From OPE Associativity to Gravitational Wave Signatures: Critical Dimensions and AdS/dS Transition

We present a framework connecting the algebraic consistency of the Operator Product Expansion (OPE) in Conformal Field Theories to observable signatures in the stochastic gravitational wave background (SGWB). By interpreting the conformal 6j symbols as crossing kernels, we extract the anomalous dimensions $\gamma_{n,J}$ of double-trace operators from the singularities of the crossing equation. Through the AdS/CFT correspondence, these algebraic shifts are holographically reinterpreted as effective mass shifts in the Anti-de Sitter bulk. We identify a critical anomalous dimension, $\gamma_c$, which marks the threshold where the effective mass violates the Breitenlohner-Freedman bound. Upon analytic continuation to de Sitter space ($L_{\text{AdS}} \to iL_{\text{dS}}$), we demonstrate that crossing this threshold ($\gamma < \gamma_c$) forces the dual scaling dimensions into the complex principal series, $\Delta_{\text{dS}} = d/2 \pm i\mu$. This complex dimensionality induces characteristic logarithmic oscillations in the four-point correlators. In the cosmological context, these oscillations manifest as a log-periodic modulation in the SGWB, characterized by a specific spacing $\Delta \ln f = 2\pi/\mu$. Detecting this log-periodic structure with next-generation interferometers such as LISA would directly measure the parameter $\mu$, closing the loop between abstract OPE associativity and observational cosmology.