Dual of shadowing stays-losing polytope
is a normal vector constraining polytope
Preferential voting rules that define rights for single voters, e.g. monotonicity, can be merged together with other rules.
A rule of the form “if X loses here then X loses there” defines a raw shadow (or a light source). Any number of those rules can be combined. To do that, the dual of the product of the duals ([[Dual polyhedron]]) of each rule is calculated. So rules are able to lose their identity: they can be merged together and pulled apart in different ways.
A voting dual transform: Dual_Of_N(x)=(All y)[N(y) implies (x*y <= 0)].
In 3-D, the preferential voting dual function of a convex pyramidal “normal vector constraining” touching shadow polyhedron is scanned with lines along its surface. Perpendicular to those lines, are planes also containing the contact point. Those planes/flats define the shadow polyhedron on the other side. To use the shadow polytope: its tip is put at at point where X is known to lose. Then X is known to lose in the body of the shadow polytope.
In the case that there is a 2 winner election and the rights of an ordered pair of candidates is being upheld, then the base 2 dissatisfaction of the 2 candidates with the winners is at the smallest at the tip of the polytope.
Extra inferring is required as multivalued shadows intersect, before the winners of the non-proportional parts of the solution can be inferred. If all goes well, the proportional solution can be non-arbitrarily dropped in and an ideal preferential voting method results.