Geometric Origin of the Lepton Mass Spectrum: The Koide Formula as a Null-Vector Condition in Split-Quaternionic Space

We propose a geometric derivation of the number of fermion generations and the charged lepton mass relation (Koide formula) within the framework of the Higher-Derivative Lee-Wick (HDLW) theory based on Split-Quaternionic Krein Algebra (PQA). First, we identify the three generations of matter with the three imaginary units {e1,e2,e3} intrinsic to the split-quaternion algebra. Second, we demonstrate that the Koide relation K = 2/3 is exactly equivalent to the condition that the mass-amplitude vector lies on the light cone (zero-norm hypersurface) of the internal generation space. This geometric constraint arises naturally from the indefinite metric signature (+,−,−) required for UV finiteness, where the single physical vacuum mode is balanced against two ghost-like geometric degrees of freedom. This framework offers a parameter-free explanation for the lepton mass structure without fine-tuning.