Robust logarithmic lower bound on shared-resource cost for $f$-routing

In one-round $f$-routing, Alice receives an $n$-bit string and an unknown qubit, while Bob receives another $n$-bit string. They exchange one simultaneous message each, and the value of $f$ determines which party must recover the qubit. Message lengths, local systems, and local operations are unrestricted. We charge only $E_{\dim}(\rho_{LR})=\log_2\min\{\operatorname{rank}\rho_L,\operatorname{rank}\rho_R\}$, the logarithm of the smaller marginal support dimension of the shared state prepared before the inputs arrive. The state may be arbitrary and mixed. For the inner product modulo $2$, we prove that every protocol with worst-case error at most $0.09$ in both routing cases, measured in the full, unhalved diamond norm, satisfies $d\log_2(2d)=\Omega(n)$ for $d=\min\{\operatorname{rank}\rho_L,\operatorname{rank}\rho_R\}$. Thus $d=\Omega(n/\log n)$ and $E_{\dim}(\rho_{LR})\ge\log_2 n-\log_2\log_2 n-O(1)$. The closest earlier growing Schmidt-rank lower bound for an explicit routing function assumes zero error in one routing case. Our proof converts correctness into a matrix whose entries have a constant gap between the two routing cases. It approximates this matrix by one whose rank depends on $d$, but not on the dimensions of messages or local systems. A sign-rank lower bound for the matrix of the inner product modulo $2$ completes the argument. A lower bound polynomial in $n$ on $E_{\dim}$ remains open.