Composite Characteristic Transport: Propagation and Separation of Riemann--Roch Defects

We study corrected characteristic transport along composite algebraic-geometric constructions. For an admissible geometric operation \( F \), a characteristic transformation \( \Phi \), and its Riemann--Roch-corrected operator \( F_{\Phi,*}^{\mathrm{RR}} \), the discrepancy \( \Delta_\Phi(F;\xi):=\Phi(F_!\xi)-F_{\Phi,*}^{\mathrm{RR}}(\Phi(\xi)) \) defines a characteristic transport defect. Starting from established one-stage Riemann--Roch, Verdier specialization, Gysin, and motivic characteristic-class formulas, we develop composition and support-propagation laws for defects along finite geometric routes. In the singular setting, Schürmann's Verdier--Riemann--Roch defect propagates through later proper, smooth, and compatible Gysin stages; for a nodal plane cubic, the vanishing-cycle contribution is computed explicitly as \( -\mathbf 1_{\{p\}}\mapsto[p]\mapsto[\mathrm{pt}] \). We organize stagewise data into comparative transport profiles and give support-graded criteria for distinguishing routes whose actions agree on a prescribed cohomological sector. For transfer--return correspondences, an explicit Jacobian family arising from finite covers of curves of genus at least two yields residuals in \( \ker(\rho_{L_A})\setminus\ker(\operatorname{cl}) \). We also establish corrected smooth--Gysin compatibility in Cartesian squares. The resulting formalism is diagnostic and factorization-sensitive rather than algebraicity-producing.