Formal Foundations of Stochastic Gradient Descent

We present a formally verified theory of stochastic gradient descent (SGD) and its variants, covering descent step bounds, convexity analysis, strong convexity, gradient Lipschitz conditions, distance contraction, momentum methods, mini-batch variance reduction, and learning rate schedules. The formalization comprises 78 verified theorems in the Platonic proof kernel, establishing rigorous foundations for the convergence guarantees that underpin modern machine learning optimization. Key results include: (1) deterministic and stochastic descent bounds under smoothness assumptions, (2) strong convexity convergence with quadratic growth, (3) variance reduction via mini-batching and SVRG-style control variates, (4) Polyak averaging and batch size scaling rules, and (5) proximal/regularized SGD shrinkage properties. All theorems are machine-checked with zero axioms and zero hypotheses, providing unconditional mathematical guarantees.