RG-Induced Spectral Diffusion in History-Dependent Gravity: Log-Normal Gravitational Wave Signatures for LISA

We derive the renormalization group running of the memory kernel in history-dependent gravity from the functional renormalization group framework. The kernel width obeys β_Λ = cξ, leading to spectral diffusion with width σ²(N) = Λ₀N + ½cξN². The resulting tensor spectrum exhibits a log-normal distortion with finite running dn_T/d ln k = -1/σ². Fisher forecasting shows LISA can detect ξ ≳ 10⁻³ at ~10σ significance.