For u ∉ Q̄ and coprime p, q ∈ Q̄[X]: if 1, u, u⁻¹ lie in the span of c, ch, ch², ck, chk with h = p(u)/q(u), then max(deg p, deg q) ≤ 2

Let $u \in \mathbb{C} \setminus \overline{\mathbb{Q}}$ and $c, k \in \mathbb{C}$ with $c \neq 0$, and suppose $1, u, u^{-1}$ lie in the $\overline{\mathbb{Q}}$-span of $c, ch, ch^2, ck, chk$. Suppose $h = p(u)/q(u)$ with $p, q \in \overline{\mathbb{Q}}[X]$ coprime and $\max(\deg p, \deg q) \geq 1$. Then $\max(\deg p, \deg q) \leq 2$.

A configuration $(1, h) \otimes Y$ whose products span a five-dimensional $W \supseteq \overline{\mathbb{Q}} + \overline{\mathbb{Q}}u + \overline{\mathbb{Q}}u^{-1}$ gives, after normalisation, $W = c\cdot\mathrm{span}(1, h, h^2, k, hk)$ (research notes R12, Q1; not formalised). So no such configuration has a ratio of degree $3$ or more.

Proof. Evaluation at $u$ embeds $\overline{\mathbb{Q}}(X)$ in $\mathbb{C}$; let $f = p/q$. Coprimality and Mathlib's RatFunc.finrank_eq_max_natDegree give $\max(\deg p, \deg q) \le [\overline{\mathbb{Q}}(X) : \overline{\mathbb{Q}}(f)]$. As $f$ is not constant, $h \notin \overline{\mathbb{Q}}$, and DiazModulus.quadratic_over_ratio_of_mem_progression_pair_span gives $A(h)u^2 + B(h)u + C(h) = 0$; by injectivity $X$ is a root of the non-zero $A(f)Y^2 + B(f)Y + C(f)$ over $\overline{\mathbb{Q}}(f)$ ($f$ is transcendental), so $[\overline{\mathbb{Q}}(X) : \overline{\mathbb{Q}}(f)] = \deg \operatorname{minpoly} X \le 2$.

Novelty. Not found in the sources read; it is short.