The One-Parameter Banach Factorization for Stable Lévy Processes: Representability Obstructions and Leibniz Defects

This preprint studies the one-parameter Banach adjoint associated with symmetric stable Levy stochastic integrals. For a stable process L, the one-parameter integral delta_L(u)=int_0^T u_t dL_t naturally acts on a Banach integrand space governed by the stable scale, rather than by Hilbert-space L2 geometry. Its adjoint D_L=delta_L^* therefore takes values in a Banach dual space and characterizes only the image of the one-parameter stable integral. The main result is a representability obstruction: one-parameter stable integrals correspond, at the underlying jump level, only to kernels of the special form h(t,z)=u_t z. General jump-size functionals require genuinely two-parameter kernels h(t,z). In particular, centered nonlinear jump functionals such as large-jump counts lie outside the one-parameter image, even though they are naturally visible to Poisson/add-a-point Malliavin calculus. The paper develops this obstruction as a Banach-adjoint factorization result, contrasts it with the Hilbert/Riesz setting, and records a Leibniz-defect formulation that detects jump nonlinearities even when one-parameter representation fails. The results are intended as the one-parameter precursor to a companion two-parameter analysis, where the obstruction becomes a distinction between nu-regular compensated-Poisson components and nu-singular stable-integral components. A Lean 4 formalization accompanies the paper at the abstract BanachEnergySpace level. The formalization verifies the algebraic adjoint, factorization, obstruction, and Leibniz-defect structure conditional on a bounded divergence map; it does not formalize the analytic construction of the stable stochastic integral itself.