The classical update and permutation extension

Define $f(x)$ by setting stage to 1, replacing the packet index by $m_{\mathrm{src}}$, decreasing permission by one, and recomputing reports. Token, clock, incidence mode, and metadata are retained. Prescribe

The number of canonical geometric inputs is

Matter states and name locations are not counted by $Q$.

Outputs are distinct because they retain $x$. Input and output sets are disjoint because their first-block stage bits differ. Exchange each input word with its prescribed output and fix every other $2B$-bit word. This defines a product $P_{\mathrm{geo}}$ of $Q$ disjoint transpositions.

Retaining input history is the elementary reversible-computation principle associated with Bennett Bennett1973; the gate basis is the classical reversible basis associated with Toffoli Toffoli1980. The bound needed here is proved directly.