7.4 Differential tests for interaction

The differential framework supplies a local test of whether two proposed ingredients interact. Let \(A,B\in \mathsf{Weil}_1^{\mathrm{tan}}\) be probe shapes realized by \(F(A)\) and \(F(B)\). The monoidal structure compares the joint probe

with the isolated responses. Depending on the application, an observer may test additivity, order sensitivity, mixed variation, bracket closure, or failure of a comparison square.

Write \(o\in \Lambda \) for an observer. An interaction witness is any registered comparison

whose null value means that the observed joint response is explained by the declared independent-composition law. A non-null value says that the two directions interact relative to that observer and law. It does not yet say that the interaction is valuable, causal, or globally integrable.

First-order probes ask whether each ingredient has a locally visible effect. Paired or higher-order Weil probes ask whether their combination introduces a mixed effect. Order-sensitive probes can distinguish \(F(A)F(B)\) from \(F(B)F(A)\) when the semantics permits such a comparison. This gives combinational creativity a diagnostic sequence:

1. verify that each ingredient has a typed local interpretation;

2. realize the joint probe through the proposed interface;

3. compare joint and isolated responses under selected observers;

4. localize any interaction or obstruction to the interface;

5. integrate the local candidate and test its finite consequences.

. A mixed derivative, commutator, or nonzero interaction residual is evidence of coupling, not a certificate of creativity. Its meaning depends on the chosen observer, baseline composition law, and semantics of the probes.