Eur. Phys. J. B (2026) 99: 8
https://doi.org/10.1140/epjb/s10051-025-01109-8

Research - Condensed Matter

Certified three-dimensional Ising critical exponents from harmonized bootstrap series and interval Monte Carlo

Peking University, Beijing, China

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Received: 23 September 2025
Accepted: 12 December 2025
Published online: 20 January 2026

Abstract

Rigorous, machine-checkable intervals for the critical exponents of the three-dimensional Ising universality class are established through a certificate-first methodology. Specifically, the objective is mathematical certainty for β, ν, η, and ω—rigorous enclosures rather than point estimates. This approach unifies conformal bootstrap, Borel-conformal resummation of perturbative and high-temperature series, and interval Monte Carlo under a single acceptance rule with machine-checkable certificates. Bootstrap islands for the leading operators are reconstructed using exact rational functionals, accompanied by feasibility or refutation logs, then mapped to exponents via exact scaling relations. Series expansions are resummed via Borel-conformal maps under explicit Gevrey control, yielding tail bounds and parameter boxes for rigorous enclosures. Monte Carlo outputs are enclosed using interval and ball arithmetic, combined with autocorrelation-aware concentration inequalities and finite-size scaling that incorporates verified crossings and sandwich fits. A topology-aware reconciliation enforces equality on overlaps and prevents cyclic tightening; acceptance requires unanimity across all anchors, subject to veto by any certified refutation. The main result is a theorem-level statement: the exponent set lies in the nonempty intersection of three independently certified boxes. The certified intervals are $\mathrm{\beta }\in [\text{0.32635,0.32650}]$, $\mathrm{\nu }\in [\text{0.62990,0.63005}]$, $\mathrm{\eta }\in [\text{0.03620,0.03640}]$, and $\mathrm{\omega }\in [\text{0.829,0.833}]$ (closed, outward-rounded). These intervals match the sharpest values in the literature while providing solver-independent proofs and reproducible artifacts; they offer a transferable blueprint for other universality classes.