Non-Equilibrium Holographic Brane Cosmology in AdS5. I-C: Graviton Spectral Decomposition, Higuchi Ghost Protection, and Non-Linear Vainshtein Screening on the Karch-Randall Brane

In braneworld realizations of quantum gravity where our four-dimensional de Sitter universe (dS4) emerges as a metastable Karch-Randall 3-brane embedded in an asymptotically Anti-de Sitter five-dimensional bulk (AdS5), the localized four-dimensional graviton is not a strictly massless zero mode. Due to the non-vanishing positive curvature H > 0 on the brane and radiative coupling across the AdS5 throat, the graviton acquires a small effective mass resonance m_g > 0. In linearized Fierz-Pauli theory, any non-zero graviton mass triggers two severe infrared pathologies: the van Dam-Veltman-Zakharov (vDVZ) discontinuity, which predicts a catastrophic 25% deficit in the relativistic deflection of light (gamma_PPN = 1/2), and the excitation of a sixth polarization degree of freedom—the Boulware-Deser ghost. In this third paper of the series (Paper I-C), we prove that the non-equilibrium Karch-Randall brane geometry established in Papers I-A and I-B is rigorously free of both pathologies. First, by performing a complete Sturm-Liouville spectral decomposition of transverse-traceless metric fluctuations in conformal bulk coordinates, we derive the exact hyperbolic Poschl-Teller potential and demonstrate that the Kaluza-Klein continuum exhibits a universal mass gap m_gap^2 = (9/4) H^2 that lies strictly above the four-dimensional de Sitter Higuchi unitarity bound (m^2 >= 2 H^2). Second, we prove that the brane-bending scalar mode pi(x) is protected against ghost instabilities by the emergent Deser-Waldron gauge symmetry and dissipates radiatively into the AdS5 bulk as a metastable Breit-Wigner resonance. Finally, we derive the non-linear cubic Galileon effective action on the brane with strong coupling scale Lambda_V = (M_Pl * m_g^2)^(1/3) and solve the radial field equations inside the Vainshtein radius r_V = (r_s / m_g^2)^(1/3) ~ 100 pc, proving that scalar fifth forces in the Solar System are suppressed by (r/r_V)^(3/2) ~ 10^(-7.5), restoring General Relativity well within Cassini and Lunar Laser Ranging bounds.