{
  "abstract": "We establish a comprehensive algebraic, geometric, and physical classification of the wave closure space generated from primitive plane wave sequences on the discrete spatial lattice $\\mathbb{Z}$. Resolving lattice degeneracies via unwrapped phase spaces, we prove that the linear wave group $\\mathcal{G}$ under pointwise product, inversion, and root extraction is an infinite divisible abelian torsion group isomorphic to $(\\mathbb{Q}/\\mathbb{Z}) \\times (\\mathbb{Q}/\\mathbb{Z})$, with canonical decomposition into Prüfer $p$-groups and maximal cyclotomic value field $\\mathbb{Q}^{\\mathrm{ab}}$. Extending to coordinate powers yields the polynomial phase group $\\mathcal{G}_{\\mathrm{poly}}$, classified by integer-valued polynomials $\\operatorname{Int}(\\mathbb{Z})$. Adjoining addition yields the group algebra $\\mathbb{C}[\\mathcal{G}_{\\mathrm{poly}}]$, for which we establish the Permutation-Symmetric Phasor Superposition Theorem, factoring superpositions into collective barycentric carriers and closed Born probability envelopes. Using algebraic sieves over roots of unity, we construct idempotent projectors and complementary Not-sieve notch filters, achieving exact algebraic synthesis of both momentum and localized Kronecker position bases. Generalizing to non-abelian quaternions $\\mathcal{V}_\\mathbb{H}$, we prove a polar decomposition into scalar envelopes and $\\mathrm{SU}(2)$ spinor rotors. Identifying coordinate advance with time, biquaternion determinants intrinsically yield Minkowski spacetime $s^2 = c^2 t^2 - \\|\\mathbf{x}\\|^2$ and the Lorentz group $\\mathrm{SO}^+(1,3)$. We demonstrate emergent vacuum zero-point jitter, topological selection of rational frequencies, and correspondences with discrete qudits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23233",
  "author_total": 1,
  "authors": [
    {
      "name": "Terence R. Smith"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b3yaxcqfs312q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23233",
  "fulltext_url": null,
  "id": "b3yaxcqfs312q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23233",
  "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.NT"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7214004791",
  "pdf_url": "https://arxiv.org/pdf/2609.23233",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b3yaxcqfs312q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.23233",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23233"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Wave Numbers: Discrete Sequence Algebras, Sieve Projectors, and Dynamical Geometry",
  "title_text": "Wave Numbers: Discrete Sequence Algebras, Sieve Projectors, and Dynamical Geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/b3yaxcqfs312q",
  "venue": "arXiv",
  "versions": []
}