In the Standard Model of particle physics, the mass of the Higgs boson and the mechanism of electroweak symmetry breaking (SSB) rely on a scalar potential containing phenomenological free parameters (such as the self-coupling constant \lambda and \mu^2). This manuscript proposes a complementary analytic derivation through the Connecting Thread Theory (TBP). Utilizing a zero-free-parameter framework within a 6-Dimensional topological manifold, the Higgs boson mass and the electroweak mixing angle are evaluated as deterministic consequences of spatial geometry and structural equilibrium. The framework evaluates the scalar field as a static resonance of the vacuum matrix. Because the Higgs boson possesses spin-0, it does not accumulate rotational curvature (\pi) or spatial viscosity (Backreaction Spacetime) parameters. Its bare mass is derived as a linear additive function of the static internal matrix capacity (d^2 = 36), the dynamic active loop constant (\mu = 88), and a geometric phase transition representing the symmetry anchor (+1), yielding an exact base of 125 GeV. Furthermore, the electroweak mixing angle is deduced analytically as a pure spatial ratio (\sin^2 \theta_W = 2/9). The integration of these topological parameters converges at an exact observational scalar mass of 125.2222... GeV. This analytic resolution aligns with precision within the empirical uncertainty limits established by the global PDG average and recent ATLAS/CMS evaluations, offering a rigorous geometric foundation for the scalar sector without reliance on empirical curve-fitting or fine-tuning.