Root-Free Numerical Inversion of Massive-Order Rational Laplace Transforms up to K=10^ 6 Poles

The numerical inversion of high-order rational Laplace transforms has historically been constrained by the ill-conditioning of polynomial root-finding and catastrophic floating-point cancellation in multiprecision quadrature algorithms. In this paper, we present an industrial-grade, 100% root-free numerical framework capable of inverting rational transfer functions with degrees up to K = 10^6 poles strictly in standard IEEE-754 double precision without calculating denominator roots. The proposed methodology establishes two complementary mathematical engines: (i) a conformal Möbius–Laguerre orthogonal projection that transforms the right half-plane into the unit disk, guaranteeing L^2[0, \infty) convergence while completely eliminating the classical "Taylor Hump" numerical instability; and (ii) an asymptotic Laurent–Stirling combinatorial recursion evaluated in logarithmic space (lgamma) to prevent floating-point underflow/overflow. Furthermore, we prove a deterministic tuning criterion based on Cauchy–Fujiwara spectral bounding for parameter selection. Comprehensive numerical benchmarks on stiff resonant networks (K = 5,000), anomalous thermal diffusion PDEs, and large-scale Chebyshev transmission lines (K = 1,000,000) demonstrate uniform L_\infty accuracy below 10^-14 and up to 50x speedup over conventional methods.