Structure Growth on an Effective Geometric Expansion Background

This paper follows a simple but necessary line of reasoning: if late–time cosmic acceleration comes from geometry rather than a new energy component, then that same geometric expansion must also allow structures to form in a sensible way. Within the Recursive Geometric Field (RGF) framework, the expansion history is treated as an effective geometric background fixed by observations, while gravity at local scales remains unchanged. No new fields are added, no modifications to the Einstein–Hilbert action are introduced, and dark energy is not assumed as a fundamental substance. Using this geometric expansion law, the standard linear growth equation for matter perturbations is solved in the late–time universe. For the observationally preferred value kβ≈1.33, the growth of structure is smooth, monotonic, and free of instabilities. Compared to ΛCDM, clustering is mildly suppressed at low redshift. This suppression is not abrupt or scale–dependent; it reflects the fact that the geometric expansion delays the full transition into acceleration, leaving slightly less time for structures to grow. When growth observables are constructed, the predicted f(z) and fσ8(z) remain well within current observational error bars. Even without adjusting gravity, the model stays compatible with redshift–space distortion measurements, and a small phenomenological correction to the effective gravitational coupling only adds flexibility, not necessity. The main takeaway is that a purely geometric origin for cosmic acceleration does not automatically fail when confronted with structure growth. At the linear level, the framework is consistent, testable, and observationally viable.