A Phenomenological Search for Geodesic-Diffusion Noise in the Einstein Telescope: Model Specification, Pipeline Validation, and Sensitivity on ET-MDC1 Data (v0.2)

We specify and validate a phenomenological search for geodesic-diffusion noise in the Einstein Telescope (ET): a stochastic strain component with one-sided power spectral density $S_{h}(f)=A_{h}(f/f_{0})^{-\gamma}$, with the random-walk index $\gamma=2$ as the primary case. The analysis is a nested likelihood-ratio (LR) test under a complex-Wishart approximation for Welch cross-spectral matrices, with spline-parameterized instrument PSDs, a low-rank environmental coherence model, and phase-only calibration nuisances. This is explicitly not a quantum-gravity derivation; it is an effective noise model confronted with data. Version 0.2 corrects a serious optimization defect in the v0.1 reference implementation—scale-inappropriate parameter bounds that pinned every fit at a data-independent corner solution, which produced identical, negative LR values across all datasets—and replaces the v0.1 empirical section entirely. On the ET-MDC1 "loudest" sample sets the corrected pipeline yields per-dataset LR statistics consistent with the null hypothesis, calibrated by a parametric bootstrap; synthetic injections into real data are recovered above an empirically measured threshold; a null-stream diagnostic confirms the three MDC1 channels carry independent noise; and a profile-likelihood upper limit on $A_{h}$ is set from a single 128 s stretch. A Fisher forecast shows that ET reaches $A_{h}\sim4\times10^{-49}\text{ Hz}^{-1}$ per 2048 s segment and $\sim3\times10^{-51}\text{ Hz}^{-1}$ per year of coincident data—12 to 16 orders of magnitude below the best published bounds for $\gamma=2$, and deep enough to decisively test the Planck-scale random-walk benchmark. All code, data manifests, and validation logs are public.