C-D20-BASE-NECESSARY

O-A: a global counterexample induces a base-level cover at every prime (necessary), but a base cover is NOT sufficient: the exact p=197 witness covers at base level yet is proper at refinement level 14. So the base-cover compatibility search is a sound one-way filter; every survivor must pass the refinement tower

In plain language

A single-prime cover is a required clue for a counterexample, but not proof of one; finer time grids can break it.

Exact statement

O-A: a global counterexample induces a base-level cover at every prime (necessary), but a base cover is NOT sufficient: the exact p=197 witness covers at base level yet is proper at refinement level 14. So the base-cover compatibility search is a sound one-way filter; every survivor must pass the refinement tower