Residuated sufficiency is the condition in which Residuation is fully satisfied between two structures sharing a relational field.
The residuation law states: Includes(Together(a, b), c) if and only if Includes(a, Induces(b, c)). When this equivalence holds, the two structures are residuated-sufficient with respect to each other — each can reconstruct the other’s contributions through the shared operations of the field. When it fails, there is a gap: the structures make demands on each other that the field cannot mediate.
The residuated sufficiency gap g measures the degree of failure:
g = 1 − V(Together(a, b)) / V(Either(a, b))
where V is a measure on the observable space. g = 0 means full sufficiency; g = 1 means the structures are disjoint. Values between 0 and 1 indicate partial residuation — partial mutual intelligibility.
This measure appears across the framework’s applications: as the reflexive calibration index in measurement theory, as the reflexive disequilibrium order parameter in cosmology, and as the diagnostic for informational malfunction in sociology.