8.7 From exploration to obstruction

Exploration can reveal three different kinds of difficulty, which should not be collapsed:

1. search failure: no path was found under the available resources;

2. model failure: a candidate in the current language conflicts with evidence;

3. language failure: the registered objects and operations cannot express the distinction demanded by the evidence.

Only evidence for the third directly motivates opening a transformational proposal. Language failure is always relative to a registered syntax and semantic class; it should be certified by benchmark construction, an expressivity result, exhaustive finite analysis, or another explicit nonrealizability argument. Counter-witnesses still matter. One can vary the search policy, enlarge the computational budget, test alternate parameterizations, or ask whether a known solution is representable. A robust obstruction survives the registered controls and localizes to a declared limitation of \(\mathfrak R_{\mathrm{diff}}\).

Nonclosure is one particularly informative signal. If admitted local moves compose to a direction not represented by the current vocabulary, the system has detected a mismatch between its local dynamics and declaration. Omitted structure is one hypothesis; observer error, estimation error, hidden state, or failure of a side law are alternatives. These must be audited before a new generator is proposed.

. Exploration warrants opening a candidate transformational trace when it produces an auditable obstruction: a reproducible failure that survives the registered in-language controls and for which an explicit argument rules out the relevant class of ordinary repairs. This warrants proposal, not admission.