Consistent orientation of surface normals is a global problem in computational geometry. Orientation is intrinsically global: local geometry determines a normal line, but not which of its two directions is outward. We study the orientation problem at the continuum level for a smooth closed surface $\Gamma=\partial\Omega$. Starting from the Newtonian single-layer potential, we consider its averaged boundary divergence and curl traces, denoted by $B$ and $C$. The classical scalar Gauss constraint $B\varphi=1/2$ alone does not determine a free vector density. We show that adding the curl condition $C\varphi=0$ makes the system complete. More precisely, for every $s\in\R$, \[ \|\varphi\|_{H^s(\Gamma;\R^3)} \le C_{\Gamma,s} \bigl(\|B\varphi\|_{H^s(\Gamma)}^2+ \|C\varphi\|_{H^s(\Gamma;\R^3)}^2\bigr)^{1/2}. \] $P=(B,C)$ is therefore injective with closed range, and $P\varphi=(1/2,0)$ has the unique solution $\varphi=n_\Omega$. The Clifford Cauchy involution $T^2=I/4$ is classical. The new result proved here is the same-order lower bound for $P=(B,C)$; its problem-specific proof step is the factorization $TM=JP$, which identifies $B$ and $C$ with the scalar and bivector parts of the same Cauchy singular integral. This factorization also yields a compatibility condition for general boundary data. If the samples are distributed according to a positive density $\dd\mu=\rho\,\dd S$, the weighted system recovers $q=n_\Omega/\rho$ uniquely, so normalization removes the unknown sampling density and returns the outward orientation. Regularized point-cloud experiments illustrate the continuum result.

We study an accelerated Langevin equation in which the classical overdamped Langevin drift is supplemented by a nonlocal gradient velocity determined through a target-weighted Poisson equation. The target \(p\) is only assumed to be strictly positive and need not be normalized. Writing the dynamics in terms of \(\omega=\log(p/q)\) yields a coupled parabolic--elliptic system whose parabolic component contains the usual Langevin diffusion, whereas the elliptic component generates a global transport. The system is invariant under multiplication of \(p\) by a positive constant. On a bounded smooth domain with homogeneous Neumann boundary conditions, we establish global existence and uniqueness of classical solutions for strictly positive \(C^{2+\alpha}\) target weights and compatible \(C^{2+\alpha}\) initial data, with full \(\alpha\)-H\"older regularity up to the initial time. For long-time behavior, the oscillation of \(\omega\) decays at the explicit rate \(e^{-t}\). For normalized initial densities, the solution density converges exponentially to the normalized target. Moreover, an exact entropy-dissipation identity holds relative to the normalized target, in which the relative Fisher information and the Jeffreys divergence give the two dissipation mechanisms associated with the Langevin and weighted-Poisson components. In particular, KL divergence also converges to zero with rate $e^{-t}$ without a convexity assumption on the target.