Fixed boone undecidability implies uniform undecidability

Hard-code the finite string describing ${\mathcal{P}}_{\mathcal{B}}$ in a wrapper machine. On input $z$ over its signed generating alphabet, the wrapper writes this constant string and the pair delimiter, copies $z$, and runs $U$. This explicitly computes $D(z)=U({\mathcal{P}}_{\mathcal{B}},z)$; the preprocessor terminates because the prefix is fixed and $z$ is finite.

Every such pair is a valid input for $U$, so $D$ halts on every $z$ and satisfies $D(z)=\mathrm{y}\mathrm{e}\mathrm{s}$ exactly when $z=1$ in $\mathcal{B}$. For instance $D(\epsilon )=U({\mathcal{P}}_{\mathcal{B}},\epsilon )=\mathrm{y}\mathrm{e}\mathrm{s}$, and $D(x{x}^{-1})=\mathrm{y}\mathrm{e}\mathrm{s}$ by cancellation. The claimed behavior for all words, rather than these two calculations alone, makes $D$ a decider forbidden by [F1]. Thus the proposed uniform algorithm cannot exist.