Variance Is Not Information: Why Priors over EEG Cannot Close the Trial-Averaging Gap

Every headline number on THINGS-EEG2, the standard benchmark for visual decoding from EEG, is computed on responses averaged over 80 repetitions of the same stimulus. The same frozen encoder scores $0.841$ at 80 repetitions and $0.158$ at one, and two reproduced 2026 baselines keep $10\%$ and $16\%$ of their headline number at one repetition. We ask whether that $0.683$ gap can be closed algorithmically and answer no, in a strong sense: classical MMSE estimators recover none of it, and the provably optimal diffusion prior over EEG, which is available in closed form and hence immune to a training-quality objection, recovers none of it and at $R\ge 4$ actively hurts. One measurement explains why. Ordering the response's principal directions by variance, the 32 leading directions carry $43.7\%$ of the estimated signal energy, retrieve at chance in isolation, and deleting them is free; keep-$k$ accuracy does not saturate until $k\approx 4096$. The discriminative signal is high-rank and anti-aligned with variance, so any prior, estimator or likelihood-maximisation step that concentrates its estimate where the variance is allocates capacity where the information is not. Consistently, the estimator $17\times$ closer to the high-SNR response in NMSE loses four points of Top-1. Both findings replicate on THINGS-MEG. The same principle governs the image side: multi-view target augmentation is worth $+0.278$, yet an operator fitted to reproduce that target reaches cosine $0.984$ on held-out images while delivering $16.7\%$ of its benefit, making it nonlinear adds $+0.0035$ Top-1, and training it on the retrieval loss instead gives a family whose optimum is the identity. Reproducing a target and reproducing its value are different problems. Progress on this benchmark is measurement-bound rather than model-bound.