History-Dependent Gravity: Nonlocal Memory Kernel, Critical Phase Boundary β_c(z), and the Signature dγ/dz > 0

We investigate a phenomenological modification of cosmological structure formation in which the growth of density perturbations de pends explicitly on the past expansion history through a nonlocal mem ory kernel. We demonstrate that even weak nonlocality produces percent-level deviations in the observable fσ8(z) while inducing a qualitatively dis tinct redshift dependence of the growth index γ(z). In particular, we find that γ(z) increases with redshift, in contrast to both ΛCDM and f(R) gravity, where γ(z) is constant or decreasing. We argue that this behavior originates from the non-stationary nature of the memory kernel and cannot be captured within standard time local effective field theory descriptions. The key parameter β controls the coupling to expansion history: β ≠ 0 implies non-stationary memory and necessarily dγ/dz > 0, distinguishing this class from EFT-based modifications. This provides a new observational discriminator between history dependent and local theories of gravity.