OUTPUT WITHOUT WARRANT: SPECTRAL DIMENSION READOUTS CANNOT CERTIFY THEIR OWN HYPOTHESES

Local intrinsic dimension is routinely read off the local covariance spectra of neural representations. We distinguish a readout's output, the index it returns, from its warrant, the grounds for reading that index as a dimension. The output is cheap: for any data law, a spectral readout recovers its own population target at a sample size set by effective rank and spectral margin. The warrant is not. The hypotheses under which that target equals the dimension are lower bounds on reach, which no procedure can certify from a window of fewer than $(2^d-1)\log (1/2\delta )$ samples under any hypothesis-independent observation channel. Moreover, the covariance does not identify dimension at all, so a stable readout can be wrong for every dimension but one. In planted windows matched to BERT and GPT-2 representations, the readout tracks where a decaying spectrum crosses the noise edge, moving by two orders of magnitude at fixed structure. We replace the dimension claim with a narrower one: a profile-depth statistic, calibrated against an anisotropy-matched null with measured power, that certifies detectable structure but not its dimension.