The Partition Theorem
The four conditions of IVP do more than describe what a compliant module looks like. They determine the module structure uniquely.
1 The claim
A modularization that satisfies all four IVP conditions must be the Γ-equivalence partition — unique up to coarsening ties.
Two elements with the same change-driver assignment must co-reside (Unification), and two elements with different assignments must not co-reside (Separation). These two constraints together fix which elements belong together: exactly the Γ-equivalence classes, and nothing else.
There is no latitude in a compliant structure. Once you have read the change-driver assignment off the domain, the partition is determined.
2 Why this is not a design choice
This is the force of the result. Classical modularization advice often treats “how to cut modules” as a judgement call, one of many defensible choices. The partition theorem says that under the change-management criterion, it is not a choice. If your goal is to contain change — to make the module that must change together and nothing else — then the Γ-equivalence partition is the answer, and it is the only answer.
This is what makes IVP diagnostic rather than suggestive. Given the domain, it reads off the structure the domain requires.
3 Up to coarsening ties
The phrase “up to coarsening ties” records one genuine freedom. Within a single Γ-equivalence class, all elements are identical with respect to change: they answer to the same drivers. You may therefore subdivide a class into smaller modules, or keep it as one, without violating any condition. The subdivision is free for readability, team structure, or packaging — it does not change the change-containment behaviour.
What is not free is merging across classes or splitting within a class so that elements respond to different drivers are mixed. The partition’s boundaries between classes are forced; its internal subdivision is not.
4 Where the proof lives
The derivation that the Γ-equivalence partition is the unique cost-optimal structure is proved in the published paper on the Independent Variation Principle [1], which gives a counting argument on the element–change-driver incidence graph showing that grouping elements by their change-driver sets minimizes change propagation cost. The cost-minimisation of the partition is further developed in the paper on cohesion and change propagation [2] and the paper on minimizing change propagation and maintenance cost [3].
The formal development. The definitions, theorems, and proofs behind this page are published in:
- The Independent Variation Principle — Zenodo, 2026 — doi:10.5281/zenodo.20794332
- Cohesion and Change Propagation: Four Necessary Conditions for Optimal Modularization — Zenodo, 2026 — doi:10.5281/zenodo.21362420
- Why Minimizing Change Propagation Minimizes Maintenance Cost — Zenodo, 2026 — doi:10.5281/zenodo.21362542