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 \((\operatorname{Cone}(u),\operatorname{Cone}(v),\operatorname{Cone}(R))\) 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.