Sharp Higher-Chain Amplification Under a Single Cover Edit in Finite Posets

Let C be a finite bounded poset and let C+ be obtained by adding exactly one new canonical Hasse cover while preserving every old cover. For k ≥ 3, write C_k(C) for the number of k-element chains and ΔC_k = C_k(C+) − C_k(C). We first derive an exact local decomposition of ΔC_k into old lower- and upper-chain profiles across the genuinely new comparability bridge. Combining this identity with the cover-preserving witness budget yields a sharp extremal formula: for every n ≥ 6 and k ≥ 3,max ΔC_k = binom(n−2,k) − binom(floor((n−4)/2)+2,k) − binom(ceil((n−4)/2)+2,k),where binom(m,k)=0 for m