RLMT: 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

The kinetic term D of the scalar perturbation vanishes at every turnaround of a dark-energy field, and the Einstein–Boltzmann solver hi_class lets the user regularize it with a constant k_s added to the kineticity. We ask when this regularization biases the results. For α_H = 0 we show that the coefficient A13 of k² in the equation of the default variable and the numerator N of the sound speed obey an exact identity, A13 = −N − B_α, and confirm by an independent derivation from a general ADM Lagrangian with matter that N/D is the sound speed. When the α functions and the expansion history come from one Lagrangian, N vanishes with D and σ8 is independent of k_s. When they are supplied separately, a residual that does not vanish with D produces at the turnaround a gradient instability with a growth rate proportional to k; the regularization then has a threshold, set by a growth of about twenty e-folds at the largest wave number, below which the code returns divergent or finite but wrong spectra, with errors of 0.1% to 100% in σ8 in controlled tests. Independently of any residual, the default variable x = −δφ/φ' has a pole at the turnaround: near- and super-horizon scales depend on k_s and on the sampling of the background, by factors up to 3×10⁴ in P(k) at k = 10⁻⁴ h/Mpc in the public code, and the limit k_s → 0 does not exist. We establish both effects with the unmodified public code for quintessence, compared with the field-perturbation formulation, and find both in a tabulated background, where removing the regularization had shifted σ8 by −0.27%. We give checks that detect them and remedies that remove them. Data and code: doi:10.5281/zenodo.23828666. Audit record of the tabulated background: doi:10.5281/zenodo.23827616.