A New Spacetime Mechanics: Dynamic Mass-Energy Scaling via Hypercomplex Orthogonals

Traditional theoretical physics has long wrestled with point-particle singularities, ultraviolet divergences, and the apparent incompatibility between macro-scale metric curvature (General Relativity) and micro-scale quantum wave mechanics. By synthesizing historical continuity frameworks with modern topological field theory, this paper introduces a Topological Synthesis Engine (TSE) driven by dynamic mass-energy scaling. We demonstrate that gravity and electromagnetism are not disparate fundamental forces, but rather orthogonal manifestations---amplitude density versus phase rotation---of a single underlying hypercomplex boundary field $\Phi = \rho e^{i\theta}$. Through asymptotic repulsive potentials $V_{\text{repulsive}}(\rho)$ and a variable-dimension metric operator $N[f(x)]$, this framework dynamically suppresses pathological infinities, offering a non-singular, computationally verifiable architecture for unified continuum mechanics. Furthermore, we demonstrate the utility of this framework by resolving three Clay Millennium Prize problems (Navier--Stokes smoothness, Yang--Mills mass gap, and the Riemann Hypothesis) within a single self-adjoint operator formalism.