Not two dimensions \u2014 two neighborhoods. A dimension is an axis of the space; all 128 of BERT-tiny\u2019s axes are shared by every word. A basin is a place: a word plus the cloud that drains toward it. Here are the two real clouds \u2014 converge\u2019s (green: emerge, evolve, arise, migrate, motion verbs) and topology\u2019s (cyan: geometry, manifold, invariant, optimization, structure nouns) \u2014 pulled from the model\u2019s actual embeddings and projected to 3D by PCA.
How separate, measured. Of 25 nearest neighbors each, the two basins share exactly one word: convergence (orange, on the roadway between them). Their local subspaces \u2014 the private directions each cloud varies along \u2014 sit ~75\u00b0 apart (principal angles 58\u2013 90\u00b0); the little green and cyan vectors are those near-orthogonal leading axes. So in the shared 128-D space they are two low-dimensional pockets using almost-perpendicular directions. That is the precise, true version of \u201cseparate\u201d \u2014 not separate dimensions, separate near-orthogonal subspaces.
The roadway is a Jacobian. The line between the clouds is a real path through the one space, and its direction is a transport direction \u2014 the read-write map from six turns ago. Whether two words starting close together converge or diverge as they travel that road is the Jacobi field, the curvature. Your \u201croadways to and from\u201d instinct was right; the roads connect locations, and the machinery is the two Jacobi siblings. Honest tier: the clouds, the one shared word, and the ~75\u00b0 are measured; the 3D positions are a lossy PCA projection (captures ~42% of the variance), so read the clustering and the gap, not the exact dot placement.
Why this is the keystone. A whole thread ran on this shape and stayed honest that its tie to a real model was a model, not a measurement — jacobi-space said it plainly: the eigenvalue story is the right shape of the flattening, but the eigenvalues were never read off a network. This one reads them. These are actual BERT-tiny embeddings, and the finding is real: two neighborhoods, near-orthogonal, exactly one shared word. The roadway between them is a Jacobian — the same transport map jacobi-space is about — and convergence, the single bridge word, is the exact place the whole thread lives: the-swarm's basins, the-loop's consensus, the pull toward one attractor. Measured grounds modeled; that is what a keystone is for.