Canonical Quantization, Relational Schrödinger Dynamics of the Quantum Universe

Within the framework of Emergent Time Theory (ETT) we investigate the Problem of Time in canonical quantum gravity. Building on the classical construction of an internal clock developed in preceding parts of this work, we examine how the Hamiltonian constraint of a homogeneous cosmological system can be reformulated to describe physical evolution with respect to an emergent relational time.

Starting from the ADM formulation and its homogeneous FLRW reduction, we analyze the canonical structure of the gravitational and scalar-field degrees of freedom, including the role of the conserved internal Noether charge. The Hamiltonian constraint is resolved on the selected regular branch, and the emergent clock is constructed locally as a phase-space variable rather than identified directly with the scalar field. Through the reduction of the original symplectic structure, local canonical variables and a true Hamiltonian governing relational dynamics are obtained.

The resulting reduced Hamiltonian framework provides the classical basis for canonical quantization and the formulation of a relational Schrödinger-like evolution equation.

The construction is restricted to the locally regular domain of the relational clock and reduced symplectic structure. The global extension of the clock, the dynamical accessibility of singular boundaries, and the full analysis of self-adjointness and unitary quantum evolution are left for subsequent investigations.