Spectral structure of a kinetic Ising chain at the rate θ*: an isolated one-particle branch, certified bounds on the gap, and the edge spectrum

Abstract. We study the single-spin-flip dynamics of the one-dimensional Ising model with the flip rates (1/2) θ*^(s_x (s_{x−1} + s_{x+1})), where θ* = √2 − 1, on the infinite chain, on the half-line with a fixed end spin, and on finite paths with fixed end spins. For the infinite chain we show that, for momenta near zero, the generator has an isolated branch of one-particle type, that this branch is real-analytic in the momentum, and that it is quadratic at its bottom, which is the gap λ_∞ of the generator. We prove that 0.075426743 ≤ λ_∞ ≤ 0.075427472. For the half-line we prove that the generator has no spectrum below λ_∞ on the orthogonal complement of the constants. For the path with the sites 0, …, k we prove that the smallest eigenvalue on the orthogonal complement of the constants is at least λ_∞ for 2 ≤ k ≤ 24, and that its upper limit as k → ∞ is at most λ_∞. The rates contain no small parameter. The proofs rest on a pointwise comparison with the Glauber rates; the numerical statements are certified by computations in exact arithmetic. Version. Version 1.0 (October 2026). Files. The record holds 14 files: the paper (paper-v1.0.pdf); the bundle of the three certificates of Appendix A (gibbs-certificates.zip), which a single command runs; a description of the files (README.md); the list of the places in which the English proofs differ in order or grouping from the reviewed sources, with the changes of notation (DEVIATIONS.md, in Japanese); the Japanese text of the wording of the disclosure (disclosure-ja.md) and the record of the facts on the role of AI systems (gb-ai-disclosure-03.json); and copies of the decision logs and of the intervention logs of the four projects named in Appendix C (8 files, in Japanese), in which the passages that reproduce the original text of messages of the author are replaced by a mark.