{
  "abstract": "The statistics of a classical flow is a problem of quantum mechanics in loop space, and not by analogy: the loop functional of any flow is the loop-ordered exponential of the covariant derivative $\\partial+\\mathrm{i}v/ν$. We follow this representation from the Navier--Stokes equation to the Euler ensemble, the fixed point of the loop equation regularized by polygons with $N$ sides. Near the turning angle $β=π$ an exact operator identity, followed by an asymptotic continuum step and without a saddle-point approximation, reduces the ensemble to two quantum pendulums, the Mathieu equation; this supersedes our earlier WKB evaluation. A Monte Carlo simulation up to $N=10^5$ confirms the reduction. Extreme turbulence has singularities at small scales, and, as in quantum field theory, we define it by this regularization and a renormalization: the leading power of $N$ is removed, and the turbulent viscosity $\\tildeν$ remains as the single physical scale. This is a proposed definition of turbulence at infinite Reynolds number; its continuum spectrum is computed under a stated convergence condition, and its identification with physical turbulence is a hypothesis. The energy spectrum is a function of $κ=k\\sqrt{\\tildeν(t+t_0)}$ that falls as $1/k$ and then as $k^{-2}$. At $κ=\\sqrt2$ it has a weak nonanalyticity: its second derivative diverges logarithmically. At a fixed wave number the decay is therefore singular at the finite time $t_*=2/(\\tildeνk^2)-t_0$, when $t_*>0$, and the vorticity correlations oscillate in space with the period $\\sqrt2\\,π\\sqrt{\\tildeν(t+t_0)}$. Of the limits of the ensemble with structure, only the pendulum limit is taken at a critical point and has a bulk dissipation that vanishes with the viscosity. A companion paper compares it with simulations and experiments.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.05745",
  "author_total": 1,
  "authors": [
    {
      "name": "Alexander Migdal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ceg4vbgcgcyqf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.05745",
  "fulltext_url": null,
  "id": "ceg4vbgcgcyqf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.05745",
  "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",
      "math.NT"
    ],
    [
      "arxiv",
      "nlin.CD"
    ],
    [
      "arxiv",
      "physics.flu-dyn"
    ],
    [
      "openalex",
      "2206|Fluid Dynamics and Turbulent Flows"
    ]
  ],
  "openalex": "W7167662880",
  "pdf_url": "https://arxiv.org/pdf/2607.05745",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ceg4vbgcgcyqf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:18Z",
      "native_id": "2607.05745",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.05745"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.flu-dyn",
      "subfield": "fluid-dynamics"
    },
    {
      "field": "physics",
      "origin": "arxiv:nlin.CD",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Reduction of Decaying Navier--Stokes Turbulence to the Quantum Pendulum",
  "title_text": "Reduction of Decaying Navier--Stokes Turbulence to the Quantum Pendulum",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/ceg4vbgcgcyqf",
  "venue": "arXiv",
  "versions": []
}