Micropolar Neutrinos from Triadic Orthogonal Calculus: Deriving Mass Suppression and the PMNS Matrix from the FCC Vacuum Triad

The Standard Model offers no dynamical explanation for the extreme mass hierarchy between neutrinos and charged leptons, nor for the striking disparity between their flavor mixing matrices (the small-angle CKM versus the large-angle PMNS) [12]. We propose a unified, mechanical origin for both phenomena based on the elasticity of a structured Face- Centered Cubic (FCC) vacuum lattice [4]. Modeling the vacuum as a Cosserat (micropolar) continuum [9, 10], we decompose the local field into two independent degrees of freedom: translational displacements u (charged fermions) and microrotational pseudo-vectors ϕ (neu- trinos). Because pure microrotations do not stretch invariant geometric bonds, neutrinos bypass the massive Bulk Modulus (Ebulk ) and couple only to the weak Micropolar Twist Modulus (γ). By projecting this 3D bulk mass onto the 2D weak interaction boundary, we derive a geometric mass suppression factor mobs ≈0.0503 eV, yielding an exact geo- metric match to the observed atmospheric mass splitting scale (∆m2 31 ≈2.53 ×10−3 eV2) [12]. Dimensional quenching along lattice axes strictly mandates the Normal Hierarchy (m3 ≫m2 > m1 ≈0), providing a falsifiable prediction for the upcoming JUNO exper- iment. Furthermore, because these unanchored microrotations propagate through the 3D lattice bulk rather than along 2D interaction surfaces, their mixing angles are governed by 3D volumetric projections. We analytically derive the PMNS Atmospheric angle θ23 = 45◦ , the Reactor angle sin2 θ13 ≈0.020 (derived from the Regge deficit gap [5]), and the Solar an- gle sin2 θ12 ≈0.319 directly from the fundamental symmetry vectors of the Cuboctahedron unit cell.