The bottom of the one-particle branch and the sign of an edge quantity for a kinetic Ising chain at the rate θ*

This note continues the paper [1] on the single-spin-flip dynamics of the one-dimensional Ising model with the flip rates (1/2) θ*^(s_x (sx−1 + sₓ₊₁)), where θ* = √2 − 1, and uses its setting and notation. It contains two results, with complete proofs. The first concerns the one-particle branch λ° of the infinite chain, whose minimum is the gap λ_∞ of the generator: we prove that the minimum is attained only at the momentum p = 0, that (3/5)·2(1 − cos p) ≤ λ°(p) − λ_∞ ≤ (5/8)·2(1 − cos p) for |p| ≤ 1/3, that λ°(p) − λ_∞ ≥ 0.066 at the remaining momenta of the branch, and that the curvature a = (1/2)(λ°)''(0) satisfies 0.618 ≤ a ≤ 0.622. This answers two questions left open in [1]. The second concerns the half-line with the end spin fixed to +1, at the threshold z = λ_∞: we prove that the equation defined by the matrix of the half-line reduction of [1] has exactly one solution of the form x + d_x with a real sequence d whose differences are square summable, we define with it a linear functional on the local functions that vanishes on the constants and on the image of the local functions under the generator shifted by λ_∞, and we prove that a number B defined through this functional satisfies −20.6 ≤ B ≤ −11.9; in particular B < 0. The numerical statements are certified by computations in exact arithmetic, which a single command repeats. No relation between B and the behaviour at late times of the paths of [1] is proved or claimed; the questions that remain are stated at the end. [1] Y. Maeda, 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, Zenodo, version 1.1, 2026. https://doi.org/10.5281/zenodo.23285424 Version. Version 1.0 (October 2026). Files. The record holds 32 files: the note (note-v1.0.pdf); the bundle of the two certificates of Appendix A (kramers-certificates.zip), which a single command runs; a description of the files (README.md); and the 29 files that Section C.5 of the note lists, which are in Japanese: the list of the differences between the note and the reviewed documents (DEVIATIONS.md), the Japanese text of the statement of Section C.5 (disclosure-ja.md), the rules of the two projects, a page of the staff, and the records of the projects ONSAGER (11 files) and KRAMERS (14 files), among them the findings of the reviews. README.md lists all files with their sizes and SHA-256 digests.