{
  "abstract": "A system with two periodic directions carries two commuting holonomies \\(a=(u,v)\\in\\R^2/\\Z^2\\). We determine their distinguished values while separating lattice, arithmetic, and observable effects. Maximizing the lowest twisted eigenvalue places \\(a\\) at a deep hole of the momentum lattice. For every rectangular torus the maximizer is antiperiodic, so complex multiplication is sufficient for torsion optima but not necessary. Let \\(G_τ^{-1}\\) be the dual metric and \\(D_τ(a)\\) the normalized zeta determinant of the twisted Laplacian. At the rotation-fixed deep holes of the square and hexagonal lattices, symmetry gives the exact determinant response \\(-\\operatorname{Hess}_a\\log D_τ=2π(\\Imτ)G_τ^{-1}\\). With spectral wavenumber \\(κ=2π\\), the lowest manifolds are fourfold and threefold, with gaps \\(2κ^2\\) and \\(4κ^2/3\\); phase errors split them linearly while their centroids remain stationary. We then give a finite-device realization: an \\(8\\times8\\) microring lattice closed by two phase-controlled seams. At a reported coupling scale of \\(16\\) GHz, its exact square-lattice spectrum has a \\(17.32\\) GHz shell gap and a \\(1.92\\) GHz doublet separation for a \\(0.1\\) holonomy error; a triangular-link configuration gives a threefold qutrit manifold with an \\(18.11\\) GHz gap. This is a quantitative spectroscopy proposal, not a claim of topological protection or a completed device.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Quantum Rep. 2026, 8(3), 88",
  "arxiv": "2608.16501",
  "author_total": 1,
  "authors": [
    {
      "name": "Michel Planat"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fy3n606z9x2kv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.16501",
  "fulltext_url": null,
  "id": "fy3n606z9x2kv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.16501",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2608.16501",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fy3n606z9x2kv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:11Z",
      "native_id": "2608.16501",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.16501"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Symmetry-Fixed Holonomies and Spectral Isolation in Two-Cycle Photonic Geometries A Square Parent Manifold for a Qubit and a Hexagonal Qutrit Manifold",
  "title_text": "Symmetry-Fixed Holonomies and Spectral Isolation in Two-Cycle Photonic Geometries A Square Parent Manifold for a Qubit and a Hexagonal Qutrit Manifold",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/fy3n606z9x2kv",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.3390/quantum8030088",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}