The generic HRL engine — a respected prior, a noise / null-reject label, and sparse higher-order factors — running right here on four problems at once. Same core each time; only the factors and prior change.
Nucleotide positions are the objects; {A,C,G,T} the labels. The top strand is seeded; the bottom starts unknown. One pairwise factor carries the Watson–Crick kernel (A–T, C–G reinforce, the rest suppress), and the field relaxes the bottom strand into the exact complement.
A carbon with four different groups is chiral: R or S. Handedness is the signed volume of the three top-priority bond vectors — the contraction of them against the Levi-Civita tensor, exactly what the engine's order-3 factor computes. Mirror the molecule and it flips. A pairwise term can't: the mirror image has identical distances.
Two communities joined by a bridge. A homophily rule (neighbours share a label) plus a single seed per side — the whole graph settles into the right two-colouring. Many free rules, few supervised anchors: the data-efficiency thesis, in miniature.
Forty real 8×8 handwritten digits (scikit-learn's digits set — MNIST's little cousin — bundled so the page stays offline). A pixel-similarity k-NN graph supplies the factors; just one seed per class is labelled. The field relaxes those labels across the graph — five seeds paint forty digits. Each frame's colour is the label it relaxed to.