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Faster-than-power-law decay of order parameter correlations in the two-dimensional RP2 model: a rigorous result by vibe-proving

A mix of human-written and AI-generated textHuman understanding: some partsmath-ph — Mathematical Physicsmath.MP — Mathematical Physics

Contributed by Masaki Oshikawa ↗

Version 1 / Oct 22, 2026 / CC BY 4.0

Abstract

We consider the RP2RP^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 RP2RP^2 (and its cousin RP3RP^3) model allows point vortex defects classified by π1(RP2)=Z2\pi_1(RP^2)=Z_2 in two dimensions. It is tempting to expect the Z2Z_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 RP2RP^2 and RP3RP^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 rr in the RP2RP^2 and RP3RP^3 models on the two-dimensional square lattice is bounded above by Cβexp⁡{−cβ[log⁡(2+r)]2}C_\beta \exp\{-c_\beta[\log(2+r)]^2\} for any finite inverse temperature β≥0\beta\ge0, with positive constants CβC_\beta and cβc_\beta depending only on β\beta. Namely, the correlation function decays faster than any power of rr. The proof is formalized and verified in Lean; the formalization is publicly available at \href{https://doi.org/10.5281/zenodo.23548847}{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 Z2Z_2 vortices are always unbound at any nonzero temperature.

Provenance statement

The human author (MO) put the question to AI (ChatGPT 6 Astra), supplying the OpenAI release (the paper in PDF and Lean formalization on GitHub) on the exponential decay of correlations in the O(n) model [12]. The proof was entirely generated by ChatGPT 6 Astra. MO then asked the AI to provide a full formalization of the proof in Lean. During the process, the bound was improved from the original bound of exp[−cβ log r log log r] to the present exp[−cβ(log r)^2]. The Lean formalization was then checked with ChatGPT 6 Astra in a separate account, and also with Claude Opus 5.5. The compilation, dependency and kernel-replay checks are recorded separately in Sec. 5. The human author (MO) wrote the title, the abstract, and Sections 1, 6, and 7. The Lean code and proof exposition were generated by AI (ChatGPT 6 Astra). The goal setting of the heuristic arguments in Section 3 was given by MO; details were generated by ChatGPT 6 Astra, and further improved by Claude Opus 5.5. MO substantially edited Section 3 and the historical overview in Sec. 2, and decided the overall structure of the paper. The paper was then finalized by MO after the critical review by ChatGPT 6 Astra and Claude Opus 5.5. As the sole human author, MO is responsible for the entire paper.

Tools used

OpenAI
ChatGPTVersion 6 Astra
Anthropic
Claude OpusVersion 5.5

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Version history

  1. v1Submitted by Masaki OshikawaInitial depositCurrentOct 22, 2026