Integral homology of strong Morse matching complexes of cycles

We determine the reduced integral homology of the full strong discrete Morse matching complex of every cycle. Apart from the five-cycle, the answer is the sum of two free abelian contributions whose degrees and ranks follow explicit formulas according to the cycle order modulo four and modulo three. The proof represents a pairing by two local order inequalities. Their conflict graph is a regular bipartite graph with a known independence complex; the strong matching complex is obtained by removing exactly two families of monochromatic vertex covers. The resulting relative chain complex splits into two copies of a simplex relative to the noncover complex of the cycle. Combinatorial Alexander duality and the long exact sequence then give the homology in all orders, extending the finite calculations of Knudson and Owens-White. The same face description gives a transfer-matrix formula for the face polynomial and a twelve-periodic reduced Euler characteristic.