Hypotheses as axes
Each hypothesis gets a supposition value between 0 and 1, and each value is a distance along its own axis (section 3.5). With three hypotheses a, b and c, every state of the system is a point in the unit cube; an interpretation is one of its corners.
Constraints as planes
A logical constraint becomes a linear inequality, and each inequality is a plane with an infeasible side, drawn here with hatching. Together they cut the cube down to a convex polyhedron of feasible states. If every corner of that polyhedron is a corner of the cube, the best corner is an interpretation. If not, the best point may be fractional, like the apex (½, ½, ½) of the triad (section 3.7). A cutting plane can slice such a vertex away.
Relaxation as forces
The total infeasibility I is half the sum of squared distances past violated planes, so each violated plane pushes the state back along its normal with a force proportional to the violation (section 3.14). A constant weak force pulls along the preference vector (section 3.16). The state settles just outside the best vertex, closer when Kp is small. Momentum Kd lets steps grow while the state slides in a steady direction (section 3.17).
F_inf here uses unit normals, so Vj is the true distance past each plane and the force is exactly the downhill gradient of I.