Type Completeness
Definition
Type completeness is the formal property of a ConstrainedType that guarantees resolution always terminates in O(1) time. A type is complete when its constraint nerve satisfies the completeness criterion IT_7d: the Euler characteristic of the Cech nerve equals the Witt level n and all Betti numbers β_k are zero.
Completeness is certified by the kernel via the
CompletenessCertificate pathway. The
certified entity is a CompleteType —
a subclass of ConstrainedType that has passed the full ψ pipeline.
Certification Pipeline
The completeness certification lifecycle proceeds in four stages:
-
Candidate registration — A
ConstrainedTypeis promoted to a CompletenessCandidate by associating it with a resolver:ResolutionState and an observable:CechNerve via the candidateNerve property. -
Witness accumulation — Each constraint application that closes at least one site produces a CompletenessWitness. The witness records the applied constraint via witnessConstraint and the sitesClosed count.
-
Resolver evaluation — A CompletenessResolver reads the cached nerveEulerCharacteristic from the ResolutionState. If χ(N(C)) = n and all β_k = 0, the kernel issues a CompletenessCertificate; otherwise it emits a resolver:RefinementSuggestion.
-
Certificate issuance — The CompletenessCertificate links to the certified CompleteType via certifiedType and records provenance via an CompletenessAuditTrail.
Termination Criterion IT_7d
IT_7d requires:
- χ(N(C)) = n (Euler characteristic of the Cech nerve equals the Witt level)
- β_k = 0 for all k ≥ 0 (no topological obstructions)
When IT_7d holds, the residual entropy S = freeRank × ln 2 drops to zero, meaning no unconstrained sites remain.