RLMT:A Numerical Audit of hi_class for MTFE/RLMT Background Tables: Consistency Identities, Regularization Thresholds, and Stable σ₈

“We audit the linear-perturbation solver hi_class when it is driven by tabulated background quantities produced by the MTFE/RLMT model… After correcting several interface inconsistencies, the perturbations become stable with a small kineticity regularization.” This manuscript presents a full numerical audit of hi_class (v3.0) when operated with MTFE/RLMT background and α-function tables. The initial runs produced unphysical perturbations (σ₈ up to 10⁶³), motivating a systematic investigation rather than parameter tuning. Main findings Identified and corrected four interface inconsistencies: (i) missing factor 3 in the generator, (ii) radiation normalization mismatch, (iii) effective neutrino number Nur inconsistency, (iv) linear interpolation of E(a) producing spurious residuals. Verified to machine precision that the Phase‑E expression for cs2 matches the standard hi_class formulation. Demonstrated that the mismatch between (ρ+p)smg and the action closure does not affect σ₈. Identified two distinct regimes where the kineticity floor ks is required: early‑time stabilization (canonical Q0 case) and late‑time turnaround (yn1e‑3 case). Obtained stable σ₈ values with numerical uncertainty ≈ 1.5×10−5. Results σ₈ = 0.863793 (+1.94% vs ΛCDM with same h, Nur) for yn1e‑3 σ₈ = 0.840053 (−0.12%) for Q0 These values reflect audit parameters, not observational fits. Limitations The regularization ks is not physically consistent with the model; its lower bound (~1–2×10⁻⁸) is empirically measured but not theoretically explained. Additional unresolved items include turnaround behavior, closure residual origins, and asymmetry in σ₈ sensitivity to δNur. Reproducibility All patched source code, diagnostics, table-generation scripts, refined tables, and audit logs are archived as supplementary files on the same Zenodo/Jxiv record as this manuscript.