Witt Levels

Witt levels W8--W32 are the four scaling tiers of the UOR The Ring Substrate substrate. Every computation, identity, and proof in UOR is valid at one or more Witt levels. Understanding Witt levels is essential for reading the algebraic identities and their associated proofs.

Level Definitions

Each level is a WittLevel named individual with:

• wittLength -- Witt vector length
• bitsWidth -- bit width 1, 2, 4, or 8
• cycleSize -- ring cardinality (2^bitsWidth)
• nextWittLevel -- the next level up (W8->W16, W16->W24, W24->W32)

W32 (the byte ring) is the primary ring for Content Addressing. W8 is the minimal boolean ring. The tower W8 < W16 < W24 < W32 is captured by the nextWittLevel chain.

The WittLevel Enum

In the generated uor-foundation Rust crate, WittLevel is an open-world newtype struct WittLevel { index: u32 } rather than a sealed enum. This is deliberate: the OWL ontology treats Witt levels as a known finite set today, but the newtype structure allows the framework to scale beyond W32 without breaking changes.

The #[non_exhaustive] attribute on the Rust type signals this openness. The four canonical instances W8--W32 are provided as associated constants.

Universal vs. Level-Specific Validity

Not all algebraic identities hold at all Witt levels. The ValidityScopeKind enum class captures four validity patterns:

• Universal -- the identity holds for all Witt levels (k in {0,1,2,3} and beyond)
• ParametricLower -- the identity holds for all k >= k_min (some minimum level)
• ParametricRange -- the identity holds for k_min <= k <= k_max
• LevelSpecific -- the identity holds only at exactly one Witt level

These scopes are enforced by the conformance validator validate_forall_scope_alignment, which checks that the universallyValid boolean is consistent with the validityKind and that validKMin/validKMax bounds are present when required.

Tower Chains

Amendments 41 and beyond introduced tower chain vocabulary to formalize the W-n scaling of inductive proofs. A LiftChain records a sequence of type lifts from level k to level k+1, and a LiftObstruction records where the lifting fails.

The LiftChainCertificate certifies that a lift chain is valid all the way to the target level. The ObstructionChain captures the failure mode -- where and why a lift cannot proceed.

Inductive proofs (InductiveProof) have three key properties (see Proofs, Derivations & Traces for details):

• baseCase -- proof at the minimal Witt level
• inductiveStep -- proof that validity at level k implies validity at level k+1
• validForKAtLeast -- the certified minimum Witt level

Connection to Content Addressing

The Witt level determines the granularity of Content Addressing. At W32 (byte width), a content address is a sequence of bytes -- the standard interpretation for file systems, network protocols, and cryptographic hash functions.

At W8 (bit width), a content address is a single bit -- useful for Boolean domains. The The Ring Substrate Z/(2)Z has only two elements (0 and 1), making W8 addresses the coarsest possible content-addressable identifiers.

The WittLevel individuals are referenced by the WittLevelBinding class, which links operations to the Witt level at which they are defined. This binding ensures that no operation is applied at a ring scale it was not designed for.