An Exact-Bayes Account of the Few-Label Protocol Gap in Tabular Foundation Models

Few-label evaluations of tabular foundation models (FMs) often label the same number of rows per class. Against random labelling, this stratified protocol raises macro-F1 and balanced accuracy and lowers accuracy. This protocol gap belongs to the protocol and the decoder; it is not specific to FMs. The exact Bayesian predictive that reads the context as i.i.d. evidence shows the same gap under argmax decoding, and under the stratified design it is the exact design-aware Bayes predictive. We split the gap into a learning term and a prior term, prove the sign of the prior term for balanced accuracy and, under random labelling, for accuracy, and derive the weak-signal limits; for two classes the macro-F1 gap changes sign at a majority share that decreases to (1+√5)/4. At uniform shares, weak signal and large budgets, the learning term for balanced accuracy is positive for three to five classes and negative for six or more. Decoders that target balanced accuracy or macro-F1 remove the prior term or the positive macro-F1 gap. In pre-registered studies, a prior-data fitted network trained on the synthetic prior reproduces this gap, and on 50 tables published after every FM checkpoint a Bayesian twin and four FMs show it. Preprint. 55 pages including appendices. A code and data archive that regenerates every number, table and figure from stored results is attached.