{
  "abstract": "Protein function may depend not only on endpoint conformations but also on the ordered deformation histories through which they are reached. This distinction is relevant to allostery, conformational switching, mutation-induced rearrangements, and epistatic effects, where different perturbation sequences may produce similar visible structures while retaining distinct internal transport histories. Current state-centered or endpoint-centered representations do not always preserve this order-sensitive information. The practical motivation is therefore to provide a foundation for future descriptors of protein deformation trajectories that can distinguish ordered histories even when endpoint conformations are similar. We propose a deformation-first geometric framework based on quaternionic frame transport along the protein backbone. Local backbone frames are lifted to quaternionic variables, with infinitesimal rotation encoded by \\(Ω(\\ell)=2\\,q(\\ell)^{-1}\\partial_\\ell q(\\ell).\\) Ordered concatenation of admissible deformation paths generates a noncommutative transport algebra, recording that deformation A followed by B need not be equivalent to B followed by A. From this ordered transport layer, we construct a spectral-response layer comprising a global Dirac-type operator, local spectral germs, a renormalized spectral density, and a mixed response form. A minimal realization on an idealized \\(α\\)-helix shows how localized pitch and bending perturbations can yield similar endpoint descriptors while producing a nonzero endpoint-derived ordered-transport discrepancy. At the formal level, the framework separates an order-sensitive transport-memory sector, lost under a commutative shadow, from a spectral-response sector that remains visible.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.29101",
  "author_total": 9,
  "authors": [
    {
      "name": "Xiaoting Chen"
    },
    {
      "name": "Chon-Fai Kam"
    },
    {
      "name": "Yu Li"
    },
    {
      "name": "David Medina-Ortiz"
    },
    {
      "name": "Cedric Damour"
    },
    {
      "name": "Jean Pierre Chabriat"
    },
    {
      "name": "Alain Miranville"
    },
    {
      "name": "Miloud Bessafi"
    },
    {
      "name": "Frederic Cadet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/21pdmaz62e8gv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.29101",
  "fulltext_url": null,
  "id": "21pdmaz62e8gv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.29101",
  "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",
      "q-bio.BM"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2607.29101",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/21pdmaz62e8gv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:21Z",
      "native_id": "2607.29101",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.29101"
    }
  ],
  "subjects": [
    {
      "field": "biochemistry",
      "origin": "arxiv:q-bio.BM",
      "subfield": "biophysics"
    }
  ],
  "title": "Quaternionic Response Geometry for Proteins: Toward a Noncommutative Theory of Ordered Deformations",
  "title_text": "Quaternionic Response Geometry for Proteins: Toward a Noncommutative Theory of Ordered Deformations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:21Z",
  "url": "https://research.draconitic.com/paper/21pdmaz62e8gv",
  "venue": "arXiv",
  "versions": []
}