Within the axiomatic framework of Arithmetic Rigid K\"ahler Geometry (ARKG), we rigorously establish the mathematical and physical equivalence among G\"odel incompleteness oscillation, spontaneous symmetry breaking, the Higgs mechanism, and Yukawa couplings. The theory identifies Axiom~2.5 (G\"odel oscillation dynamics) as the unique, primordial mass-generation mechanism without any additional assumptions. Neither the Higgs field nor Yukawa couplings are taken as fundamental postulates; instead, they emerge as low-energy effective descriptions of the G\"odel oscillation. We give a precise definition of the Scaleon field (the spatial incarnation of the G\"odel oscillation) and derive its dynamical equation, thereby unifying mass scales, energy hierarchies, scale symmetry breaking, and the singlet force oscillation. We prove that the Higgs and Yukawa terms in the ARKG Lagrangian are uniquely derived from the axioms, rendering the system self-consistent, free of redundancy, and free of dual mechanisms. The entire construction strictly follows the original ARKG axioms, geometry, and unified field equations, completing a first-principles closure from mathematical incompleteness to the origin of particle masses.
Yaao Wang (Mon,) studied this question.