We address the dynamics of large-scale structures, specifically the mass discrepancy in galaxy clusters (the Dark Matter problem), by extending the Virial Theorem to open thermodynamic systems embedded in a higher-dimensional Bulk. We demonstrate that the standard Virial Theorem (2K + U = 0) is fundamentally incomplete for brane-world geometries subject to external boundary pressure. Starting from the Collisionless Boltzmann Equation, we rigorously derive a Modified Virial Equation incorporating a boundary tension term exerted by the gravitational interaction with a parallel Shadow Brane. We show that at large radii, the effective gravitational potential deviates from the Newtonian 1/r behavior, acquiring a confining Linear Tension term (V proportional to r) due to Brane Bending. Crucially, we resolve the ambiguity between coherent galaxies (dominated by Bulk Gravitomagnetism) and virialized clusters (dominated by Scalar Tension) via a vector cancellation mechanism in the phase space ensemble. Finally, we provide an analytical derivation of the empirical Constant Surface Density law (SigmaDM approx 140 solar masses per pc²) observed in Dark Matter halos. We show that this universal constant is a direct measure of the inter-brane tension, offering a unified geometric explanation for the Dark Sector without requiring non-baryonic particles.
Giovanni Frisina (Thu,) studied this question.