversarial reviews, checked the mathematical outputs, and revised the manuscript after each finding. An earlier AI-agent package supplied preliminary computational observations; the workflow independently rederived and checked them before use.",
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          "url": "https://github.com/aconsciousfractal/FCIG-Autocorrelation-Phase-Lattices-on-Cyclic-Groups",
          "kind": "code",
          "license": null
        },
        {
          "label": "Agulnick and Busick-Warner, Higher-Order Autocorrelations on Finite Abelian Groups (arXiv:2604.13310)",
          "url": "https://arxiv.org/abs/2604.13310",
          "kind": "paper",
          "license": null
        },
        {
          "label": "Gate-Disciplined Computational Mathematics - the author's own working protocol",
          "url": "https://github.com/aconsciousfractal/Gate-Disciplined-Computational-Mathematics",
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      ],
      "sources": [
        {
          "label": "Autocorrelation Phase Lattices on Cyclic Groups: Unit Supports and the 2pq Orbit-Stable Classification",
          "url": "https://github.com/aconsciousfractal/FCIG-Autocorrelation-Phase-Lattices-on-Cyclic-Groups/blob/b4bbfccdd508caa4a5a6e145abd78ac082195c16/paper/Autocorrelation_Phase_Lattices_on_Cyclic_Groups.pdf",
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          "label": "VibeMathed: Complete rational classification of fifth-order autocorrelation ambiguities on $U_{30}$",
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      "independence": "unknown"
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  "verifications": [
    {
      "id": "vibemathed:complete-rational-classification-of-fifth-order-autocorrelation-ambiguities-on-u:verification",
      "problem_id": "vibemathed:complete-rational-classification-of-fifth-order-autocorrelation-ambiguities-on-u",
      "solution_event_id": "vibemathed:complete-rational-classification-of-fifth-order-autocorrelation-ambiguities-on-u:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "No named independent domain expert has endorsed the theorem. The repository carries exact symbolic checks, fail-closed verification scripts, frozen manifests and an adversarial review report, but those are author-side and agent-side assurance, so this stays Unreviewed and Candidate.\n\nThis site ran its own checks, written from the statement rather than from the repository's scripts. Building a pair from a chosen $\\alpha$ and $z$: both inverse transforms are rational at all 30 points, the support is exactly $U_{30}$, the pair agrees through order five and differs at order six, and replacing $z$ by a sixth root of unity restores order-six agreement - the claimed $z^6=1$ boundary, exactly. The phase lattice was recomputed independently by Smith normal form: $\\mathbb Z^8/L$ has free rank 1 through order five and rank 0 at order six, so a one-parameter ambiguity survives order five and dies at six. And the Agulnick-Busick-Warner pair itself fits the classification - its Fourier ratio is Galois-equivariant with $z+\\bar z=13/7$ and $z\\bar z=1$, so $z=(13\\pm3\\sqrt{-3})/14$ lies in $\\mathbb Q(\\zeta_6)$.\n\nNot checked: completeness itself, which is the novelty. The converse direction, the lattice skeleton and the known example are all consistent with it without establishing it.",
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          "label": "VibeMathed: Complete rational classification of fifth-order autocorrelation ambiguities on $U_{30}$",
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          "label": "Autocorrelation Phase Lattices on Cyclic Groups: Unit Supports and the 2pq Orbit-Stable Classification",
          "url": "https://github.com/aconsciousfractal/FCIG-Autocorrelation-Phase-Lattices-on-Cyclic-Groups/blob/b4bbfccdd508caa4a5a6e145abd78ac082195c16/paper/Autocorrelation_Phase_Lattices_on_Cyclic_Groups.pdf",
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      "label": "Autocorrelation Phase Lattices on Cyclic Groups: Unit Supports and the 2pq Orbit-Stable Classification",
      "url": "https://github.com/aconsciousfractal/FCIG-Autocorrelation-Phase-Lattices-on-Cyclic-Groups/blob/b4bbfccdd508caa4a5a6e145abd78ac082195c16/paper/Autocorrelation_Phase_Lattices_on_Cyclic_Groups.pdf",
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      "generated_at": "2026-09-13T16:28:40.178Z"
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    "The source status is candidate and must not be represented as solved.",
    "VibeMath has not independently verified the mathematical claim.",
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    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
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  "generated_at": "2026-09-13T16:28:40.178Z"
}