Zero-coupling and duplicate-label controls

For every universe-polymorphic index type $\iota$, every complex number $\beta$, and every function $w:\iota\to\mathbb Z$ that is not injective, two statements hold simultaneously: for every integer $n$, $\exp((n:\mathbb C)\cdot0)=1$; and the $\iota$-indexed family $\bigl(\exp((w(i):\mathbb C)\beta)\bigr)_{i\in\iota}$ is not linearly independent over the algebraic closure of $\mathbb Q$ inside $\mathbb C$. The first conjunct includes negative and zero $n$ and is independent of $\beta$, $w$, and $\iota$. The second imposes no algebraicity, transcendence, or nonzeroness hypothesis on $\beta$. If $\iota$ is empty or a subsingleton, no noninjective $w$ exists, so the hypothesis cannot be supplied and the theorem is vacuous for such index types.