Spencer-Riemann-Roch Theory: Mirror Symmetry of Hodge Decompositions and Characteristic Classes in Constrained Geometry

The discovery of mirror symmetry in compatible pair Spencer complex theory brings new theoretical tools to the study of constrained geometry. Inspired by classical Spencer theory and modern Hodge theory, this paper establishes Spencer-Riemann-Roch theory in the context of constrained geometry, systematically studying the mirror symmetry of Spencer-Hodge decompositions and their manifestations in algebraic geometry. We utilize Serre's GAGA principle to algebraic geometrize Spencer complexes, establish coherent sheaf formulations, and reveal the topological essence of mirror symmetry through characteristic class theory. Main results include: Riemann-Roch type Euler characteristic computation formulas for Spencer complexes, equivalence theorems for mirror symmetry of Hodge decompositions at the characteristic class level, and verification of these theories in concrete geometric constructions. Research shows that algebraic geometric methods can not only reproduce deep results from differential geometry, but also reveal the intrinsic structure of mirror symmetry in constrained geometry through characteristic class analysis, opening new directions for applications of Spencer theory in constrained geometry.