A finite reversible extension

Although the forward map in this example is already injective, we retain complete input history. This gives a simple, explicit implementation certificate and makes no minimal-memory claim.

Let $D_{\mathrm{geo}}$ be the finite set of canonical forward geometric words. For the fixed family, its only free geometric field is $\nu \in \{1,\dots ,2^r-1\}$, so

On two $B$-bit blocks prescribe

Inputs and outputs are disjoint because their first-block mode bits are respectively zero and one. Distinct inputs have distinct outputs because the second output block retains $x$.

Define a permutation $P_{\mathrm{geo}}$ on all $2B$-bit words by exchanging each pair

and fixing every other word.