IndisputableMonolith.Patterns.GrayCodeAxioms

The GrayCodeAxioms module defines the inverse Gray code map that recovers a natural number from its binary-reflected representation through cumulative XOR. Researchers building Hamiltonian cycles on hypercubes cite it to guarantee bijectivity in pattern constructions. The module consists of targeted definitions and supporting facts that establish the required invertibility properties.

claimThe inverse Gray code satisfies $g^{-1}(g) = g_0 + g_1 + g_2 + g_3 + g_4 + g_5 + g_6 + g_7 + g_8 + g_9 + g_{10} + g_{11} + g_{12} + g_{13} + g_{14} + g_{15} + g_{16} + g_{17} + g_{18} + g_{19} + g_{20} + g_{21} + g_{22} + g_{23} + g_{24} + g_{25} + g_{26} + g_{27} + g_{28} + g_{29} + g_{30} + g_{31} + g_{32} + g_{33} + g_{34} + g_{35} + g_{36} + g_{37} + g_{38} + g_{39} + g_{40} + g_{41} + g_{42} + g_{43} + g_{44} + g_{45} + g_{46} + g_{47} + g_{48} + g_{49} + g_{50} + g_{51} + g_{52} + g_{53} + g_{54} + g_{55} + g_{56} + g_{57} + g_{58} + g_{59} + g_{60} + g_{61} + g_{62} + g_{63} + g_{64} + g_{65} + g_{66} + g_{67} + g_{68} + g_{69} + g_{70} + g_{71} + g_{72} + g_{73} + g_{74} + g_{75} + g_{76} + g_{77} + g_{78} + g_{79} + g_{80} + g_{81} + g_{82} + g_{83} + g_{84} + g_{85} + g_{86} + g_