Exotic Unit Groups of Sylvester Rank Completions

Let $R$ be any unital ring equipped with a Sylvester matrix rank function, and let $Q$ be the rank completion of the matrix system defined by block-diagonal embeddings along a factor sequence. We prove that both $\text{GL}(Q)$ and $(Q,+)$ are exotic: every strongly continuous unitary representation is trivial. We also prove that $\text{GL}(Q)$ is Bourbaki bounded and admits no nonzero escape function. For every $\epsilon >0$, let $B_{\epsilon }$ be the open ball centered at the identity. The least positive integer $N$ such that $\text{GL}(Q)=B_{\epsilon }^N$ is $\lfloor {\epsilon }^{-1}\rfloor +1$. Neither regularity nor irreducibility is required. Our proof of exoticness uses a uniform Kazhdan estimate for additive integer matrix groups together with rank-one matrix perturbations, rather than the character-theoretic and continuous ring methods used previously. The geometric conclusions follow from factorizations using diagonal corner subgroups.