17.3.2 Program B: obstruction-theoretic repair

17.3.2 Program B: obstruction-theoretic repair

Chapters 14–15 classify repair spaces and prove finite preservation results, but do not yet decide when a failed horn is repairable inside the current declaration. The next step is an obstruction theory for registered repair problems. A vanishing class should certify an admissible tangent or coherent filler; a nonvanishing class should distinguish boundary repair from genuine doctrinal accommodation.

Ordinary cohomology in a group such as \(H^2(N_\bullet E;A)\) is a useful first candidate for low-dimensional extension problems, not a general answer: the coefficient object and the degree of the obstruction must be derived from the horn, fibration, and settled obligations. The target theorem should connect vanishing obstructions to functorial comparison-guided localization. A complementary amplification theorem should state when weak learners freely compose into strong structure without destroying the repairs and invariants already certified.