Primordial Coherence Theory (PCT): Formal Structure, Geometric Foundations, and Emergent Topology of the Field \phi

This work presents a complete and foundational formulation of the Primordial Coherence Theory (PCT), introducing the scalar field \phi as a temporally independent, spatially coherent foundational medium from which observable matter emerges. We develop the full geometric and topological structure induced by \phi, from local gradient behavior to metric deformation, curvature tensors, and global topological invariants. We show that the emergence of the visible field \psi occurs as a localized instability in the coherent dynamics of \phi, characterized by energetic interference and curvature concentration. The formalism proceeds from classical differential geometry to Riemannian curvature, and culminates in a topological analysis through Gauss–Bonnet integrals and cohomological invariants. Finally, the theory is reconstructed in a formal logical setting using Type Theory and Homotopy Type Theory (HoTT), where \phi is treated as a generative object in a constructive system. Coherence and emergence are thus revealed as provable, internal properties of the field \phi within a purely mathematical framework.