Meaning
A complete defect ledger detects exactly when all required realization conditions hold.
The ledger assigns nonnegative defect coordinates to the physical tests. With positive weights, their sum q is complete on the declared candidate class when q(x)=0 if and only if x determines an admissible physical realization. Completeness requires both directions: every admitted realization has zero defect, and every zero-defect candidate is admitted.
Example (illustrative)
If a toy ledger checks typing and positivity but omits sector survival, a candidate with a missing sector might score zero. That ledger is incomplete even if each recorded defect is nonnegative.
Scope and limits
A sum of nonnegative tests is not automatically complete. Nor does an abstract truth-valued test necessarily provide an effective decision algorithm.
Source
GAT 0 — Finite Cohomological and Coercive Recognition. GAT 0, Definition 4.5.