The Structural Formulation of IVP

This is the statement of the Independent Variation Principle itself. Everything else in the core theory — the partition, purity and completeness, the partition theorem — is either an ingredient the principle needs or a consequence it produces. This page states the principle.

1 The principle in one sentence

For a software system $S = (F, \kappa_F, E, \text{Coupling}, C, \Gamma)$ , IVP prescribes the partition of the element set $E$ induced by the equivalence relation

$e_1 \sim_\Gamma e_2 \iff \Gamma(e_1) = \Gamma(e_2).$

The modules of an IVP-compliant system are the Γ-equivalence classes of the element set: elements assigned the same set of change drivers belong in the same module, and elements assigned different sets belong in different modules.

That single sentence is the whole principle. The four conditions below are not additional rules — they are that sentence stated as four obligations a module structure must satisfy.

2 The four conditions

IVP-1 — Element Admissibility. Every element has at least one change driver: $|\Gamma(e)| \geq 1$ . No element floats free of the change structure.
IVP-2 — Change-Driver Assignment / Element Form. Every element is pure ( $|\Gamma(e)| = 1$ ) or irreducibly composite. The model admits only elements whose driver set is exactly what the domain requires.
IVP-3 — Unit Purity / Separation. Elements with different driver assignments reside in different modules: $\Gamma(e_1) \neq \Gamma(e_2) \Rightarrow M(e_1) \neq M(e_2)$ .
IVP-4 — Unit Completeness / Unification. Elements with the same driver assignment reside in the same module: $\Gamma(e_1) = \Gamma(e_2) \Rightarrow M(e_1) = M(e_2)$ .

Separation (IVP-3) is the “do not mix” half: elements that answer to different change drivers must not co-reside. Unification (IVP-4) is the “do not split” half: elements that answer to the same change drivers must co-reside. Together they force exactly one module structure — the Γ-equivalence partition.

3 Why the two halves matter

Each half corrects one classic failure.

A module that violates separation mixes elements that change for different reasons. When one change driver activates, the module must change in part, dragging unrelated elements into the blast radius.

A module that violates unification splits elements that change for the same reason. When that change driver activates, the change must be made in several places, and keeping them consistent is coordination cost.

Both are placement errors. Both are named by a condition, which makes them diagnosable.

4 Admissibility and element form set the model

The first two conditions govern what counts as an element at all. IVP-1 says every element belongs to the change structure. IVP-2 says every element’s driver set is exactly what the domain requires — an element is single-driver, or multi-driver in a way that is irreducible, never accidentally bloated.

These two prepare the element set $E$ for the grouping that IVP-3 and IVP-4 then prescribe.

5 The principle is a criterion, not a recipe

IVP is stated over a system tuple that is read off the domain. Once $\Gamma$ is fixed, the principle determines the structure. This is what makes IVP a criterion: it does not suggest a structure, it names the structure the change drivers require.

6 How the rest of the core theory follows

The Γ-equivalence partition is the object the principle produces.
Purity and completeness are the two properties a module gets from the partition.
The partition theorem says the compliant structure is uniquely this partition.

The statement above is the foundation; the other core-theory pages develop what it implies.

7 Where the derivation lives

The derivation that this is the unique cost-optimal structure under the change-management hypothesis is proved in the published paper on the Independent Variation Principle [1], and the proof that the four conditions — Admissibility, Element Form, Separation, Unification — are necessary and jointly exhaustive is developed in the paper on cohesion and change propagation [2]. The purity and completeness properties the conditions produce are defined formally in the paper on the nature of cohesion [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
On the Nature of Cohesion: Cohesion as Correctness Relative to a Partitioning Rule — Zenodo, 2026 — doi:10.5281/zenodo.21703193

All published papers

References

[1]

Y. Loth, “The independent variation principle.” Zenodo, Jun. 2026. doi: 10.5281/zenodo.20794332.

[2]

Y. Loth, “Cohesion and change propagation: Four necessary conditions for optimal modularization.” Zenodo, Jul. 2026. doi: 10.5281/zenodo.21362420.

[3]

Y. Loth, “On the nature of cohesion: Cohesion as correctness relative to a partitioning rule.” Zenodo, Jul. 2026. doi: 10.5281/zenodo.21703193.