Scope and Validity Conditions

The partition theorem is a conditional result. It holds for systems that meet a set of premises and preconditions, and it returns a stronger conclusion when two further premises hold. This page states all of them in one place: the assumptions the derivation rests on, the preconditions that must hold for the result to apply, and the gates that decide whether IVP is the right tool for a given system.

The principle itself is stated on the structural formulation page, and the object it produces is described on the Γ-equivalence partition and partition theorem pages. This page records the conditions those results assume.

1 The three unconditional premises

Three premises are required for the derivation to apply at all.

The change-driver hypothesis. Elements change because external conditions change. Each element’s driver assignment $\Gamma(e)$ captures that causal structure: $\gamma \in \Gamma(e)$ exactly when activating $\gamma$ creates a requirement to modify $e$ , established by the counterfactual test against the domain artifact that anchors $\gamma$ .
A causal domain model. The system has a functional purpose $F$ and relevant causal domain knowledge $\kappa_F$ , and every element embodies some part of $\kappa_F$ .
Change-propagation minimisation as the objective. Cost is measured as access cost plus interference, so the target is the modularisation that minimises the scope read and modified when a change driver activates. The change-management hypothesis supplies this objective — minimise the lifetime cost of change — and change-propagation minimisation is how the derivation measures it.

Premises 1 and 2 together support Admissibility and Element Form. Premises 1 and 3 support Separation and Unification.

2 The two conditional premises

Two further premises are required only for the stronger forms of the result.

The contamination premise — strict Separation and uniqueness. A $\gamma$ -irrelevant co-resident in a touched module imposes non-zero verification overhead. Under this premise, Separation is strictly forced between classes with disjoint driver sets, and the optimal partition is unique. Without it, Separation is a propagation-neutral tie, and the Γ-equivalence partition is one propagation-minimal modularisation but not the only one.
The coefficient condition $(\star)$ — total-maintenance-cost optimality. For every irreducibly composite element $e^{*}$ and every pure element $e$ whose driver set is a proper subset of $\Gamma(e^{*})$ , write $S = \Gamma(e)$ for the shared drivers and $U = \Gamma(e^{*}) \setminus \Gamma(e)$ for the unshared drivers. The condition is

$(\star)\quad \alpha \sum_{\gamma \in U} \lambda(\gamma) + 2\omega_c \;\ge\; \sum_{\gamma \in S} \lambda(\gamma),$

where $\lambda(\gamma)$ is a change driver’s activation frequency, $\alpha$ is the interference coefficient (reading plus disturbance penalty per irrelevant co-resident), and $\omega_c$ is the cognitive cost per co-located pair. When $(\star)$ holds, the propagation minimum also minimises total maintenance cost. When it fails at some composite/sharer pair, a coarsening of the Γ-equivalence partition achieves lower total cost, and that pair needs a direct cost comparison.

3 Precondition: driver independence

The companion results assume that distinct change drivers are independent: their knowledge slices are disjoint, and their activation events are logically independent. The load-bearing requirement is weaker. The partition prescription stays valid whenever no pair of drivers has both overlapping knowledge slices and mutual activation-coupling.

If two drivers always activate together in both directions, they are not two drivers: no element can be governed by one without the other. Merge them into a single driver, and the partition is correct for the merged driver. A pair with overlapping slices but no mutual coupling is harmless: an element that carries both drivers becomes an irreducible composite in its equivalence class, and the overlap is orthogonal to partition correctness.

Driver independence is therefore not an external stipulation. It is a well-formedness condition on the driver set $C$ .

4 Precondition: decisional autonomy

The organisational unit that owns a module must be able to modify, test, and deploy within its scope without cross-unit coordination. If every change needs external sign-off, even a pure and complete module carries coordination overhead that the cost model does not capture.

Decisional autonomy is an organisational property, not a property of the code or the causal structure. The theory derives the module boundaries that minimise change propagation; it does not guarantee that those boundaries align with decision-making authority. When autonomy fails, real cost is change-propagation cost plus coordination overhead, and the partition that minimises the first may not minimise the sum. Two remedies are available: align team boundaries to the Γ-equivalence classes (reverse Conway’s Law), or add the organisational objective to the analysis and accept a composite optimum.

5 Does IVP apply to your system?

Two gates decide applicability: is the system maintenance-dominated, and is its causal structure known?

Maintenance-dominated?	Causal structure known?	Result
Yes	Yes	The Γ-equivalence partition unconditionally minimises change-propagation cost; under $(\star)$ it also minimises total maintenance cost; under the contamination premise the partition is unique.
Yes	No	IVP applies in principle; driver identification is the blocking step. Proceed with domain analysis.
No	Yes	IVP is not cost-optimal for this system’s dominant objective. Parameterise the analysis with the actual objective and derive the analogous conditions.
No	No	Neither gate is open. IVP is inapplicable as a cost-minimisation result, though it may still supply useful structural vocabulary.

Within the yes/yes cell, the preconditions and conditional premises above further qualify the result. When driver independence holds and decisional autonomy holds, the partition minimises propagation cost directly, and minimises total cost when $(\star)$ holds. When the driver set contains mutually activation-coupled overlapping pairs, merge them. When decisional autonomy fails, extend the objective or restructure the teams. When $(\star)$ fails at a composite/sharer pair, use a direct cost comparison over candidate coarsenings for that pair.

6 What the derivation does not supply

Driver identification. The assignment $\Gamma$ is assumed available; the derivation does not perform domain analysis for any specific system. Driver identification is domain analysis, grounded in the counterfactual test against domain artifacts.
Irreducibility determination. Reducibility judgments assume that the element’s purpose specification is available; the derivation does not decide reducibility from code alone.
Driver coverage. A system whose causal structure has not been documented is not a system that defies IVP — it is a system awaiting domain analysis. Whether every such system yields to driver analysis is an empirical question the theory does not settle.

7 Where the analysis lives

The three unconditional premises, the two conditional premises, and the two preconditions are examined in the paper on the Independent Variation Principle [1]. The four conditions as necessary and jointly exhaustive, and the contamination premise for strict Separation, are developed in the paper on cohesion and change propagation [2]. The coefficient condition $(\star)$ and the total-maintenance-cost result are developed in the paper on minimising change propagation and maintenance cost [3].

What IVP does not address at all — runtime behaviour, performance, security, teams, process, and tooling — is recorded on the consequences and application page.

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

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, “Why minimizing change propagation minimizes maintenance cost.” Zenodo, Jul. 2026. doi: 10.5281/zenodo.21362542.