Adelic Wave–Coherence Field Theory (AWCFT): A Prospective Mathematical Completion of the Unified Flow–Coherence Theory (UFCT) and the Wave Interference Theory of Energy and Mass (WITEM)

Adelic Wave–Coherence Field Theory (AWCFT) presents a prospective mathematical completion of the Unified Flow–Coherence Theory (UFCT) and the Wave Interference Theory of Energy and Mass (WITEM). Rather than introducing a new physical theory, AWCFT develops a deeper mathematical framework intended to derive the effective macroscopic equations of UFCT from modern mathematical physics. The framework combines adelic harmonic analysis, Tomita–Takesaki modular theory, persistent sheaf cohomology, metric-affine geometry, Einstein–Cartan torsion, noncommutative spectral geometry, and variational principles into a single hierarchical construction. Within this framework, spacetime, time, coherent flow, matter, gravitation, halo structures, and weak gravitational lensing are treated as successive emergent levels of coherent informational organization. The manuscript is organized as a mathematical research program whose physical relevance ultimately depends on future observational validation through the previously developed UFCT observational framework. This work is intended as a mathematical foundation rather than a claim of observational confirmation.