From Finite-Node Conifold Geometry to BPS Structures I: Algebraic State Data

Abstract: Let $π:X\to Δ$ be a one-parameter degeneration whose central fiber $X_0$ is a complex threefold with finitely many ordinary double points $Σ={p_1,\dots,p_r}\subset X_0$. Associated with this degeneration is the corrected finite-node perverse extension, together with its mixed-Hodge-module refinement and a finite-node schober datum whose perverse-sheaf shadow is identified with the corrected perverse sheaf $\mathcal P$. The purpose of the present paper is to extract from these finite-node geometric, extension-theoretic, mixed-Hodge, and categorical inputs the intrinsic algebraic state data carried by the degeneration. More precisely, we isolate the finite localized quotient $Q_Σ:=\bigoplus_{k=1}^r i_{k*}\Q_{{p_k}}$, the nodewise coupling space $E_Σ:=\Ext<sup>1_{\Perv(X_0;\Q)}(Q_Σ,IC_{X_0})$, its canonical nodewise decomposition $E_Σ\cong\bigoplus_{k=1}^r \Q e_k$, and the coefficient vector $c_Σ=(c_1,\dots,c_r)\in\Q<sup>r$ defined by $$[\mathcal P]<em>{\mathrm{perv}}=\sum</em>{k=1}^r c_k e_k$$. We then prove that these state variables are compatible with both the mixed-Hodge-module lift and the schober realization of $\mathcal P$, so that the same finite-node architecture appears simultaneously in perverse, mixed-Hodge, and categorical form. The resulting package $(V_Σ,E_Σ,c_Σ)$ is the intrinsic algebraic state data attached to the finite-node conifold degeneration. It provides the first algebraic layer in the passage from finite-node geometry to later incidence, quiver, stability, BPS-spectral, and wall-crossing structures.