Empirical Scaling of Statistical Distinguishability Under Controlled Spectral Overlap in Atmospheric Retrieval Architectures

This technical note develops the quantitative extension of a previously published controlled synthetic retrieval experiment. Using the same architecture and the same controlled overlap scan between a true CH4 basis and an alternative distorted basis, the note studies how the statistical distinguishability between retrieval models scales with the spectral-overlap deficit, defined as ε = 1 − overlap. The main result is that distinguishability follows a robust empirical scaling law: a linear approximation remains useful as a first-order operational description, but raw-level validation shows that a weakly superlinear power-law model provides the best overall empirical fit. Cluster-aware validation further supports that the departure from strict linearity is small but systematic. The note does not propose a universal law of atmospheric retrieval; rather, it quantifies the internal scaling structure of the controlled degeneracy regime already identified in the previous experiment.