G-Active Structural Inertia in Neural Networks: Numerical Evidence from Gradient-Flow Systems and Neural Network Training

This record contains the main article and reproducibility appendix for IDT Programme Working Paper NN-01. The work studies relaxation dynamics in natural-gradient systems and shows that the classical Hessian-only critical slowing down (CSD) law is not generally valid when the Fisher-Rao metric G is non-trivial. The standard prediction \tau = 1/\lambda_{\min}(H) holds only in the Euclidean special case G = cI. The paper demonstrates both theoretically and numerically that the correct relaxation law for natural-gradient flow \dot{w} = -G^{-1}\nabla \Phi is governed by the generalized eigenvalue problem H v = \lambda G v . The central numerical experiment holds the Hessian H fixed and varies only the metric G. In this controlled setup the observed relaxation time changes by a factor of 32×, while the Hessian-only prediction remains constant. The record contains two files: 1. Main article Concise publication version presenting the theoretical framework, numerical experiments, and conclusions. 2. Technical appendix / working paper Full reproducibility material including: deterministic CPU experiments neural network experiments reproducible Python code GPU test suite for ResNet-20 / CIFAR-10 The experiments confirm that relaxation dynamics in natural-gradient systems depend on the generalized spectrum of G^{-1}H rather than on the Hessian spectrum alone.