15.2 When invariance fails

Four recurrent failures should be kept distinct:

1. trajectory failure: the algorithms choose different points;

2. observer failure: decisions agree but their audit or consequence maps do not;

3. tangent failure: base decisions agree while sensitivities or repair directions differ; and

4. domain failure: the comparison leaves the registered class, as when projection, nonsmoothness, delay, or adaptive state breaks an unconstrained equivalence.

Presentation invariance is therefore a typed theorem, never permission to rename two methods because their formulas look similar.