Time Geometrization of Manifold G: A Research Roadmap for High-Dimensional Gauge Theories, Lattice QCD, and the Early Universe

This paper proposes a systematic research program within the framework oftime-structured manifolds, integrating high-dimensional gauge theories, non-perturbativelattice computations, and early universe cosmology. The core idea is to treat timegeometry as a dynamical entity whose curvature and topological structure (e.g.,cohomology dimension) are coupled to the effective dynamics of physical systems.The program achieves a three-layer closed loop from mathematical constructionto physical prediction: First, in four-dimensional non-Abelian gauge theories, werigorously prove the existence of a Temporalized Geometric Fixed Point (TGP)on the θ = 0 plane, which simultaneously satisfies time-geometric flatness andglobal causal uniqueness, and we derive its scaling laws. Second, we design anon-perturbative scheme for measuring the corresponding geometric structures inlattice Quantum Chromodynamics via gradient flow and effective actions, and propose constraining the effective cosmological constant through basin transition rates.Third, by treating the TGP as an attractor for inflationary dynamics, we deriveunique imprints testable by next-generation cosmic microwave background observations, including a small correction to the tensor-to-scalar ratio (∆r ≈ −1.7×10−3)and specific primordial non-Gaussianity signals. The paper aims to provide a unified explanatory framework for a series of fundamental physical problems based onthe geometric and topological attributes of spacetime itself, and establishes clearfalsifiable testing paths.