1.9 A minimal worked example

Suppose a sketch declares two routes

and a realization \(D\) fails to factor through that equation. A parameter presentation \(P\to M\) contains redundant coordinates, and a cover \(\{ U_i\} \) localizes the discrepancy.

The repair calculus organizes the reasoning:

1. apply R2 to remove parameter motion that does not descend to \(M\);

2. apply R3 to diagnose the failure on the \(U_i\);

3. propose a target edit of \(p\) and use R5 to preserve the declared behavior of every non-target path;

4. use R4 to test whether the repaired path remains coherent under admissible perturbations;

5. if the available repair language cannot resolve the obstruction, use R6 to propose a conservative extension \(\mathbb S^+\); and

6. apply the admission rule on held-out observations before replacing \(D\) by \(D'\).

No scalar loss is required for these transformations. A scalar observer may rank candidates, but it does not supply the descent, invariance, conservativity, or admission premises.