{
  "abstract": "We consider the RP$^2$ model on the two-dimensional square lattice with nearest-neighbor interactions. It is obtained by replacing the spin in the standard classical Heisenberg model with a director, which does not distinguish two opposite directions. Unlike the standard classical Heisenberg model or the O(3) nonlinear sigma model, the RP$^2$ (and its cousin RP$^3$) model allows point vortex defects classified by $π_1(RP^2)= Z_2$ in two dimensions. It is tempting to expect the $Z_2$ vortices to induce a phase transition analogous to the well-established Berezinskii-Kosterlitz-Thouless transition in the O(2) model (XY model) in two dimensions. Indeed, the possibilities of a critical phase or a critical point with power-law correlations of the order parameter in the RP$^2$ and RP$^3$ models have been discussed in the literature. However, the challenging nature of the problem has prevented a definitive conclusion on the phase diagram so far. In this paper, we report a rigorous mathematical proof by Artificial Intelligence (ChatGPT 6 Astra) that the absolute value of the two-point correlation function of order parameters at distance $r$ in the RP$^2$ and RP$^3$ models on the two-dimensional square lattice is bounded above by $C_β\\exp{(-c_β[\\log(2+r)]^2)}$ for any finite inverse temperature $β\\ge0$, with positive constants $C_β$ and $c_β$ depending only on $β$. Namely, the correlation function decays faster than any power of $r$. The proof is formalized and verified in Lean; the formalization is publicly available at doi:10.5281/zenodo.23548847 . A heuristic argument based on a one-parameter effective theory ansatz at large distances combined with the rigorous correlation bound suggests (but does not prove) that the $Z_2$ vortices are always unbound at any nonzero temperature.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41256",
  "author_total": 1,
  "authors": [
    {
      "name": "Masaki Oshikawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6mhd12dzs05md.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41256",
  "fulltext_url": null,
  "id": "6mhd12dzs05md",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41256",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "hep-lat"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41256",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6mhd12dzs05md.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:15Z",
      "native_id": "2610.41256",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41256"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-lat",
      "subfield": "high-energy"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Faster-than-power-law decay of order parameter correlations in the two-dimensional RP$^2$ model: a rigorous result by vibe-proving",
  "title_text": "Faster-than-power-law decay of order parameter correlations in the two-dimensional RP^2 model: a rigorous result by vibe-proving",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/6mhd12dzs05md",
  "venue": "arXiv",
  "versions": []
}