Hodge Atoms at Conifold Degenerations: F-Bundles, Limiting Mixed Hodge Modules, and the Rigid-Flexible Decomposition

We extend the Hodge atoms framework of Katzarkov–Kontsevich–Pantev–Yu to one-parameter conifold degenerations of Calabi–Yau threefolds. For a degeneration $\pi \text{\hspace{0.17em}⁣}:X\to \mathrm{\Delta }$ whose central fiber $X_0$ has $r$ ordinary double points, we construct a canonical rigid-flexible decomposition of the Hodge atoms of the nearby smooth fiber attached to the corrected degeneration object. The rigid atom $A({\text{IC}}_{X_0}^H)$ is preserved across the degeneration, while the flexible atoms $A(i_{k\ast }{\text{QQ}}_{\{p_k\}}^H(-1))$ are rank-one contributions, one for each vanishing cycle. The total degeneration atom $A(P^H)$ is the atom of the corrected mixed Hodge module $P^H\in \text{MHM}(X_0)$ and fits into an exact sequence of atoms whose non-split structure is controlled by the intersection matrix $(⟨{\delta }_i,{\delta }_j⟩)$. The technical core is the Stokes–Extension Identification, which identifies the Stokes matrix of the Dubrovin connection at the conifold locus with the matrix of the variation morphism $\text{varF}\text{\hspace{0.17em}⁣}:{\phi }_{\pi }(F)\to {\psi }_{\pi }(F)$ under mixed Hodge module realization.