For u ∉ Q̄ and coprime p, q ∈ Q̄[X]: a five-dimensional Q̄ + Q̄u + Q̄u⁻¹ + Σ Q̄ c (p(u)/q(u))ᵏ contains u², u⁻² or some 1/(u − b), or is R9's involution space

Let $u \in \mathbb{C} \setminus \overline{\mathbb{Q}}$, let $p, q \in \overline{\mathbb{Q}}[X]$ be coprime, $h = p(u)/q(u)$, let $c \in \mathbb{C}$ be non-zero, and let $W$ be the $\overline{\mathbb{Q}}$-span of $1, u, u^{-1}$ and $c, ch, ch^2, ch^3$. Suppose $\dim W = 5$. If $\max(\deg p, \deg q) \geq 1$, then $W$ contains $u^2$, $u^{-2}$ or $1/(u - b)$ for some non-zero $b \in \overline{\mathbb{Q}}$, or $W$ is the $\overline{\mathbb{Q}}$-span of $1, u, u^{-1}$ and $p(u)^iq(u)^{3-i}/(u\,l_1(u)l_2(u))$, $i = 0, \dots, 3$, for some non-zero $l_1, l_2 \in \overline{\mathbb{Q}}p + \overline{\mathbb{Q}}q$.

Together with DiazModulus.mobius_progression_space_contains_square_or_reciprocal (ratio of degree $1$) and R9's DiazModulus.involution_pencil_configuration_invisible_to_four_dimensional_extensions (ratio of degree $2$), this classifies the five-dimensional spaces $W \supseteq \overline{\mathbb{Q}} + \overline{\mathbb{Q}}u + \overline{\mathbb{Q}}u^{-1}$ that contain a $4$-term geometric progression with rational ratio: either $W$ contains $u^2$, $u^{-2}$ or some $1/(u - b)$, or $W$ is one of R9's involution spaces, which contain none of them.

Proof. By DiazModulus.progression_space_ratio_degree_le_two the ratio has degree $1$ or $2$. In degree $1$, $h = (\alpha u + \beta)/(\gamma u + \delta)$ with $\alpha\delta - \beta\gamma \neq 0$ by coprimality, and DiazModulus.mobius_progression_space_contains_square_or_reciprocal gives $u^2$, $u^{-2}$ or some $1/(u - b)$ in $W$. In degree $2$, DiazModulus.quadratic_ratio_progression_space_eq_involution_pencil_space gives R9's form.

Novelty. Not found in the sources read; it is short. That a configuration whose six products span four dimensions is such a progression is Bachoc–Serra–Zémor (2017, Lemma 4); here the progression is a hypothesis.