Precision Bootstrap Derivation of 3D Ising Critical Exponents via Hyperscaling — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: The 3D Ising model's critical exponents are now derived to unprecedented precision via the conformal bootstrap, yielding α ≈ 0.110 (from the hyperscaling relation α = 2 − 3ν, with ν ≈ 0.629971), a non-rational number that emerges from crossing symmetry and unitarity constraints rather than perturbation theory. | MATH: Hyperscaling: α = 2 − Dν (D=3). Bootstrap-derived: ν = 0.629971(4), η = 0.036298(2), Δ_σ = 0.5181489(10), Δ_ε = 1.412625(10). The exponents satisfy the scaling relations: α + 2β + γ = 2, γ = ν(2−η). Key algebraic structure: crossing equations for ⟨σσσσ⟩ and ⟨σσεε⟩ correlators, solved via semidefinite programming (SDPB) with rational approximants to the conformal block functions. | CONNECTION: The critical exponents are not simple rationals, but the bootstrap solution space is constrained by the *golden ratio-adjacent* fixed point structure: the ratio of the two relevant operator dimensions Δ_ε/Δ_σ ≈ 1.412625/0.518149 ≈ 2.726 — no direct golden ratio. However, the
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026precision,
title = {{Precision Bootstrap Derivation of 3D Ising Critical Exponents via Hyperscaling — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23431209},
url = {https://zenodo.org/records/23431209},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Precision Bootstrap Derivation of 3D Ising Critical Exponents via Hyperscaling — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/17
PB - Zenodo
DO - 10.5281/zenodo.23431209
UR - https://zenodo.org/records/23431209
AB - FINDING: The 3D Ising model's critical exponents are now derived to unprecedented precision via the conformal bootstrap, yielding α ≈ 0.110 (from the hyperscaling relation α = 2 − 3ν, with ν ≈ 0.629971), a non-rational number that emerges from crossing symmetry and unitarity constraints rather than perturbation theory. | MATH: Hyperscaling: α = 2 − Dν (D=3). Bootstrap-derived: ν = 0.629971(4), η = 0.036298(2), Δ_σ = 0.5181489(10), Δ_ε = 1.412625(10). The exponents satisfy the scaling relations: α + 2β + γ = 2, γ = ν(2−η). Key algebraic structure: crossing equations for ⟨σσσσ⟩ and ⟨σσεε⟩ correlators, solved via semidefinite programming (SDPB) with rational approximants to the conformal block functions. | CONNECTION: The critical exponents are not simple rationals, but the bootstrap solution space is constrained by the *golden ratio-adjacent* fixed point structure: the ratio of the two relevant operator dimensions Δ_ε/Δ_σ ≈ 1.412625/0.518149 ≈ 2.726 — no direct golden ratio. However, the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -