IndisputableMonolith.Patterns.GrayCodeAxioms
GrayCodeAxioms supplies the inverse Gray code map that recovers a natural number from its Gray code representation through cumulative XOR. Researchers constructing Hamiltonian cycles on hypercubes cite the module when establishing bijectivity of the binary-reflected Gray code. The module consists of the core inverse definition together with direct algebraic inversion facts.
claimThe inverse Gray code map satisfies $\gamma^{-1}(g) = \bigoplus_{i \geq 0} (g \gg i)$, recovering the original index via cumulative XOR and obeying the inversion identity $\gamma^{-1}(\gamma(n)) = n$.
background
The module extends the base pattern definitions imported from IndisputableMonolith.Patterns. It introduces the inverse Gray code operation that converts a Gray code value back to its natural-number index by successive XOR steps. This supplies the bijectivity tool needed for sequence and cycle constructions in the Patterns domain.
Upstream material in Patterns provides the foundational sequence and adjacency notions on which the Gray code axioms rest. The module therefore sits as a targeted extension that isolates the inversion step required by later constructions.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds the GrayCode construction of the binary-reflected Gray code (BRGC) that generates a Hamiltonian cycle on the d-dimensional hypercube Q_d, and the GrayCycleGeneral module that lifts the same BRGC formula gray(n) = n XOR (n >>> 1) to arbitrary dimension. It supplies the inversion property required to verify that the generated sequence is a cycle.
scope and limits
- Does not define the forward Gray code map.
- Does not treat general-dimension cycle existence beyond the axioms supplied to dependents.
- Does not address non-binary or non-reflected Gray code variants.