The Grand Consistency Unification Theorem

This paper formulates the Grand Consistency Unification Theorem as the mathematical synthesis of Grand Consistency Theory (GCT). Building on the sector-specific results developed in Papers 1–8, it establishes a common deductive architecture connecting the primitive consistency structure, the Constraint Laplacian, its spectral data, infrared sector maps, and physical observables. The framework distinguishes operator-level sector projections from spectral sector-extraction maps and develops compatibility, refinement, and commutativity conditions required for a unified microscopic origin. A global prediction map and cross-sector Jacobian are introduced to characterize parameter reduction, predictive closure, and independent relations among observables from distinct physical sectors. Under explicit assumptions of spectral regularity, refinement convergence, sector compatibility, and observable correspondence, the Grand Consistency Unification Theorem provides a unified spectral structure for geometry, quantum dynamics, matter, gauge interactions, gravitation, cosmology, particle spectra, and fundamental constants. The paper concludes with a numerical and experimental programme for testing the resulting cross-sector consistency relations from finite microscopic realizations.