Omegaon Model v3.0 — Gauge-Invariant Scalar and Tensor Perturbations Across a Non-Singular Bounce

This paper develops a fully gauge-invariant treatment of scalar and tensor perturbations in the Omegaon Model using the Mukhanov–Sasaki formalism. We derive the effective perturbation equations on the non-singular Ω-core background and analyse mode evolution across the bounce. We demonstrate that the Mukhanov–Sasaki potential remains finite and localised in time, defining a well-posed scattering problem for cosmological perturbations. Both scalar and tensor modes propagate smoothly through the high-curvature regime without instability or loss of predictability. These results confirm that the non-singular bounce is compatible with gauge-invariant perturbation theory and reinforce the physical viability of the Omegaon framework beyond background-level dynamics.