A Unified Geometric, Variational, and Minkowski-Embedded Perspective on the Koide Leptonic Mass Relation

The Koide relation for charged leptons, [Formula: see text], 2/3, remains one of the most precise empirical regularities among Standard Model mass parameters. We examine its structure through three complementary perspectives: (i) a geometric formulation in which the vector of lepton square-root masses forms a [Formula: see text] angle with the democratic direction in [Formula: see text]; (ii) a Minkowski-type embedding in which the Koide condition corresponds to a null-cone constraint in a (1, 2) pseudo-Riemannian space; and (iii) a variational perspective in which the Koide condition is treated as a distinguished level set of a scale-invariant functional. A perturbative analysis clarifies the local geometry of the Koide condition: the relation is exactly invariant under uniform rescaling of all three masses (the radial/scale direction in [Formula: see text]-space), while it is first-order sensitive to angle-changing perturbations that rotate [Formula: see text] relative to the democratic axis, thereby distinguishing soft (scale) and rigid (angle-changing) directions in mass space. Extensions to other fermionic sectors demonstrate that neither up-type nor down-type quark triplets satisfy Koide-like relations with comparable precision, and neutrino and composite baryon masses lie in distinct angular regions within the same framework. These comparisons underscore the uniqueness of the charged lepton case and establish a coherent geometric and variational basis for future interpretations of fermion mass patterns.