Finite-Node Perverse Schobers and Corrected Extensions for Conifold Degenerations

Abstract: We study one-parameter conifold degenerations whose central fiber has finitely many ordinary double points. Working within a minimal finite-node schober datum formalism, we define a local ordinary-double-point schober block at each node and prove that these local blocks assemble into a finite-node schober datum with one localized categorical sector per node. We further show that the decategorified shadow of this finite-node schober datum is the corrected finite-node perverse extension previously identified from nearby- and vanishing-cycle methods, and that the resulting finite-sector architecture is compatible with the mixed-Hodge-module refinement established in earlier work. In particular, the present paper provides a theorem-level categorical bulk / localized-sector framework for finite-node conifold degenerations, together with its quiver shadow and the first precursor of a later wall-crossing formalism.