{
  "abstract": "Let $N=\\binom n2=\\dim\\mathrm{SO}(n)$. We prove that the coordinate-plane Kac walk on $\\mathrm{SO}(n)$ has total-variation mixing time of order $N$: for every fixed $0<\\varepsilon<1$, \\[ t_{\\mathrm{mix}}^{(n)}(\\varepsilon)=Θ_\\varepsilon(n^2). \\] The lower bound is the dimensional singularity obstruction before $N$ steps. The upper bound removes the final logarithm from the previously known $O(n^2\\log n)$ estimate. The proof combines the discrete Malliavin coupling and low-degree pseudo-mixing inputs with a new log-free analysis of the derivative shells. Its static core is a circuit-anchored, arbitrary-spectrum root/pass identity for the physical five-box prime. Keeping one normalization base per original circuit permits simultaneous scalar regluing without paying for artificial cuts. Its temporal core is an exact chronological calculus: passive singleton runs acquire a coboundary/Riesz gain, while root-interrupted components are allocated by vertex-labelled packets before absolute values are taken. The curvature split into pure-Weyl and Ricci parts is kept at its physical tensor type. All-Weyl packets retain a full $N^{-1}$ resource; mixed packets contain a typed $O(n^{-1/2})$ Ricci debit; and the final packetless Ricci cell is closed by a joint invariant-column estimate on its two root-hit circuits and an exact causal restoration of the marked root time. These estimates yield an $O(n)$ squared first-derivative shell and a summable all-order marked-shell expansion through logarithmic degree. The resulting score energy is $O(n/c^2)$ after $cN$ steps. A weighted submersion integration-by-parts argument and the Haar log-Sobolev inequality then give the uniform total-variation upper bound. No cutoff profile or cutoff window is asserted.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29403",
  "author_total": 1,
  "authors": [
    {
      "name": "Tianle Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3je3twp7b9xxp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29403",
  "fulltext_url": null,
  "id": "3je3twp7b9xxp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29403",
  "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.PR"
    ],
    [
      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
    ]
  ],
  "openalex": "W7204953190",
  "pdf_url": "https://arxiv.org/pdf/2608.29403",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3je3twp7b9xxp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2608.29403",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29403"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Kac's Walk on Rotation Matrices Mixes in $\\boldsymbol{Θ(n^2)}$ Steps: A Proof Discovered with AI",
  "title_text": "Kac's Walk on Rotation Matrices Mixes in Θ(n^2) Steps: A Proof Discovered with AI",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/3je3twp7b9xxp",
  "venue": "arXiv",
  "versions": []
}