Defect Triangles and Intersection-Space Hodge Atom Shadows for Calabi--Yau Conifolds

We prove a projection-triangle statement for projective Calabi–Yau threefold conifold degenerations and use it to organize an intersection-space Hodge atom shadow package. For a nodal central fiber $X_0$, assume the relevant Banagl–Budur–Maxim, multi-node gluing, mixed-Hodge-module, and specialization-splitting hypotheses, so that ${\psi }_{\pi }(F)\simeq {\mathcal{I}\mathcal{S}}_{X_0}^H\oplus {\mathcal{C}}_{\mathrm{\Sigma }}^H$. Projection of the variation morphism to the intersection-space summand defines ${\mathrm{var}}_I:{\phi }_{\pi }(F)\to {\mathcal{I}\mathcal{S}}_{X_0}^H$, and the octahedral axiom gives $P^H\to P_I^H\to {\mathcal{C}}_{\mathrm{\Sigma }}^H\stackrel{+1}{\to }$, where $P^H=\mathrm{Cone}(\mathrm{var})[-1]$ and $P_I^H=\mathrm{Cone}({\mathrm{var}}_I)[-1]$. This realizes the intersection-space atom shadow package ${\mathsf{H}\mathsf{A}}^I(X_0)$ and compares it with the intersection-homology package ${\mathsf{H}\mathsf{A}}^{IH}(X_0)$. Under the self-dual specialization-splitting hypothesis, the projected variation object satisfies $\mathbb{D}{P}_I^H\simeq Q_I^H(3)$, where $Q_I^H=\mathrm{Cone}({\mathrm{can}}_I)[-1]$. Under the mixed-Hodge realization of Banagl’s middle exact sequence, we isolate a rigid–vanishing filtration and identify the IIB vanishing atom with the realized kernel. For the classical $125$-node quintic, the middle-degree IC–intersection-space defect has rank $202$. The construction remains at Hodge-realization level and identifies ${\mathcal{C}}_{\mathrm{\Sigma }}^H$ and ${\mathrm{\Delta }}_{I\mathrm{/}IC}(X_0)$ as geometry-side handoff objects for future DT/BPS comparisons.