title: Inhabitance Verdict category: concepts

Inhabitance Verdict

The v0.2.1 release introduces carrier non-emptiness as a first-class verdict primitive sibling to cert:CompletenessCertificate. Together they answer two distinct questions about a type:ConstrainedType:

• cert:CompletenessCertificate: does freeRank = 0 on the minimal basis? (IT_7d: all Betti numbers zero, χ = n.)
• cert:InhabitanceCertificate: is the carrier non-empty? (At least one value tuple satisfies the constraint system.)

For constraint systems that admit multiple satisfying value tuples, an InhabitanceCertificate.verified may be true while the corresponding CompletenessCertificate.verified is false — the carrier is inhabited but the minimal basis leaves sites free.

Dispatch routing

predicate:InhabitanceDispatchTable routes every input to one of three polynomial deciders based on the constraint system's shape:

1. predicate:Is2SatShape → resolver:TwoSatDecider (priority 0): Aspvall-Plass-Tarjan SCC decision in O(n+m).
2. predicate:IsHornShape → resolver:HornSatDecider (priority 1): unit propagation in O(n+m).
3. predicate:IsResidualFragment → resolver:ResidualVerdictResolver (priority 2, catch-all): returns Err(InhabitanceImpossibilityWitness) with reduction:ConvergenceStall for non-polynomial residual fragments.

Coverage is exhaustive by construction; reduction:DispatchMiss is unreachable for this table.

Identities

• op:IH_1 (soundness): InhabitanceCertificate(T).verified ⇔ carrier(T) ≠ ∅.
• op:IH_2a (2-SAT cost, restricted): O(n+m) via classical 2-SAT.
• op:IH_2b (Horn-SAT cost, restricted): O(n+m) via unit propagation.
• op:IH_3 (carrier preservation): basis reduction preserves the carrier.

Unrestricted IH_2 would be equivalent to P = NP and is not adopted.

One-liner consumer API

use uor_foundation::enforcement::prelude::*;
use uor_foundation_macros::ConstrainedType;

#[derive(ConstrainedType, Default)]
#[uor(residue = 255, hamming = 8)]
struct Shape;

let cert: Validated<InhabitanceCertificate> =
    InhabitanceResolver::new().certify(&Shape)?;
match cert.witness() {
    Some(witness_bytes) => println!("inhabited: {witness_bytes:?}"),
    None => println!("inhabited (verified, witness elided)"),
}

The Lean 4 counterpart exposes the same shape via UOR.Enforcement.Certify and UOR.Pipeline.runInhabitance.

See also

• Grounded Wrapper — compile-time ground-state guarantee.
• Certify Trait — one-liner verdict API across all four resolver façades.
• Ontology: cert:InhabitanceCertificate, proof:InhabitanceImpossibilityWitness, predicate:InhabitanceDispatchTable.