A Self-Contained Derivation of the Optimal Modularization from Coupling

In the companion paper “On the Nature of Coupling” [1], coupling is defined as access to a shared resource. From that definition alone — without change drivers, Γ, or IVP — it follows that grouping elements by identical resource-access sets (𝑅(𝑒1) = 𝑅(𝑒2)) minimises accidental coupling cost. This paper proves that result. The paper makes three contributions. First, it formalises the tripartite structure that mediation makes visible: 𝐸 ⟶ℛ︀ ⟶ 𝐶. Every element accesses a set of resources; every resource is governed by a set of change drivers; the element’s change-driver set is derived, not primitive — Γ(𝑒) = ⋃𝑟∈𝑅(𝑒) 𝐷(𝑟). Second, it formalises the element-as-resource-realisation layer and the governance chain, yielding what is, to the authors’ knowledge, the first structural account of leaky abstractions: an abstraction leaks through the change drivers the implementer cannot absorb. Third, it proves that the resource-set partition is the unique minimiser of accidental coupling cost among all partitions — using only the coupling definition and the premise that accidental coupling costs positive. In the pure-resource regime, the resource-set partition refines the Γ partition. The two derivations, coupling-side and cohesion-side, are two ends of the same refinement chain.