A Phenomenological Search for Geodesic-Diffusion Noise in the Einstein Telescope: Model Derivation, Identifiability, Pipeline Validation, and Sensitivity on ET-MDC1 Data (v0.3.3)

We specify and validate a phenomenological search for geodesic-diffusion noise in the Einstein Telescope (ET): a stochastic optical-path-noise component with one-sided strain power spectral density Sₕ(f)=Aₕ(f/f₀)-γ, with the random-walk index γ=2 as the primary case. Version 0.3 is a major revision. First, the channel covariance template M(ρ), treated as ad hoc in earlier versions, is derived from a shared-tunnel path-noise model, giving the correlation parameter ρ a physical meaning. Second, we prove a gauge-invariant identifiability statement: the product of the three off-diagonal cross-spectra is invariant under per-channel phase calibration, is non-negative for any diagonal-plus-rank-1 environmental model, and is negative for the signal, and we show this protection provably fails at environmental rank 2. Third, the empirical program is rebuilt: an external referee review exposed two further optimization defects in the v0.2 reference implementation (absolute finite-difference steps applied to parameters twenty orders of magnitude smaller, and insufficient iteration of the nested-fit symmetrization), as well as an analysis-bin layout that discarded 99.6% of the spectral data and starved the search band. All v0.2 numbers are withdrawn and replaced by results from the corrected pipeline on log-spaced averaged bins. On the ET-MDC1 "loudest" sample sets all eight groups are consistent with the null in both detection channels. The v0.3.1 second-round revision closes the gaps a further referee review identified and one of the resulting upgrades overturns v0.3's central negative finding. (i) A normalization inconsistency in the covariance derivation is fixed: Aₕ is the per-path amplitude and the per-channel signal PSD is 2Aₕ(f/f₀)-γ; no numerical result changes, because the reference implementation was consistent throughout. (ii) Finite-difference gradients are replaced by closed-form analytic Wishart scores (verified against finite differences, ~13× faster), and the fit runs in scale-normalized variables; the raw parameterization spans 26 orders of magnitude, and its conditioning had been silently stalling every fit hundreds of log-likelihood units short of the optimum. With both fixed, the v0.3 claim that an unconstrained environmental model absorbs injections completely up to 256σF does not survive: absorption is real but incomplete, the likelihood-ratio test recovers injections with unbiased amplitudes above a measured threshold of ~30σF (64s: 50% detection), and the earlier "no power" result is reclassified as the third optimizer artifact in this paper's history. The honest quantitative statement, from a noiseless Asimov ladder run through the production evaluator: the nuisance model absorbs 98–99.9% of the injected template deviance, a ~30x amplitude penalty relative to the naive Fisher scale—severe, but not the blindness v0.3 reported. (iii) The sign obstruction is promoted from a theorem to a statistic: a fit-free, calibration- and convention-invariant sign-channel statistic on the binned cross-spectra, with a universal (optimization-free) null calibration that the sign theorem itself makes conservative against the entire admissible environmental class; its measured power (50% detection near 70σF, exact false rate at the null) makes it the robust confirmation channel beside the more powerful but nuisance-exposed likelihood ratio. (iv) The optimizer-noise floor that forced v0.3's convergence-inflated upper limit is gone; restart scatter collapses from O(50--80) to ≤ 6.9 log-likelihood units and the 95% profile upper limit becomes statistics-limited: Aₕ95%=7.8×10⁻⁴⁷ Hz⁻¹ (128s, template ρ=0.5, per-path convention). A third referee round sharpened the boundaries of these claims, and this document (v0.3.2) folds the answers in as measurements: the sign channel's universal calibration is exact for independent channels and conservative in the mean for the admissible class, but is not uniformly conservative in the tail at finite m (worst measured false rate 0.24 at nominal 0.05 under strongly coherent rank-1 nulls), so its validity is now gated by a fit-free coherence check that the analyzed data pass; an isotropic gravitational-wave background in the triangle also produces Re t<0 (pairwise DC correlation coefficient -1/2), so the sign channel establishes non-instrumental correlated power, not its geodesic origin—that separation belongs to the null stream and the spectral index; a joint CBC-foreground amplitude nuisance costs a further factor 2.45 in the profiled Fisher scale rather than collapsing the measurement; the optimization noise on Aₕ itself is negligible (std 0.2 at threshold amplitudes against Λ≈30); a first Wishart-replicate coverage check of the upper limit gives 17/20 and all sensitivity curves are explicitly labeled as path-template ($R=I$) forecasts, with the strain-reading forecast provisional pending full-response injections . A fourth round of edge-case questions is answered mostly by measurement (this document, v0.3.3): a frequency-decaying transverse correlation κ(f) is recovered without bias by the spline-ρ pipeline (the constant-ρ statement applies only to trials-controlled templates, not the model class); cross-block spectral leakage on the log grid is bounded analytically and numerically (≤ 0.11 correlation between the narrowest adjacent blocks, falling as 1/n²); the sign channel's detection amplitude scales as Tobs-1/2 like the Fisher channels (demonstrated, not assumed: threshold multipliers 69 and 64 in σF units at 64 and 128s), so the two-channel discovery reach is a fixed multiple of the Fisher curve; the sign statistic's universal calibration holds at the production p=10⁻³ tail under line forests and broadband-coherent rank-1 nulls at moderate coherence (worst measured zero events in 3×10⁴ draws); and the fixed-amplitude exclusion bound generalizes to non-unitary calibration with a (1-ε)² prefactor, moving the limit by twice the amplitude-calibration uncertainty at most . We retain the forecast, the existing-bounds comparison distinguishing the strain-noise and path-length-noise readings (under the latter, the paper's own Planck mapping implies short-baseline experiments already constrain the benchmark to α≤10⁻⁷, so the ET case rests on the strain reading), the stochastic-background confusion analysis, and the pre-declared detection rule, now a two-channel coincidence. All code, data manifests, validation logs, and all post-mortems are public.