Superceded version Paper T22: The Coherence–Unitarity Theorem in Angular Phase Evolution: A Real-Valued Formalism for Deterministic Quantum Dynamics

This version has been superseded by the February 8, 2026 revision. Readers should consult the updated version for the authoritative formulation, refined proofs, and corrected statements. This document is retained for archival and developmental transparency only. In conventional quantum mechanics, unitary time evolution is guaranteed by thestructure of complex Hilbert space. In Holosphere Theory, however, phase evolution ismodeled using real-valued coherence vectors that evolve through angular strain propa-gation in a discrete rotating lattice. This paper formalizes the conditions under whichsuch real-valued evolution preserves norm and coherence amplitude. We derive a gen-eral Coherence–Unitarity Theorem, applicable to both dyadic (R90) and triadic (R123)rotation operators, and show that norm conservation arises from specific antisymmet-ric Hamiltonian couplings. This establishes a rigorous foundation for deterministic,real-valued quantum dynamics without the use of complex numbers, and provides themathematical basis for subsequent extensions to entanglement, measurement, and fieldtheory.