The Stitching Defect
A Mayer–Vietoris Decomposition of the Coherence Fee Under Composition
fee(G₁ ∪_B G₂) = fee(G₁) + fee(G₂) + rank(δ_B); residual(A, B; S) = fee(A, B) − max_C |D(A) ∩ D(B) ∖ D(C)|; 33/33 empirical match on tier3 corpus
A gluing of two exact-regime compositions whose fee disagrees with the Mayer–Vietoris formula, or an MCP corpus pair where the structural-residual closed form deviates from the empirical residual
Abstract
When two compositions of MCP servers are joined at a shared tool, the coherence fee of the joined composition is not in general the sum of the parts. We prove an exact formula: fee(G₁ ∪_B G₂) = fee(G₁) + fee(G₂) + rank(δ_B), where δ_B is the connecting map of the Mayer–Vietoris long exact sequence for the seam cochain complex. We characterize rank(δ_B) as a dimension gap and identify a sufficient condition—safe stitching—under which it vanishes. On a 13-server MCP corpus, no pair admits a bypass route reducing the path-fee to zero, and only 6 of 33 positive-fee pairs admit any reduction at all. We then prove a closed form for the structural residual: residual(A, B; S) = fee(A, B) − max_C |D(A) ∩ D(B) ∖ D(C)|, where D(T) is the set of named convention dimensions in server T. The formula matches the empirical residual on 33/33 pairs and turns the bypass analysis from O(|S|·|G|^ω) path search into O(|S|) lookup over named-dimension sets.