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 $A_Σ:= (V_Σ,E_Σ,c_Σ)$, where $V_Σ$ is the finite node-indexed vertex set, $E_Σ$ is the nodewise coupling space, and $c_Σ$ 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 $$S_Σ:= (\mathcal C_{\mathrm{bulk}},{\mathcal C_{p_k}}<em>{k=1}^{r,{Φ_k,Ψ_k}}</em>{k=1}<sup>r,Sh(S_Σ))$$, we define the extended vertex set $$V_Σ^{{\mathrm{ext}}} := V_Σ\sqcup {v_{\mathrm{bulk}}}$$, the functorial coupling relation determined by the attachment functors, the resulting functorial incidence package $$\mathfrak{I}<em>Σ:= (V</em>Σ<sup>{\mathrm{ext}},\rightsquigarrow_Σ)$$, and its canonical binary decategorification $\mathcal I_Σ:= (V_Σ<sup>{\mathrm{ext}},I_Σ)$. From these data we assemble the finite quiver-theoretic package $\mathfrak Q_Σ:= (V_Σ,E_Σ,c_Σ,\mathcal F_Σ,I_Σ)$, where $\mathcal F_Σ:= {(Φ<em>k,Ψ_k)}</em>{k=1}<sup>r$ 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.