Phase Structure and Isotropization Theorem in Nonlocal Gravity with Temporal Memory (Paper XXXIX)

Nonlocal gravitational theories with temporal memory admit a sharp classification based on the integrability of the memory kernel. This property fully determines the infrared fate of anisotropic degrees of freedom and defines distinct universality classes. We prove a general isotropization theorem: for any causal memory kernel with finite norm, ∫₀^∞ K(t) dt < ∞, the shear necessarily decays to zero at late times. Gravitational memory in this regime acts as a non-Markovian friction mechanism that enforces isotropy. Persistent anisotropy arises exclusively at a critical boundary defined by the scale-invariant kernel K(t) ~ 1/t, where the kernel norm diverges. This regime exhibits logarithmic relaxation and exact scale invariance, corresponding to a nontrivial fixed point of the renormalization group flow in the space of nonlocal theories. These results establish that the space of nonlocal gravitational theories is organized by renormalization group flow, reinterpreting persistent anisotropy as a critical phenomenon rather than a stable phase.