= Acyclicity of a finite Dynkin diagram
The underlying graph of a finite <Dynkin diagram> is a forest. Normalize every <simple root> to length one. Each bonded pair has inner product at most $-1/2$, and every other distinct pair has nonpositive inner product. A cycle with $m$ vertices would make the squared norm of their sum at most $m-m=0$, contradicting their <linear independence> in an <inner-product space>. Multiple bonds denote root-length information, not graph-theoretic parallel-edge cycles.
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