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Low-temperature transfer-gap asymptotics of the square-lattice O(4) model

The question is whether for the nearest-neighbor square-lattice O(4) model, the full Osterwalder–Schrader transfer gap, including rotation-invariant local-observable sectors and measured per original lattice time step, is asymptotic to 32 exp(π/4−1/2) √β exp(−πβ) as β tends to infinity.

Why it matters and what counts as progress

Why it matters: It seeks the exact lattice prefactor of the O(4) mass scale, beyond its leading exponential behaviour.

What counts as progress: Prove the exact low-temperature gap asymptotic including its lattice-time normalization and all observable sectors, rather than only its exponential scale.

How the problem developed

  1. 1988 · Numerical context [1]

    M. Heller Reported numerical renormalization-group evidence for the mass scale of the two-dimensional O(4) model; this did not give an exact full-lattice prefactor.

  2. 1997 · Continuum context [2]

    Martin Lüscher, Peter Weisz and Ulli Wolff Determined the mass gap in the O(n) sigma model in the continuum scaling context, distinct from the exact lattice transfer-gap asymptotic.

  3. 5 Oct 2026 · Asymptotic claim [3]

    OpenAI Claimed an exact low-temperature asymptotic for the full Osterwalder–Schrader transfer gap of the nearest-neighbor square-lattice O(4) model, including rotation-invariant local-observable sectors.

AI contribution

Sources

  1. M. Heller, A determination of the mass spectrum of the O(4) nonlinear sigma model
  2. Martin Lüscher, Peter Weisz and Ulli Wolff, A determination of the mass gap in the O(n) sigma model
  3. OpenAI O(4) exact-mass manuscript
  4. Exact mass asymptotics for the two-dimensional O(4) lattice model
  5. OpenAI mathematics release
  6. Scope of the O(n) Lean proof
  7. Hasenfratz–Maggiore–Niedermayer mass prediction
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