Coupling as a Consequence of Cohesion

The classical software design imperative — “maximise cohesion, minimise coupling” ([1], [2]) — treats the two as independent objectives requiring a trade-off. This paper proves otherwise: within any single dimension of coupling, coupling is a consequence of cohesion, and the coupling categories that modularization controls are diagnostic of cohesion defects, not independent design variables. The paper builds on [3], which established that cohesion is correctness relative to a partitioning rule — a 2𝑘-tuple of (purity, completeness) pairs, one per rule, for𝑘 rules. We decompose structural coupling into eight mutually exclusive categories — the product of three binary partitions (internal/external, same-class/cross-class, necessary/not) — parameterized by a partitioning rule. Within each single dimension, purity eliminates internal cross-class coupling and completeness eliminates external same-class coupling. At maximal cohesion (purity = completeness = 1), all modularization-controlled coupling categories are empty. We then establish the coupling floor: the irreducible essential coupling the domain demands, given both an unweighted (edge-count) and a weighted (change-propagation-cost) form, and characterize the necessary dependency graph as its embodiment. At correct boundaries the floor equals the necessary dependency graph exactly; every edge above it is a boundary defect or an accidental edge. Because coupling is measured against a partitioning rule, necessary coupling is rule-relative: the same edge that one concern demands be internal is, under another, demanded external. The imperative to “minimise coupling” therefore has no fixed referent until the rule is fixed, and a raw count cannot separate domain-mandated coupling from placement error. The decomposition extends beyond structural dependency. Each of 𝑘 coupling concerns (structural, temporal, concurrency, security, platform, language) induces a partitioning rule and carries its own dependency relation, so maximal cohesion under one rule does not bound coupling in another. The result: the normative decomposition simultaneously maximises cohesionand minimises avoidable coupling — one optimum, not two.