Numerical Companion to Processing Proxies II: Robustness of the Normalized Accretion-Like Sigma-Response Constraint

This numerical companion documents the R4b robustness tests associated with Processing Proxies II. The original Processing Proxies II calculation tested whether a normalized accretion-like source history could drive the Kernel-Cosmology II response variable sigma toward the reference neighborhood sigma_* = 2/3. The present companion does not introduce a new cosmological model and does not physically calibrate the microscopic coupling. Its purpose is narrower: to test the numerical robustness of the normalized sigma-response constraint.

The response equation studied is the decoupled first-order closure

d sigma / dx = [A Sigma_dyn(x) - sigma] / Theta_sigma,

with x = ln a = -ln(1+z), and with all source histories normalized by max Sigma_dyn = 1. The companion tests sensitivity to the initial redshift range z_max = 30, 100, 300, 1000, to interpolation choices including linear interpolation in redshift, linear interpolation in x, and PCHIP interpolation in x, and to useful-window clipping around the inherited Kernel-II operational range.

The main result is that the normalized PP-II sigma-response constraint is numerically robust, but not physically calibrated. Across the tested accretion-shaped pilot reconstructions, all cases with Theta_sigma = 0.8, 1.0, and 1.5 remain operational survivors under the criteria 1 <= A_sigma* <= 3 and no upper useful-window overshoot before reaching sigma_*. The fast-response branch Theta_sigma = 0.5 is disfavored for sharper accretion-shaped histories because the response overshoots the useful-window upper bound before reaching the present-day target.

The result should therefore be read as a robustness statement about the normalized PP-II test: accretion-shaped source histories continue to require only order-unity normalized amplitudes for the Kernel-II sigma target over the tested response range, but the calculation does not derive the microscopic coupling, does not propagate observational uncertainty bands, and does not solve the fully coupled Kernel-II system. A physical calibration of the map from accretion-rate density to Sigma_dyn remains a task for future work.