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

Abstract 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 (+ p 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 atT ₑ₇ = 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.