Why individual failure would not suffice

Failure of one selected extension, one correction, or one ordinary product criterion would not exclude all permitted constructions. Conversely, arbitrary Hahn–Banach extensions or algebraic corrections would not establish a construction in the frozen class.

The full decision requires either:

1. an actual admissible full map, with its defining operations and all required bounds proved; or

2. an incompatibility theorem in a class proved to exhaust the permitted maps.

Neither is supplied by the present local tests.