Linear Growth of Cosmic Structure in a Recursive Geometric Field Framework

This work asks a simple but necessary question: if cosmic acceleration comes from geometry rather than dark energy, can structure still grow in a physically consistent way? Within the Recursive Geometric Field framework, late–time acceleration is treated as an effective background property of spacetime, not as a new dynamical field. Local gravity is left untouched, and the standard linear perturbation equation is used. The only input is the expansion history already constrained by supernova data. The goal is not to redesign gravity, but to check whether matter perturbations behave sensibly inside this geometric expansion regime. The result is reassuring. Matter fluctuations grow stably, with a mild and scale–independent suppression compared to ΛCDM, but without instabilities or pathologies. The predicted growth rate and fσ8(z) remain compatible with current observations, and the framework stays partially degenerate with ΛCDM at present precision. This shows that a purely geometric origin of acceleration does not break structure formation. The Recursive Geometric Field remains internally consistent, observationally viable, and fully falsifiable at the level of linear cosmic growth.