Weakly Convex Distinguishability Loss Under Controlled Spectral Overlap: A Synthetic Retrieval Stress Test

In this technical note, I introduce a controlled synthetic retrieval stress test designed to quantify how distinguishability degrades as spectral overlap increases between a target absorber basis and an alternative competing basis. The protocol generates noisy synthetic transit-like spectra, compares two fixed retrieval architectures across repeated realizations, and maps the resulting fit-quality gap Δχ² to the empirical overlap deficit ε = 1 − overlap_mean. The main result is deliberately local and architecture-dependent. Within the explored overlap range, distinguishability loss is monotonic and structurally regular, but it is not best described by either a simple power law or a purely linear model. Instead, direct model comparison, leave-one-out validation, bootstrap analysis, and subset robustness tests support a weakly convex local law, well approximated by a quadratic correction. The notebook and accompanying output files included in this record reproduce the full pipeline: synthetic data generation, overlap scan, raw replicate tables, phase-diagram figures, and local-curvature analysis. The claim is diagnostic rather than universal: the note does not propose a domain-independent scaling law, but isolates a reproducible local geometric feature of distinguishability loss under controlled spectral overlap.