Exact Gradient Explosion Time, Optimal Clipping Threshold, and Topological Invariants of Training Dynamics via the Fλ Class

We apply the F_lambda class to gradient explosion in deep neural networks. The model dg/dt = g·exp(λg) yields a closed-form explosion time T* = E1(λg0), where E1 is the exponential integral. This gives an exact gradient clipping formula, a hierarchy of feedback mechanisms classifying four architecture types (MLP, RNN, Transformer, ResNet), a PDE model of backpropagation with exact solution via Gaussian convolution, and a topological invariant Ω = π/2 − Si(μg0) for oscillatory training dynamics under cyclic learning rates. The single parameter λ encodes the per-architecture feedback strength and is calibrated from early training dynamics. Calibrated values range from λ ≈ 0.096 (MLP, strong feedback) to λ ≈ 0 (ResNet, near-linear regime).