№ 27 needed two probes; the torus gives both from one walker. As it winds, the poloidal angle carries it across the inner wall (IN, toward the hole) and the outer wall (OUT) every half-turn — self-quadrature, in-frame and out-frame, 90° apart, one body. Two more forces fold onto the same ring. GRAVITY: the torus's Gaussian curvature flips sign with that very angle — a well on the outer wall (K>0, geodesics focus), a saddle on the inner (K<0, they defocus) — yet the total ∮∮K dA = 0 (Gauss–Bonnet, χ=0): well and saddle exactly cancel. ELECTRICAL: the ring is cut once — a cut torus is the electrode gap, anode on one lip, cathode on the other, the walker carrying charge across the break; the toroidal turns link flux through the hole — the transformer / tokamak.
REAL — geometry: poloidal v=0 and v=90° are a true quadrature pair (outer wall v=0, top v=90, inner wall v=180). A walker winding (p:q) visits both faces on a fixed cadence, so it IS its own two-probe rig. An irrational ratio never closes and densely fills the surface — one walker that, given forever, measures everywhere.
REAL — gravity: the Gaussian curvature is exact, K = cos v / (r(R+r cos v)) — verified: a well (K>0, geodesics focus) on the outer wall, a saddle (K<0, they defocus) on the inner, flat (K=0) at top and bottom. And the total is a topological invariant: ∮∮K dA = 2πχ = 0 for a torus — the outer well and the inner saddle exactly cancel (K·dA = cos v·du·dv, integrating to zero). Local gravity is frame-like; total gravity is absolute — the same lesson as № 27, now in curvature.
REAL — electrical: cutting the torus once leaves a clean open gap with two lips — the electrode pair, anode ‖ cathode, the walker carrying charge across the break. The toroidal turns link flux through the central hole: current around → flux through, the transformer / tokamak (the toroid of Sheet 7).
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 (the 90° offset and the curvature sign are geometry, unfakeable) but not a fault in the traveler. The ring checks its own phase; it cannot check its own walker. That gap stays open, like the cut. And the curvature/flux are exact geometry, not a fabbed device — no SPICE, no metric tensor solved; it is the topology made legible.