Minimal persistence grids and homological Morse bounds

The persistent homology of a finite simplicial complex is determined by a canonical grid whose coordinate values occur in minimal generators and relations. We prove that the number of values in each coordinate is bounded by the homological Morse lower bound defined from the multigraded Betti tables. The result holds over every field, for every number of parameters, and for filters with coincident coordinate values. It establishes the conjecture in Remark 5.12 of Harrington, Tillmann and Torras-Pérez. The proof selects a minimal presentation degree on each coordinate slice. At that degree a local Koszul calculation eliminates higher syzygies; injectivity of top-dimensional homology handles the final homology degree.