A Closed-Form Saddle-Node Threshold for Power-Rate Reaching-Law Sliding Mode Control Under Stiffness Mismatch

This corrected Version 2 studies the loss of the desired equilibrium in power-rate reaching-law sliding-mode control under stiffness mismatch. The paper derives the exact nonzero-mismatch equilibrium relation and the closed-form saddle-node threshold for the power-rate reaching law. The corrected Version 2 supersedes the preliminary exact-tracking interpretation of the earlier release, preserves the corrected fold expression and base numerical threshold, and clarifies why finite-horizon simulations can detect the transition slightly above the analytical fold. The revision includes: the correct nonzero-mismatch hypothesis in the equilibrium proposition; the corrected x_d exponent in the fold formula; an explicit explanation of the 1747.0 N/m versus 1732.1 N/m transition offset through saddle-node critical slowing down; a clarification that the numerical offset is a finite-horizon bottleneck effect rather than an error in the analytical fold. A detailed revision note is included in the deposit. This Version 2 is the authoritative public version and supersedes the earlier preliminary interpretation.