This manuscript presents version 6 of the Supra–Omega Resonance Theory (SORT), a mathematically closed operator-projection framework that provides a unified structural foundation for cross-domain system analysis. SORT is constructed around a closed set of 22 idempotent resonance operators, a global projector Ĥ, and a calibrated projection kernel κ ( k ) , forming an invariant mathematical core capable of addressing fundamental anomalies across cosmology, artificial intelligence, quantum systems, and complex systems within a single coherent architecture. Version 6 establishes the complete algebraic structure and modular domain architecture that will enable systematic empirical validation in subsequent versions. The framework introduces a clear separation between a publicly documented core API and an internal validation engine, ensuring reproducibility and controlled extension. Structural consistency is verified via calibrated reference parameters, including kernel parameter σ 0 = 0.00190643, Jacobi residual below 5 × 10 −15, and idempotency residual below 10 −12 . The framework is validated within the MOCK v4 environment using a three-layer verification architecture. Version 6 focuses on architectural completeness, modular extensibility, and preparation for high-performance computing deployment. The explicit formulation of operator transition laws and adjacency validation provides the mathematical machinery for addressing phenomena ranging from the Hubble tension to early galaxy formation within a unified projection-based description. Empirical validation against observational data is scheduled for version 7.