First-Principles Derivation of Ground-State Inertial Density, the Higgs Mechanism, and Yukawa Couplings from the Substrate

Abstract The Synthesis Framework establishes the ground-state inertial density of the Substrate, the electroweak vacuum expectation scale, the Higgs potential, spontaneous symmetry breaking, and fermion Yukawa couplings through a unified first-principles derivation. This article develops the mathematical connection between the triadic structural organization of the Substrate and the physical mechanisms governing inertial response and particle-mass generation. The derivation begins with the three primary substrate fields—the Stabilizing Field, Conserving Field, and Motive Field—which generate one collective mode and two independent relative modes. Their projector structure determines the normalized relative-mode contribution as $2/3$. Through the triadic closure relation, the normalized ground-state inertial density follows as $\rho_{0,\mathrm{norm}}=K-2/3$. Using the independently derived vacuum stiffness $K=11.350$, the resulting normalized inertial density is $10.6833333$. Its dimensional value is established through the substrate propagation equation as $1.18837\times10^{-10}\ \mathrm{s^2/km^2}$. This ground-state inertial foundation supplies the basis for deriving the electroweak energy scale $\Omega_{\mathrm{EW}}\approx68.7183\ \mathrm{GeV}$ through the topological structure of the triadic substrate manifold. Combining this scale with the foundational substrate invariants establishes the substrate vacuum expectation scale $v_{\mathrm{sub}}=245.9891\ \mathrm{GeV}$. The effective Higgs potential is reconstructed from vacuum stiffness, synthesis pressure, and normalized inertial density, establishing spontaneous symmetry breaking as a mechanical phase condition of the Substrate. The Higgs mechanism is thereby derived as the field-theoretic manifestation of the Substrate’s underlying elastic structure and recursive equilibrium. Fermion Yukawa couplings are derived as geometric modal overlap integrals involving the substrate mass operator and stored equilibrium mass. The resulting electron and proton couplings are $2.9378\times10^{-6}$ and $5.3942\times10^{-3}$, respectively. Together, these results establish a continuous first-principles account connecting triadic substrate organization, inertial density, electroweak symmetry breaking, and fermion mass generation through the common mechanical structure of the Substrate.