Critical Identity

Statement

The critical identity is the foundational theorem of the UOR ring substrate:

neg(bnot(x)) = succ(x) for all x ∈ R_n

This states that the composition of the two canonical involutions of Z/(2^n)Z equals the successor operation.

Mathematical Proof

For x ∈ Z/(2^n)Z:

  1. bnot(x) = (2^n - 1) - x (bitwise complement)
  2. neg(bnot(x)) = -(2^n - 1 - x) mod 2^n = x + 1 mod 2^n
  3. succ(x) = x + 1 mod 2^n

Therefore neg(bnot(x)) = succ(x). ∎

Ontology Representation

The critical identity is represented by the named individual Critical Identity:

op:criticalIdentity
    a           owl:NamedIndividual, op:Identity ;
    op:lhs      op:succ ;
    op:rhs      op:neg ;
    op:forAll   "x ∈ R_n" .

Properties involved:

  • lhs: the left-hand side of the identity
  • rhs: the right-hand side
  • forAll: quantification domain

Proof Representation

The proof is captured by the class CriticalIdentityProof, a subclass of Proof.

The property provesIdentity links a CriticalIdentityProof to the op:criticalIdentity individual:

<https://uor.foundation/instance/proof-critical-id>
    a                       proof:CriticalIdentityProof ;
    proof:provesIdentity    op:criticalIdentity ;
    proof:valid             true .

Significance

The critical identity reveals:

  1. Successor is not primitive — it is derived from the two involutions
  2. Dihedral structure — neg and bnot generate the full dihedral group D_{2^n} captured by DihedralGroup
  3. Universal computation — any computable function on R_n can be expressed in terms of neg, bnot, and their compositions via composedOf

Composition

The successor operation succ is defined as the composition of neg and bnot in that order:

op:succ  op:composedOf  ( op:neg  op:bnot ) .

The list preserves application order: bnot is applied first, then neg.