A Residual-Excess Framework for Defect Resolutions: Finite-Node Saito Gluing and Integral Hodge Obstructions

Defect data attached to a morphism admit two orthogonal refinements. The first is factorization sensitive: a factorization \( A\xrightarrow{u}B\xrightarrow{v}A^\dagger \) with vu = R, gives a residual triple (Cone(u), Cone(v), Cone(R)) constrained by the octahedral axiom. The second is coefficient-sensitive: an excess object is the cone of a comparison morphism between integral or finite-coefficient residual data and its rational or mixed-Hodge realization. We organize these two directions into a residual–excess matrix. The main residual witness is a finite ordinary-double-point conifold degeneration. Saito divisor gluing produces a node-supported Hodge interface \( W^H_\Sigma=\bigoplus_{p_k\in\Sigma}i_{k*}\mathbb Q^H_{\{p_k\}}(-1) \) factoring finite-node monodromy through \( \psi^H_{\pi,1}\to W^H_\Sigma\to\psi^H_{\pi,1}(-1) \). The variation cone yields the corrected extension \( 0\to IC^H_{X_0}\to P^H_{\mathrm{var},\Sigma}\to W^H_\Sigma\to0 \), whose class decomposes into nodewise Ext residual classes. Ordinary double points calibrate the zero-excess regime because their Milnor fibers are integrally torsion-free. Diaz’s Enriques-product Bockstein mechanism calibrates the nonzero-excess regime through integral torsion killed by rationalization. The resulting framework compares finite-node Saito gluing with integral Hodge obstruction channels without conflating rational residual data with coefficient-change defects.