Three guards for subspace claims: chance, stability, and metric in the spontaneous-activity debate

Whether evoked neural activity lies within the space of spontaneous activity has been answered in incompatible ways, and the question is now standardly operationalized as subspace containment: the fraction of evoked variance falling in the top-$k$ spontaneous components. We show this operationalization requires three guards. Chance: a random $k$-dimensional subspace captures exactly $k/n$ of any signal's variance in expectation—a 10–30-point floor at classic population sizes. Stability: we screen top-$k$ projector conditioning by the spectral boundary gap $\sigma_k/\sigma_{k+1}$; across 60 Neuropixels populations (7 mice), approximately 10,000-neuron two-photon V1, and monkey DMFC, measured spectra are locally flatter than their power-law fits and median gaps fall below 1.05 beyond approximately $k=3$–$5$ (a pure $\alpha \approx 1$ power law crosses only near $k \approx 11$)—and below approximately 100 neurons the screen itself needs a finite-size null. Metric: containment is not invariant to per-unit rescaling, and the literature's pipelines sit on opposite sides. Rate normalization attenuates the large cross-state effects (hippocampal maze-in-sleep excess 0.32–0.61 to 0.02–0.19; two rats), while substantial within-state excess remains in all five animals (soft-z 0.22–0.51 at $k=10$) and V1 evoked-in-spontaneous excess stays small in both metrics (raw 0.01–0.20 across three sessions). Under the guards, the disagreement decomposes into chance-floor inflation, poorly conditioned reference projectors, and undeclared metric plus state mixing.