The Geometric Architecture of Vacuum: Dual Integration, Dimensional Transmutation, and the Origin of Flavor

【20260128v1.2】I moved to a different GitHub repository.This paper presents the ``GAV Mass-Generation Scheme,'' a unified, near-parameter-free framework for the origin of flavor structures in the Standard Model, strongly constrained by discrete geometric structures. We demonstrate that mass generation is not a mere scalar accumulation but forms an exponential hierarchy based on the eigenvalues of Casimir operators, resulting from ``Dimensional Transmutation'' associated with vacuum phase transitions. Analysis of the geometric index $K$ relative to the electron mass reveals that lepton mass generation halts at $K \approx 24$, corresponding to the $SU(5)$ Grand Unified symmetry. In contrast, quarks bypass this limit via the Casimir selection rule, traversing the exceptional group $G_2$ ($K \approx 14$) to reach the deep layer of $SO(32)$ ($K \approx 32$) associated with superstring theory. A crucial, testable prediction of this theory is the absolute mass of the third-generation neutrino, identified as $m_{\nu 3} \approx 0.030$ eV. This is derived from the ``Geometric Seesaw Mechanism'' based on the self-duality of the tetrahedron (index $K=-48$). This value is consistent with current cosmological upper limits and is specific enough to be directly verified by near-future experiments. Furthermore, we derive a Higgs self-coupling constant of $\lambda \approx 1/8$ from the geometric relationship between the Higgs mass and the vacuum expectation value. These results strongly suggest that the observed mass spectrum of all fermions is determined almost uniquely by the discrete geometric symmetry and stability of the vacuum.