Shall I rephrase your ideas more formally:
Suppose under constaint C, the precise summary is S. But we could hardly get S usually. We could generate 3 kinds of summary:
S1: what must happens, i.e., S1 is contained by S.
S2: what could happen, but I know that nothing more can happen, i.e. S2 contains S.
S3: what could happen, but I don’t know if other can happen, i.e., the intersection of S3 and S is not empty, but no one contains another.