Alpha-Omega Theory: A Geometric Nonlocal Framework for UV-Complete Gravity and Emergent Scalar Structure

We propose the Alpha-Omega (AO) Theory, a geometric nonlocal framework for gravity that achieves ultraviolet (UV) regularization while preserving infrared (IR) consistency with General Relativity. The theory is constructed from a ghost-free nonlocal form factor acting on curvature invariants, leading to exponential suppression of high-energy modes. A central result is the emergence of a scalar geometric structure, , defined via a covariant kernel representation of spacetime curvature. Distinct physical sectors arise naturally through scale-dependent projections of this structure. The framework yields a UV-complete effective description of gravity, avoids additional ghost degrees of freedom, and predicts mild observational deviations, including suppression of primordial tensor modes. The Alpha-Omega Theory provides a minimal and internally consistent candidate for geometric unification across scales.