Injective winding labels yield independent algebraic phases

Assume the proposition that, for every complex number $\beta$, algebraicity of $\beta$ over $\mathbb Q$ together with $\beta\ne0$ implies that $\exp(\beta)$ is transcendental over $\mathbb Q$. Let $\alpha\in\mathbb C$ be algebraic over $\mathbb Q$ and nonzero; let $\iota$ be any universe-polymorphic type; and let $w:\iota\to\mathbb Z$ be injective. Then the $\iota$-indexed family $\left(\exp\!\left((w(i):\mathbb C)(\mathrm i\alpha)\right)\right)_{i\in\iota}$ is linearly independent over the algebraic closure of $\mathbb Q$ inside $\mathbb C$. Equivalently, every finitely supported family of coefficients $a_i$ from that algebraic closure satisfying $\sum_{i\in\iota}a_i\exp((w(i):\mathbb C)(\mathrm i\alpha))=0$ has $a_i=0$ for every $i$. Injectivity is the only condition on $w$: it need not be surjective, and its integer values may be negative, zero, or positive. The type $\iota$ need not be finite or nonempty; when $\iota$ is empty, its unique map to $\mathbb Z$ is injective and the linear-independence conclusion is vacuous, although the hypotheses concerning the global transcendence proposition and $\alpha$ remain required. No reality condition is imposed on $\alpha$.