Exact Periodic Degrees, Degree Spectra, and Sharp Thresholds for Cyclic-Polytope Colorings

Ryabichev asked whether the periodic coloring of the boundary of a cyclic (n+1)-polytope maximizes the degree among simplicial maps from k-vertex triangulated n-spheres to the boundary of the (n+1)-simplex. We derive a closed product formula for the degree of this periodic map for every n and k, determine its exact quasi-polynomial behavior on residue classes, and prove a sharp global envelope. We establish diagonal stabilization of the complete attainable absolute-degree spectrum and a linear stable-range formula for the minimum number of vertices realizing a fixed degree within the integral simplicial homology-manifold class. We also prove balanced color-class bounds, determine the complete near-threshold topological-sphere spectra, identify the first failure of periodic extremality, and give the literal first counterexample at (n,k)=(2,5). Finally, an exact census computation classifies the nine-vertex three-dimensional case: the maximum absolute degree is 3, attained by exactly 37 of 1,296 combinatorial sphere types. The census classification is finite-census dependent, and no literature-wide novelty or priority claim is made.