Lohe dynamics: oscillators on the hypersphereThe summit of the curriculum — coupling with room to move, wiring that chooses itself, and tokens as paths

Everything on this site rests on one move: look at your neighbors, feel how far you are from agreement, slide a little toward them. The beginner tier taught that move on a circle. This module — the most advanced on the track — lets it loose on a sphere of any dimension, where the 2009 generalization by Max Lohe lives. Three sections, each interactive end to end: the dynamics themselves, the coupling matrix as a living object, and the payoff the whole curriculum has been climbing toward — a token's journey through a transformer as a path on the hypersphere.

Assumed background: phase and coupling (primer 00), the dot product (01), the parts of a transformer (03), and — helpful, not required — rotation groups (07).

The sphere: Kuramoto with room to moveThe circle and the sphere side by side under one coupling knob; then forty oscillators on S² with three wiring regimes — collapse, structured sync, and clusters condensing from geometry alone.
Effective coupling: the wiring that chooses itselfThe coupling matrix as a live object: watch K_ij re-rank as clusters form, click a bond to see it on the sphere, and quench the wiring to feel the difference between a living network and a frozen one.
Tokens as paths: where you end up is meaningOne token bending through a held context, with its path statistics live; two nearby tokens parting ways at a watershed; and the handoff to the essays that measure all of this on real models.

The curriculum Start: The sphere