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

This preprint studies degree extremality for periodic simplicial maps from boundaries of cyclic polytopes to simplex boundaries. It gives an exact product formula for every periodic degree, determines the first two coefficients on each nonzero quasi-polynomial residue class, and proves the sharp envelope |deg p| ≤ (k/(D+1))^{floor(D/2)}. For integral simplicial homology-manifold spheres, it establishes stabilization of the complete attainable absolute-degree spectrum, a stable formula for the minimum vertex number realizing any fixed degree in the stated range, and an integer-balanced color-class upper bound. In the topological-sphere class, it determines the complete spectra on n+5 and n+6 vertices and locates the first failure of periodic degree extremality for every n ≥ 3, together with the literal failure at (n,k)=(2,5). An exact census of nine-vertex combinatorial 3-spheres shows that the maximum absolute degree is 3 and that exactly 37 of the 1,296 sphere types attain it; the maximum bound is census-free, while the exact type count retains its finite-census dependence.