A Three-Bit Boundary for Cyclic Boolean–Modular Inversion

For fixed w-bit words c and k, consider F(x) = ((ROR1(x) XOR ROR3(x)) + (x AND c) + k) modulo 2^w. For every w at least four, each inverse fiber is determined by at most eight choices of the first three input bits. A carry recurrence reconstructs each candidate in w local bit updates, and three closure equations select exactly the preimages. We prove completeness, uniqueness within each boundary choice, and a sharp bound of eight preimages. An explicit family attains the bound at every width divisible by seven. A rotation of r positions gives an r-bit boundary and a 2^r w update bound.