Addressing S_8 and H_0 Tensions via Conformal Attractor in Horndeski Gravity: Stability and GW Predictions

This research presents a formal derivation of cosmological dynamics within the cubic Horndeski gravity framework. By identifying a conformal attractor phase, we bridge the gap between phenomenological ansätze and fundamental field theory. Key highlights of this version: Theoretical Foundation: Derivation of the field equations from a non-minimally coupled Lagrangian. S8 Tension Resolution: Explanation of how modified cosmic friction suppresses structure growth. Numerical Validation: Python-based integration confirming a 2.80% gravitational wave attenuation signature at z=0.5. Observational Consistency: Alignment with recent DESI 2024 results. This document provides a falsifiable prediction for future GW observatories such as LISA and the Einstein Telescope