BOREL-MOORE HOMOLOGY OF SZEMERÉDI COMPLEXES IN THE CRITICAL PHASE OF GALACTIC VOID NUCLEATION

Mathematical Applications of Science Fiction We present a rigorous topological framework for analyzing the large-scale structure of the universe during the critical phase of galactic void nucleation. By modeling the cosmic web as a hypergraph, we construct the associated Szemerédi complex through the hypergraph regularity lemma. We investigate the Borel–Moore homology groups of this complex as the matter density parameter approaches a critical threshold. We prove that at this critical point, the Borel–Moore homology undergoes a phase transition characterized by the divergence of the Betti numbers, corresponding to the percolation of infinite voids. We derive a spectral sequence that converges to the homology of the limiting void manifold and establish an exact expression for the nucleation density in terms of the Euler characteristic of the Szemerédi partition. This framework connects arithmetic combinatorics, homological algebra, and physical cosmology into a unified mathematical model of cosmic topology