Cohesion and Change Propagation: Four Necessary Conditions for Optimal Modularization

The companion papers [1] and [2] established that cohesion is a 2𝑘-tuple parameterized by 𝑘 partitioning rules, and instantiated it with one concrete principle — change-driver-assignment identity (Γ(𝑒1) = Γ(𝑒2)) — without arguing that driver-equivalence is the right principle. This paper pins the free parameter. Under the change-driver hypothesis and the objective of minimizing change propagation, four conditions are necessary and structurally exhaustive:1. Admissibility — every element has at least one driver.2. Element Form — every element is pure or irreducibly composite.3. Separation — different-driver elements belong in different modules.4. Unification — same-driver elements belong in the same module. The first two constrain the element set; the last two constrain the partition.The first three hold strictly under any faithful propagation measure (satisfying structural monotonicity properties S1–S4); Separation holds strictly under the contamination premise. Together they select the unique partition 𝐸∗/ ∼Γ . When 𝑘 = 1, Separation and Unification are exactly the purity and completeness axes of 𝐻causal; the four conditions reduce to the two-axis criterion that 𝐻causal measures.