Geometric Unification of Lepton Mass Structure and Vacuum Stability via Pseudo-Quaternion Algebra

Description: [Abstract] The replication of fermion generations remains an open question in the Standard Model. We propose a framework where the fundamental geometry of spacetime is governed by an algebra isomorphic to the Split-Quaternions ($\mathbb{H}_\eta$). We argue that the irreducible representations of this algebra naturally motivate the Koide mass formula ($Q=2/3$) as a geometric null condition. Furthermore, demanding local gauge invariance in this geometry leads to a higher-derivative scalar dynamics. Based on this structure, we derive an intermediate effective scale $M \approx 11.3$ TeV, which satisfies a Generalized Veltman Condition for the cancellation of quadratic divergences. Crucially, we show that the metric signature required for the mass hierarchy mathematically implies a topological Berry phase of $\pi$. We argue that this geometric feature allows the scalar progenitor to mimic Fermi statistics in the infrared limit. Finally, we show that this framework offers a quantitative candidate for the resolution of the proton radius puzzle ($\delta r_p \sim 0.003$ fm) and provides an electroweak oblique correction of the order required to explain the W-boson mass anomaly ($m_W \approx 80.357$ GeV). [Significance & Related Work] This work presents a geometric derivation of the scale $M \approx 11.3$ TeV, which aligns with the dynamical derivation based on universal vacuum-energy cancellations presented in our companion paper: "Uniqueness of the Higher-Derivative Operator Class for Universal Vacuum-Energy Cancellations and Higgs Naturalness" (arXiv:2512.16955). The convergence of these two independent approaches—geometric (mass structure) and dynamical (vacuum stability)—onto a single energy scale suggests a robust underlying structure beyond the Standard Model.