Scale Invariance of Computational Impedance Z and Stability Conditions for the Universal Substrate

This paper builds upon the Computation Resource Finiteness Hypothesis (CRFH), which models physical laws as computational processes executed on a finite execution platform referred to as the ``Universal Substrate,'' and introduces a new dimensionless quantity termed the \emph{computational impedance} $Z$. The quantity $Z$ is neither a new force nor a physical field, but an operational state variable that characterizes the allocation of computational resources and the efficiency of information transfer during the execution of physical laws. In previous works, a stable operating point $Z_{\mathrm{quant}} = 1/\sqrt{2}$ was derived in the quantum regime from the conservation condition of complex probability amplitudes, while in the macroscopic gravitational regime, $Z_{\mathrm{comp}}$ was shown to converge to the same value based on orbital data from all planets in the Solar System. In this paper, these results are unified, and the possibility that the computational impedance $Z$ functions as a scale-invariant stability condition across physical hierarchies is examined theoretically. Furthermore, in strong gravitational fields, a scaling claim is proposed whereby saturation of computational load leads to a redistribution of resources from geometric curvature updates of spacetime to rotational degrees of freedom, yielding consistency with observed black hole dimensionless spin parameters $a_* \approx 0.6$--$0.7$. From the perspectives of information theory and impedance matching, it is shown that $Z \approx 1/\sqrt{2}$ minimizes information reflection and suppresses computational dissipation, representing an optimal operating point of the Universal Substrate. This work does not aim to replace existing formulations of quantum mechanics or general relativity, but rather to provide an auxiliary framework that characterizes the operational conditions under which these theories are realized. The scale invariance of the computational impedance $Z$ is thus positioned as a theoretical prediction that is testable by future observations and experiments.