Data and code for "When does the kineticity regularization of hi_class bias σ8? An identity for the k² coefficient and two failures of the x formulation at a field turnaround"

Data and code that reproduce the numerical results of the paper "When does the kineticity regularization of hi_class bias σ8? An identity for the k² coefficient and two failures of the x formulation at a field turnaround" (T. Tohi, submitted to Physical Review D). The paper studies the kineticity regularization (kineticity_safe_smg = k_s) of the Einstein–Boltzmann solver hi_class at a turnaround of the dark-energy field, where the kinetic term D of the scalar perturbation vanishes. It shows (i) an exact identity between the k² coefficient A13 of the scalar equation and the numerator of the sound speed, (ii) that a residual of this identity produces a gradient instability with a threshold in k_s, below which finite but wrong spectra are returned, and (iii) that the default variable x = −δφ/φ' has a pole at the turnaround, so that near- and super-horizon scales depend on k_s and on the sampling of the background. The archive contains: the one-line injection patch for the public hi_class (commit ece81a5), inactive by default; the input files of the quintessence models with a late turnaround (Ω_smg = 0.05 and 0.69); the scan driver and the results of all 426 runs (σ8 against k_s, the injected residual, its width and k_max); the P(k) and C_ℓ spectra, including the δφ-formulation references and the dependence on the background sampling; the fine-grid background near the turnaround; the plotting scripts and figures; and the sympy scripts of the independent derivation of the sound speed from a general ADM Lagrangian with matter. The audited tabulated background of Sec. V is archived separately (doi:10.5281/zenodo.23827616). See README.md in the archive.