From Finite-Node Conifold Geometry to BPS Structures II: Functorial Incidence and Quiver Assembly

Abstract

In previous work, we extracted the intrinsic finite algebraic state data of a finite-node conifold degeneration in the form $A_\Sigma := (V_\Sigma,E_\Sigma,c_\Sigma)$, where $V_\Sigma$ is the finite node-indexed vertex set, $E_\Sigma$ is the nodewise coupling space, and $c_\Sigma$ is the coefficient vector of the corrected global extension class. The purpose of the present paper is to construct the corresponding interaction and incidence layer. Starting from the finite-node schober package $S_\Sigma := (\mathcal C_{\mathrm{bulk}},\{\mathcal C_{p_k}\}_{k=1}^r,\{\Phi_k,\Psi_k\}_{k=1}^r,Sh(S_\Sigma))$, we define the extended vertex set $V_\Sigma^{\mathrm{ext}} := V_\Sigma \sqcup \{v_{\mathrm{bulk}}\}$, the functorial coupling relation determined by the attachment functors, the resulting functorial incidence package $\mathfrak{I}_\Sigma := (V_\Sigma^{\mathrm{ext}},\rightsquigarrow_\Sigma)$, and its canonical binary decategorification $\mathcal I_\Sigma := (V_\Sigma^{\mathrm{ext}},I_\Sigma)$. From these data we assemble the finite quiver-theoretic package $\mathfrak Q_\Sigma := (V_\Sigma,E_\Sigma,c_\Sigma,\mathcal F_\Sigma,I_\Sigma)$, where $\mathcal F_\Sigma := \{(\Phi_k,\Psi_k)\}_{k=1}^r$ is the functorial coupling datum. We prove that this package is canonically determined by the finite-node schober datum, compatible with the corrected perverse extension and its mixed-Hodge-module refinement, and invariant under equivalence of finite-node schober realizations. This yields the interaction and incidence layer required for later graded interaction, stability, BPS, and wall-crossing structures.