Rational Betti Numbers, Cohesive Equivalence, and Higher Chern A Framework for Particle Emergence as Coherence

We present a unified framework in which rational Betti numbers, formulated intrinsically in a cohesive (∞, 1)-topos, encode the rational coherence data determined by the shape modality. We prove invariance under cohesive equivalence and establish a Sullivan correspondence that renders these invariants computable via rational homotopy theory. Integrating this construction into a higher Chern character framework, we demonstrate that elementary particles emerge as coherence gaps: topological defects whose charge sectors are classified by rational cohomology, with quantization arising as the integrality condition required for defect data to extend coherently through the Postnikov tower. This provides a structural explanation for discrete charge spectra and particle multiplicity, unifying classical topology, rational homotopy, and higher gauge theory within a single foundational narrative.