Topological resolution of conical intersection seams and the coupled cluster bifurcation via mixed Hodge modules

The rigorous description of Conical Intersections (CIs) remains the central challenge of non-adiabatic quantum chemistry. While the ``Yarkony Seam'' -- the $(3N-8)$-dimensional manifold of degeneracy -- is well-understood geometrically, its accurate characterization by high-level electronic structure methods is plagued by numerical instabilities. Specifically, standard Coupled Cluster (CC) theory suffers from root bifurcations near Ground State CIs, rendering the ``Gold Standard'' of chemistry inapplicable where it is needed most. Here, we present \textbf{QuMorpheus}, an open-source computational package that resolves these singularities by implementing a topological framework based on Dissipative Mixed Hodge Modules (DMHM) [P. Saurabh, arXiv:2512.19487 (2025)]. By algorithmically mapping the CC polynomial equations to a spectral sheaf, we compute the exact Monodromy ($\mu$) invariants of the intersection. We demonstrate that this automated algebraic geometry approach correctly identifies the physical ground state topology in the K\"{o}hn-Tajti model and resolves the intersection seams of realistic chemical systems, including Ethylene and the Chloronium ion ($\mathrm{H_2Cl^+}$). Furthermore, we apply QuMorpheus to the photoisomerization of Previtamin D, proving that the experimentally observed Woodward-Hoffmann selection rules are a direct consequence of a topological ``Monodromy Wall'' ($\mu=1, \gamma=\pi$) rather than purely energetic barriers. This establishes a general software solution to the ``Yarkony Problem,'' enabling the robust, automated mapping of global intersection seams in complex molecular systems. The topological stability of these intersections allows for the control protocols discussed in Ref.[P. Saurabh, Submitted to Phys. Rev. X (2025)].