Observational Notes on Anomaly Measures and Gravitational Amplification

Definitions Let μ_pk denote the baryonic surface density evaluated at the Einstein radius and normalized by a fixed critical surface density Σ_c. Nlayer=ln⁡(Rout/RE)Nlayer=ln(Rout/RE) be the logarithmic boundary accumulation depth. Ξ represent an effective gravitational amplification factor, such as a time-delay distance ratio or an effective convergence ratio relative to ΛCDM expectations. A be an anomaly score obtained from an unsupervised morphological or variability-based detection method. Prediction 1: Anomaly–amplification relation For galaxy and lens samples drawn from wide-field optical surveys: In subsamples restricted to μ_pk values close to 3, objects with larger anomaly scores A are expected to exhibit systematically larger values of Ξ. When the same analysis is performed for subsamples with μ_pk well below or well above this range, no statistically significant dependence of Ξ on A is expected. Prediction 2: μ-dependent scaling of boundary accumulation For strong gravitational lens samples: If lenses are grouped according to μ_pk, the dependence of Ξ on NlayerNlayer is expected to change qualitatively across μ_pk. For μ_pk below the critical range, Ξ should show at most a weak (constant or approximately linear) dependence on NlayerNlayer. In a narrow interval of μ_pk centered around μ_pk ≈ 3, Ξ is expected to increase approximately exponentially with NlayerNlayer. For μ_pk significantly above this interval, the exponential trend is expected to weaken or saturate. Falsifiability The framework is disfavored if no systematic association between anomaly measures and gravitational amplification is found near μ_pk ≈ 3, or if an exponential dependence of Ξ on NlayerNlayer persists independently of μ_pk. Note on timing These statements were formulated prior to the availability of large-sample public data from next-generation wide-field surveys.