Theorem: Forward-domain implementation

For every $b,d\ge 0$, $r\ge 1$, and fixed finite original-name list, there is an exact NOT–CNOT–Toffoli circuit $U$ with the bit bound (eq:totalbits) such that

for every canonical forward event code $c$ with clean history and work registers.

The circuit retains input history, returns all work bits to zero, and acts on the internal matter space as

Its geometric output and geometric control are independent of spectator number and spectator state.

Proof. The geometric permutation sends $(x,0_B)$ to $(f(x),x)$ on the forward domain. Proposition prop:decoder identifies its decoded geometric output with the geometric part of $G$.

The location registers are unchanged, which implements $\tau$. Each name history receives its original location. The copied histories are ignored by the decoder. Lemma lem:synthesis returns geometric work to zero, and the name-copy gates require no additional work.

The geometric circuit acts only on geometric and geometric-history registers. The subsequent name-copy gates do not modify any geometric register. No gate acts on the internal matter factor.

The asserted decoder identity is restricted to the declared forward domain. No geometric meaning is assigned to intermediate gate states or to arbitrary inputs of the permutation completion.