Galileon Bounce and Cyclic Cosmology in the Dual‑Brane Quantum Tunneling Model

We complete the Dual-Brane Quantum Tunneling Model (DBQTM) by constructing anon-singular cyclic cosmology in the Randall--Sundrum framework augmented witha cubic Galileon self-interaction on the visible brane. The radion field,which describes the inter-brane distance, simultaneously drives the ekpyroticcontraction (w 57) and mediates a ghost-free bounce through transientviolation of the Null Energy Condition. We solve the coupledFriedmann--Galileon equations numerically and demonstrate that the bouncesatisfies all stability requirements: NEC violation (ρ + p < 0),ghost-freedom (Qₛ > 0), and gradient stability (cₛ² > 0),the latter verified within the DHOST (Degenerate Higher-Order Scalar-Tensor)framework.Post-bounce, the universe transitions through kinetic domination, reheating atTRH = 3.1 × 10⁹~GeV (Paper~III), radiation, matter, and darkenergy epochs, reproducing the standard thermal history. The ekpyroticsteepness $c = 13.2$ yields nₛ = 0.965, consistent with Planck. The tensorspectrum is blue-tilted (nₜ = +0.070), providing a decisiveobservational discriminant against inflationary red tilt. The three-throatKlebanov--Strassler geometry predicts gravitational-wave echoes from theinflation throat (kd₃ = 3.49) at ~ 3% of the main signal amplitude,detectable by the Einstein Telescope. Together with Papers~I--III, thisestablishes the DBQTM as a complete, falsifiable framework specified by sevenintegers and one continuous parameter.