Part I: Dynamic Geometry of Spacetime and Universal Constant

We introduce the scalar deformation function C(v), which captures how local geometric structures respond to motion at velocity v. This function exhibits well-defined analytic behavior and defines a critical velocity vc beyond which geometry undergoes compression. By extending C(v) into an evolution flow C(v, τ), we construct a scalar analogue of Ricci flow that provides a smooth, singularity-free mechanism for geometric evolution toward symmetric configurations. While this scalar flow suggests a pathway toward topological classification of three-manifolds through conformal smoothing and energy minimization, full spectral rigidity and topological completeness are addressed in subsequent parts of this series. The framework presented here establishes the analytic foundation and geometric intuition underlying the spectral formalism developed in the following works, with potential applications to cosmology, quantum geometry, and spectral analysis via the operator Ĉ.<br><br>