Sheet 10 needed two probes. The torus gives you both from one walker. As it winds, the poloidal angle (around the tube) carries it across the inner wall (the IN face, toward the hole) and the outer wall (the OUT face) every half-turn — so the single rider samples both vantages itself. That's self-quadrature: in-frame and out-frame, ninety degrees apart, one body. The ring is cut once — and a cut torus is the electrode gap: anode on one lip, cathode on the other, the walker carrying charge across the break that closes the loop. Both sides connected, current around, flux through the hole.
REAL: the geometry is exact. Poloidal v=0 and v=90° are a true quadrature pair — verified: outer wall at v=0, top at v=90, inner wall at v=180. A walker winding (p:q) visits inner and outer faces on a fixed cadence, so it IS its own two-probe rig, sampling in-frame and out-frame from one trajectory. Cutting a torus once yields a cylinder (χ stays 0) — the cut is a clean open gap with two lips: the electrode pair. Toroidal current links flux through the central hole — the transformer/tokamak action from Sheet 7. And a (p:q) winding closes only when the trip count completes; irrational-ratio winding never closes and densely fills the surface — a single walker that, given forever, measures everywhere.
DOESN'T: self-quadrature has a catch the two-independent-probe rig didn't — both readings come from one walker, so a bias in the rider rides along into both faces. The torus cancels the frame (geometry guarantees the 90° offset) but not a fault in the traveler. It is the strongest thing one body can do — measure itself from two angles — and still not a substitute for a second, independent rider. The ring checks its own phase. It cannot check its own walker. That gap stays open, like the cut.