Witt Universality
Definition
Witt universality is the property of an algebraic identity that holds for all Witt levels n ≥ 1, not just at a specific W8 ring. An identity is universally valid when it is provable symbolically from ring axioms rather than verified exhaustively at one ring size.
The universallyValid boolean property
on an Identity individual declares this
status. The critical identity neg(bnot(x)) = succ(x) is the canonical
example: it holds in Z/(2^n)Z for every n ≥ 1 and carries
op:universallyValid true.
Witt Levels
The WittLevel newtype struct defines an open class of Witt levels. Named levels include:
- W8 — the base Witt level used for exhaustive verification (ring size = 2^8 in the UOR Foundation reference implementation).
- W16 — the concrete ring Z/(2^16)Z, now formally typed as W16Ring with W16bitWidth = 16 and W16capacity = 65,536.
- W24, W32, ... — higher levels declared via the
schema:nextWittLevelchain.
WittLevelBinding
A WittLevelBinding record links an
op:Identity individual to a specific Witt level at which the identity
has been verified. Because identities may be verified at multiple levels, the
verifiedAtLevel property is
non-functional: one binding per (Identity, WittLevel) pair.
Each binding carries a bindingLevel pointing to the relevant WittLevel individual.
Universal Identity Groups (QL_ series)
Amendment 26 adds seven QL_ identity individuals (QL_1 through QL_7) that
generalize key algebraic, thermodynamic, topological, and pipeline identities
to all n ≥ 1. Each carries op:universallyValid true and a
op:verificationDomain typed assertion.
| Identity | Statement |
|---|---|
| QL_1 | neg(bnot(x)) = succ(x) in Z/(2^n)Z for all n ≥ 1 |
| QL_2 | Ring carrier size is exactly 2^n |
| QL_3 | Landauer erasure cost scales as n × k_B T ln 2 |
| QL_4 | Dihedral group D_{2^n} action is faithful at all n |
| QL_5 | Canonical form rewriting terminates at all levels |
| QL_6 | χ(N(C)) = n completeness condition generalizes |
| QL_7 | Euler characteristic of the ring topology = 1 − n |