One B6 braid approximates an encoded two-qubit gate

Since $U$ is unitary, $\mathrm{\mid }\det U\mathrm{\mid }=1$. Each displayed extension is unitary on the orthogonal direct sum. Its determinant is $\det U\cdot (\det U{)}^{-1}=1$, so $T_1\in SU(5)$ and $T_{\tau }\in SU(8)$. Both preserve the encoded four-dimensional subspace and act there as $U$, by [F1].

The product of open operator-norm $\delta$-balls around $(T_1,T_{\tau })$ is a neighborhood in $SU(5)\times SU(8)$. By [A1] and [F2], it contains the image of one common normalized braid word. This proves both block inequalities at once; the word is not chosen separately for the two sectors. For certified Cauchy targets, [F5] gives a terminating search with rational strict certificates (choose any rational tolerance below $\delta$).

Apply [F3] in each charge block to the normalized braid matrix and its code-preserving extension $T_a$. The normalized compression to the code differs from $U$ by at most $\delta$, and the off-diagonal block from code to leakage has norm at most $\delta$. Write $\zeta =R_1^{e(b)}$ for the common unit scalar from [F4]. The physical compression differs from the phase-matched target $\zeta U$ by $\mathrm{\parallel }\zeta P{B}_a(b)P-\zeta \tilde{U}\mathrm{\parallel }=\mathrm{\parallel }P{B}_a(b)P-\tilde{U}\mathrm{\parallel }\le \delta$; its leakage amplitude is $\mathrm{\parallel }(I-P)\zeta B_a(b)P\mathrm{\parallel }=\mathrm{\parallel }(I-P)B_a(b)P\mathrm{\parallel }\le \delta$. The exponent-sum phase must still be retained under coherent control.