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 p₁ < p₂ under a singular pole q. We demonstrate that apparent deviations from the single-mode scaling law α = p - q do not signify a breakdown of the asymptotic physics, but reflect a crossover dynamics governed by a universal scaling function Φ(x) = 1 + xp₂ - p₁, where x = δ / δ_×. The characteristic crossover scale follows a strict power law δ_× ∝ (a_* - ac)^μ, where the exponent μ is dictated either by mode spacing μ = (p₂ - p₁)⁻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 (⁴He-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.