Invariant Projections on Cycle Spaces of Polyhedral Graphs: An Orbit-Count Formula and a Characterisation of the Regular Solids

Dilworth, Kutzarova and Ostrovskii showed that the invariant projection of the edge space onto the cycle space of a graph is unique precisely when the natural representations of the automorphism group on the cycle space and the cut space have no equivalent constituents, and asked, as their Problem 7, for which graphs this holds. We answer the question for every convex polyhedron. For the skeleton Γ of a convex polyhedron define the coupling capacity c(Γ) as the dimension of the space of Aut(Γ)-equivariant linear maps from the cycle space to the cut space, so that c(Γ) = 0 is exactly the uniqueness condition. We prove that c is an orbit count over face–vertex pairs, corrected by a term that is nonzero exactly for chiral polyhedra; that it is invariant under polyhedral duality; and, as the main theorem, that c vanishes if and only if the polyhedron is regular. The five Platonic solids are thereby characterised among all convex polyhedra by the vanishing of a homological invariant, and Problem 7 is answered in every case. The proof reduces the vanishing of the equivariant Hom space to a covering condition on mirror planes, which forces the vertices and face centres onto rotation axes, which in turn makes the question finite; a case analysis over the classification of finite subgroups of O(3) then closes it. Along the way we give a census of thirty-two polyhedra with every capacity computed twice by independent routes, an out-of-sample confirmation of the correction term, and a parity selection rule showing that in the centrally symmetric case the surviving channels are not forbidden but forced to be pseudoscalar. Finally we place the theorem: it is specific to the sphere. On the torus there are flag-transitive maps of positive capacity, so the forward direction fails; on the projective plane the hemi-cuboctahedron is irregular with vanishing capacity, so the converse fails. Each half is broken by a different surface, and the sphere is the only one of the three on which the two conditions coincide. A chiral sub-argument, by contrast, is genus-independent and verified on two hundred fifty-seven maps through genus 16; on the torus, where the reflexible square-lattice case is known [10, Thm. 24], we evaluate the chiral capacity in closed form, proving it to be ⌊(b² + c² − 1)/4⌋ for the square class and ⌊(b² + bc + c² − 1)/3⌋ for the triangular one, so the unbounded growth there is a theorem rather than an observation. The invariant arose from an attempt to give quantitative form to Curie's principle on symmetry and phenomena; §8.3 records briefly what the theorem does and does not settle there, and nothing in the mathematics depends on that reading.