The Operator Derivative in Continuous Stochastic Calculus: A Hilbert Energy Space Framework

A Lean 4 formalization of the operator derivative framework for continuous stochastic calculus, using Mathlib. The operator derivative D := δ* is defined as the Hilbert space adjoint of the stochastic integral, reducing stochastic calculus to functional analysis. The formalization comprises 4,600+ lines with zero sorry and one axiom (the Bakry-Émery semigroup interpolation theorem). Verified results include: the Clark-Ocone representation, Leibniz rule, product rule, chain rule (polynomial and smooth extension), abstract and concrete Itô formulas, representer rigidity, stochastic volatility obstruction, PRP equivalence with ker(D) ⊆ constants, Gubinelli base-invariance, Leibniz for fractional Brownian motion (all Hurst parameters H ∈ (0,1)), Lp pointwise multiplication via Hölder's inequality, and the complete Gaussian integration-by-parts chain from the Gaussian PDF derivative through Stein's lemma to the Itô correction. The Hilbert space formulation is what makes formalization tractable: the abstract pipeline compiles against Mathlib's existing functional analysis library without requiring filtrations, stopping times, or measure-theoretic martingale theory. This is a companion artifact to the paper "The Operator Derivative in Continuous Stochastic Calculus: A Hilbert Energy Space Framework" (Fontes, 2026). Lean toolchain: leanprover/lean4:v4.29.0-rc6. Verify with: lake build.