The Boundary of Hedgeability: Pricing and Hedging in Volterra-Lévy Markets

We develop the pricing and hedging theory for financial markets driven by Volterra-Lévy processes, encompassing rough volatility models with jumps. The operator derivative D = δ*, defined by duality with the stochastic integral on a combined Hilbert ⊕ Banach energy space, provides a unified calculus for the continuous and jump components without Gaussian assumptions. The central result is a two-parameter Volterra risk-neutral constraint (VRNC): risk-neutral pricing in markets with memory and jumps requires solving a Volterra integral equation for an effective drift rate that couples the Brownian and jump risk premia through the kernel. The roughness parameter H is proved to be measure-invariant (calibration-invariant), establishing it as a structural feature of the market. The Leibniz defect of the operator derivative — the non-Gaussian content invisible to Malliavin calculus — equals the structurally unhedgeable component of risk. Its Lp norm scales multiplicatively in the roughness and jump parameters: T^{2H−1+1/p} · (2p−γ)^{−1/p}. The scaling is sharp for the variance swap payoff (two-sided BDG bounds). The defect vanishes if and only if the driver has no jumps, recovering perfect hedging in pure rough volatility models. The algebraic spine of the paper is formally verified in the Lean 4 proof assistant (zero sorry, zero axioms). An interactive blueprint is available at github.com/QuijoticResearch/volterra-levy.