BECU-OΛON V6: Stable Phase-Triggered Geometric Response in Structure Growth

We present Version 6 of a phase-triggered geometric response model for cosmic structure growth. The framework introduces a minimal modification to the standard growth equation through a structure-dependent response function μ(k,a), controlled by a critical structural parameter K. The model preserves ΛCDM-like behavior at early times while allowing a weak, controlled deviation at late times through a smooth transition mechanism. Numerical integration shows that the growth rate fσ8(z) remains consistent with ΛCDM predictions, while the gravitational potential exhibits a mild enhancement. The ISW kernel remains stable and free of instabilities. The model is validated through comparison with observational data, demonstrating consistency within current uncertainties. This establishes a stable and minimal baseline framework for structure-dependent gravitational response without introducing additional matter components. Future work will explore stronger scale dependence and improved agreement with observational constraints.