Universal Scaling Function and Crossover Scale in Multi-Mode Finite-Resolution Transmission Systems

We establish a rigorous scaling framework describing power-law exponent transitions in physical systems governed by competing transmission modes $p_1 < p_2$ under a singular pole $q$. We demonstrate that apparent deviations from the single-mode scaling law $\\alpha = p - q$ do not signify a breakdown of the asymptotic physics, but reflect a crossover dynamics governed by a universal scaling function $\\Phi(x) = 1 + x^{p_2 - p_1}$, where $x = \\delta / \\delta_\\times$. The characteristic crossover scale follows a strict power law $\\delta_\\times \\propto (a_* - a_c)^\\mu$, where the exponent $\\mu$ is dictated either by mode spacing $\\mu = (p_2 - p_1)^{-1}$ for geometric critical points, or by system-specific coupling relations. Numerical simulations confirm this scaling with machine precision (relative error $< 10^{-14}$) across multiple physical domains, including condensed matter transitions ($^4\\text{He-Vycor}$) and quantum gravity regimes (black hole/string transitions). This framework provides a predictive tool for data collapse and exponent extraction in finite-resolution experimental data.