{
  "abstract": "We formalize the algebraic structure of rational polynomial phase sequences (discrete chirps) over the integer lattice $\\mathbb{Z}$. By constructing a convolution algebra generated by polynomial phases with coefficients in $\\mathbb{Q}$, we establish a rigorous operational calculus for discrete difference equations. We prove an exact orthogonality relation for these sequences over finite periodic windows, yielding a generalized discrete Fourier isometry. Furthermore, we demonstrate that the continuum limit of this discrete framework can be rigorously defined via the inductive limit of finite-dimensional $C^*$-algebras (an AF-algebra). This framework provides a rigorous algebraic substrate for finite-mode lattice approximations in quantum mechanics and the construction of mutually unbiased bases in finite-dimensional quantum information theory.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.01650",
  "author_total": 1,
  "authors": [
    {
      "name": "Terence R. Smith"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a5cxghe5n2n6v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.01650",
  "fulltext_url": null,
  "id": "a5cxghe5n2n6v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.01650",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7172443888",
  "pdf_url": "https://arxiv.org/pdf/2608.01650",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a5cxghe5n2n6v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2608.01650",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.01650"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "Algebraic Structure and Orthogonality Properties of Rational Polynomial Phase Rings on Discrete Lattices",
  "title_text": "Algebraic Structure and Orthogonality Properties of Rational Polynomial Phase Rings on Discrete Lattices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/a5cxghe5n2n6v",
  "venue": "arXiv",
  "versions": []
}