Spectral Monitors Detect Change, Not Damage, in LLM Post-Training

Can hidden-state spectra distinguish harmful post-training updates from benign adaptation? A fixed-readout construction shows that identical spectra can coexist with arbitrarily different predictive losses—an identifiability limit, not an impossibility theorem for empirical alarms. We test the empirical question with pre-registered shared-prefix stress tests on three open models, run on a corrected training/probe split with fresh seeds. Extreme duplication damages held-out NLL by $+0.3$ to $+1.8$ nats while raw RankMe rises in 9/9 model–seeds, and the same high-LR damage moves RankMe down on OLMo-2-1B but up on Qwen3-1.7B: the sign of a spectral response tracks the (regime, architecture, layer) tuple, not damage. A trajectory-level split-conformal protocol then compares spectral and loss monitors at an equal false-alarm budget on the development model (Qwen3-0.6B): on abrupt damage every spectral family alarms at the same measurement as a plain loss detector; on a gradual learning-rate ramp spectra alarm 80–110 steps earlier, but CUSUM and Page–Hinkley detectors on the same loss stream recover about half (CUSUM) and a quarter (Page–Hinkley) of that lead, and all of these detectors also fire on a non-damaging duplication ramp and on a benign control that improves its target ($-0.38$ nats) with no significant task-accuracy change. Only a loss monitor with a non-degenerate threshold separates damage from benign change; a pre-registered matched-threshold control shows the sidedness of the detector is immaterial. An earlier campaign affected by a training/probe exposure defect is disclosed, repaired, and reported side by side; its structure replicates cleanly. In these settings, the tested spectral diagnostics measure change, not damage.