Implementation theorem

The theorem is an existence construction with explicit resource bounds, not an efficient compilation claim or an execution record. It neither acquires nor verifies the token. Intermediate gate words and arbitrary words in the permutation completion need not decode as geometric configurations.

table example. For $q=6$, $K_6=1$ and ${\mathcal{T}}_6=\{-1,0,1\}$. Take old index $0$, token $+1$, stage $0$, and permission $1$. The promise is $\mathrm{Tr}(\varrho Z)=6\arctan (1/64)$. The update installs $\pi /2+6\arctan (1/64)$, sets stage to $1$ and permission to $0$, recomputes reports at $s=-1/2$, and retains the original word in history. Matter and locations are unchanged.