Non-Abelian Dielectric Branes, TT-bar Cutoff Geometry, and Moduli Stability in Warped Throats

We investigate the non-perturbative dynamics of a stack of N coincident Dirichlet 3-branes (D3-branes) in 10-dimensional Type IIB superstring theory embedded within a flux-compactified warped throat. First, we analyze the non-Abelian Dirac-Born-Infeld and Wess-Zumino actions in the non-linear coupling regime 2\pi\alpha'_{eff} F_{ab} ~ O(1). In the presence of background Ramond-Ramond 3-form flux, non-Abelian scalar commutators induce dielectric brane polarization (the Myers effect). Worldvolume instanton and center-vortex relaxation energetically favor the decomposition of the transverse gauge algebra into irreducible fuzzy-sphere sectors labeled by integer Chern numbers p \in N, generating primitive periodic closed-string geodesics of proper length L_p = l_{AdS} \ln p. Second, we implement a finite radial ultraviolet boundary r = r_c via an irrelevant TT-bar-like deformation of the dual boundary field theory. In this finite-cutoff geometry, bulk gauge and graviton fluctuations satisfy a well-posed regular Sturm-Liouville boundary value problem. We demonstrate the holographic generation of a strictly positive, discrete mass gap \Delta = m_1 > 0 for color-singlet glueball excitations, bounded below by the infrared dielectric sphere radius. Third, we evaluate the cumulative one-loop Casimir stress-energy of closed-string throat modes and its gravitational backreaction on the compactification moduli of the ambient Calabi-Yau manifold. Requiring the preservation of the Euclidean Coleman-De Luccia bounce action against catastrophic non-perturbative vacuum decay (\delta B \ge 0) imposes a rigorous spectral constraint on the radial dilation operator, preventing secular parametric amplification. Finally, we discuss the mathematical properties of the resulting discrete spectrum, noting its structural resemblance to hyperbolic trace formulas and spectral determinants.