Proofs, Derivations & Traces

The UOR Proofs, UOR Derivations, and UOR Computation Traces namespaces implement the certification pathway of the PRISM pipeline. Every algebraic identity must be proved, every proof must be derived from axioms, and every derivation must be traced for reproducibility.

Proof Species

UOR defines four fundamental proof types, each corresponding to a different verification method:

• ComputationCertificate -- verified by exhaustive computation. The identity is checked for every element at a given Witt Levels scale.
• AxiomaticDerivation -- derived from ring axioms. The identity follows logically from the algebraic laws of Z/(2^n)Z without needing to enumerate elements.
• InductiveProof -- proved by induction over Witt levels. The base case holds at some minimal level, and the inductive step lifts validity from level k to level k+1.
• EmpiricalVerification -- verified by quantum measurement. Used for identities that involve superposition or measurement outcomes.

Proof Properties

Every proof individual carries key properties:

• atWittLevel -- the Witt level at which the proof is valid.
• universalScope -- whether the proof holds at all Witt levels (true for Universal scope).
• derivationWitness -- a link to the derivation that produced this proof.

Inductive proofs additionally carry:

• baseCase -- proof at the minimal Witt level.
• inductiveStep -- proof that validity lifts from level k to k+1.
• validForKAtLeast -- the certified minimum Witt level.

Derivation Chains

The Derivation class records how one identity follows from another. Derivation chains trace the logical dependency graph of the 624 named identities, showing which identities are axioms and which are derived theorems.

Computation Traces

The ComputationTrace class records every step of a computation, creating an auditable log. The GeodesicTrace records the shortest-path trace -- the most efficient computation route through the derivation space.

Traces enable independent replay: given a trace, any verifier can re-execute the computation and confirm the result without trusting the original prover.

Certificates

Proofs flow into the UOR Certificates namespace, which issues formal certificates:

• CompletenessAuditTrail -- certifies that a completeness resolution is exhaustive (no cases were missed).
• GeodesicCertificate -- certifies that a computation followed the geodesic (shortest) path.
• GroundingCertificate -- certifies that the evaluation context is fully grounded (all bindings resolved).

Connection to the PRISM Pipeline

The proof system spans the Resolve-to-Certify boundary. Proofs and derivations are Resolve-stage objects that produce evidence; certificates are Certify-stage objects that attest to that evidence. This is the pathway through which the 624 algebraic identities are formally verified and certified.

See Witt Levels for how proofs are scoped to specific ring scales, and Resolution & Queries for how the resolution machinery produces the evidence that proofs reference.