{
  "abstract": "The macroscopic collective motion of active continuum on curved manifolds is conventionally addressed through perturbative dynamic renormalization or finite-element simulations, often obscuring the underlying geometric mechanisms. Here, an exact algebraic framework is established to reformulate the active phase transition on hyperbolic spaces $\\mathbb{H}^2$. By rigorously expanding the covariant Navier-Stokes-like equations and applying the Weitzenböck identity, we derive the exact critical threshold $α_c = \\frac{5}{4}Dκ^2$ for macroscopic polarization, which is dictated by the geometric mass gap of the Hodge-de Rham Laplacian. We strictly define the parameter subspace $α= 2Dκ^2$ where the topological free energy reaches the Bogomolny-Prasad-Sommerfield (BPS) limit. This enables the reduction of the complex velocity field to Blaschke products via Möbius gauge symmetry. The flat-space limit ($κ\\to 0$) exactly degenerates to the topological phase of the classical O(2) model, demonstrating that the constant negative curvature acts as an un-perturbative infrared regularization for non-linear amplitude saturation. Furthermore, mapping the non-variational active convective modes onto the defect translational zero-modes yields an intrinsically non-reciprocal interaction matrix. By analytically extending the singular integral operator of the dynamically condensed defect ring, we identify a macroscopic second-order exceptional point (EP2) characterized by a strictly algebraic dynamic scaling law $τ\\sim |Δν|^{-1/2}$. This closed-form theoretical paradigm provides exact solutions for geometric frustration and non-Hermitian topology in soft mechanics.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03500",
  "author_total": 1,
  "authors": [
    {
      "name": "Yu-Xin Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fnqwf8mttsgbx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03500",
  "fulltext_url": null,
  "id": "fnqwf8mttsgbx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03500",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.soft"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "3107|Quantum Mechanics and Non-Hermitian Physics"
    ]
  ],
  "openalex": "W7208777305",
  "pdf_url": "https://arxiv.org/pdf/2609.03500",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fnqwf8mttsgbx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:31Z",
      "native_id": "2609.03500",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03500"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.soft",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Exact Scaling Laws and Non-Hermitian Topological Phase Transitions of Active Continuum on Hyperbolic Manifolds",
  "title_text": "Exact Scaling Laws and Non-Hermitian Topological Phase Transitions of Active Continuum on Hyperbolic Manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/fnqwf8mttsgbx",
  "venue": "arXiv",
  "versions": []
}