On a Class of Non-Local Elliptic-Transport Systems in Score-Guided Probability Flow on $SE(3)$: Score-Singularity Resolution, Space-Time ADN Shift, and Markov Mean-Field Games

We establish mathematical foundations for optimal spatiotemporal vector fields governing continuous score-guided probability flows over bounded submanifolds M ⊂ SE(3) on the Special Euclidean group SE(3) R³ × SO(3). Modeling dynamic density evolution via a score-perturbed Lie-algebraic continuity equation under zero-flux boundary conditions, we resolve the non-local score drift singularity s(g,t) = ∇SE(3) log ρ(g,t) as ρ→ 0⁺ via weighted parabolic flux dissipation bounds. Using left-invariant exponential group mollifiers, we derive boundary-preserving DiPerna--Lions commutator estimates tracking structure constants ᵢⱼᵏ\. Second, interpreting the optimality system as a 7D space-time elliptic operator L7D = -ΔSE(3) - ∂ₜ² with zero-order corner compatibility on ∂ M × \0,T\, we establish Agmon--Douglis--Nirenberg Wₚ²(QT) shift estimates tracking Lie algebra constants, boundary geometry \|∂ M\|C², and Ricci curvature \|RicSE(3)\|L^∞, yielding a full C^∞(Q̄T) Schauder bootstrap expansion. Third, we extend this system to non-cooperative Markov Mean-Field Games (MFGs) on $SE(3)$, establishing score-guided Nash equilibria via Schauder--Tychonoff fixed points. Crucially, the constant diagonal Ricci curvature RicSE(3) = diag(0,0,0,1/2,1/2,1/2) from $SO(3)$ induces a sharp Bochner--Weitzenböck spectral gap boost λ₁(-ΔSE(3)) ≥ λ₁(-ΔR³) + 1/2 > 0. This positive rotational curvature enhances the Lie-algebraic Poincaré coercivity constant, demonstrating that intrinsic group curvature stabilizes score-guided flows against dispersion and expands the critical Nash uniqueness horizon TSE(3)^* beyond flat Euclidean limits.