Certified Inference and Training for Deep Equilibrium Networks: A Continuation Framework with Polynomial Complexity Guarantees

We develop a certified continuation framework for equilibrium computation and for training deep equilibrium networks (DEQs), with training formulated as interpolation to accuracy $2^{-b}$. For inference, compact input homotopy selects a unique branch from a supplied start root, and a rounded Newton tracker follows it under certified boundary, conditioning, derivative, and tube-radius bounds. For training, we augment local-plus-low-rank recurrence with programmable dormant bilinear rank-one channels. Loaded Tikhonov solves diagnose a failed interpolation pass without spectral decomposition; an output-preserving repair aligned with the pass residual supplies the required direction. Training requires certified gate realization and column stability on each pass region, well-posed inference, and finite-update error budgets. With polynomial geometric, encoding, precision, and complete backend budgets, both certified inference and training have bit cost $O(\mathrm{poly}(L+b))$, where $L$ is the encoded instance length. The trainer uses $O(b+\mathrm{\ell })$ passes and reserve channels from an initial residual bounded by $2^{\mathrm{\ell }}$. These guarantees concern a certified promise class. Lean 4 verifies the quantitative core and concrete inference backend; numerical comparisons illustrate the loaded mechanism.