MSFCM Paper VI: Velocity Field Anisotropy in Cosmic Filaments — The Q₂₂ Tensor as the Physical Origin of the Four-Quadrant Jet–Filament Alignment Flip

We demonstrate analytically that the NFW virial identity forces vᵢnf/σₑff = constant for any velocity scaling, ruling out potential-geometric mechanisms as the origin of the four-quadrant flip detected in Paper IV. We resolve this by direct measurement of the gas velocity field anisotropy from LoTSS DR2 × Tempel 2014 (N=80, 695 pairs). The second-order alignment tensor Q₂₂ (ρ, r⊥) ≡ ⟨cos2δ⟩ changes sign at ρₜhresh=2. 09 and is independent of r⊥ across 0, 3 Mpc, confirming a bulk anisotropy of the filament gas velocity field. The excess of perpendicular orientations in Q3 relative to Q1 is Δf⊥=+0. 0975, detected at p=7×10⁻⁶⁹ (KS) and p=2×10⁻²⁴⁵ (χ²). Prediction for Paper VII: MBH ∝ σ⁴fil, testable with JWST at z>7.