11.5 Nonclosure as a discovery trigger

The closure residual can play a second role beyond screening. When

the visible response distribution has failed to contain an interaction that its own fields generate. This is an empirical obstruction to the declared causal geometry. It creates a discovery question: what must change for the response family to become adequate?

The first generated enlargement is

A nontrivial quotient \(Q^{(1)}\) records directions required for closure but missing from the fitted visible span. Iterating the construction gives the Lie closure

This completion is a mathematical hypothesis about missing directions, not a semantic identification of hidden causes.

Several explanations can produce the same quotient. A latent variable may supply an omitted direction. So may a measured but excluded coordinate, history dependence, an incorrectly pooled regime, a changing intervention target, support failure, or field-estimation error. The discovery phase must therefore branch across candidate declaration changes rather than map every residual to a latent node. One branch enlarges the state space; another refines the cover of regimes; another changes the transport model; still another rejects the fitted fields.

This is a small, executable instance of the outer discovery loop developed in Chapter 22. Nonclosure proposes that the current declaration is incomplete. Controlled additions of variables, contexts, or memory then generate competing completions. Admission asks which completion contracts the held-out residual while preserving calibrated influences and producing new testable consequences.