Cohomological Invariants of Frobenius Splittings and the Spectral Decomposition of the Cartier Algebra on Post-Singularity Varieties

Mathematical Applications of Science Fiction We construct a rigorous cohomological framework for analyzing the spectral properties of the Cartier algebra on a class of schemes designated as Post-Singularity Varieties. By extending the classical theory of Frobenius splittings to the derived category of coherent sheaves, we define the Void-Walker Filtration on local cohomology modules. We prove that for a Post-Singularity Variety admitting a compatible Frobenius splitting, the spectral sequence associated with the Cartier operator degenerates at the second page, yielding a canonical decomposition of the intersection cohomology. Furthermore, we establish the existence of the Theron Invariant, a numerical quantity derived from the fractal dimension of the F-signature, which governs the topological stability of the variety under void-field perturbations. This work provides the algebraic foundation for understanding the quantization of geometric singularities in high-energy void dynamics.