A five-dimensional space of rational functions of u containing 1, u, u⁻¹ carries a 2×3 configuration iff it is D1, D2a or D2b

Let $u \in \mathbb{C}$ be transcendental, that is, $u \notin \overline{\mathbb{Q}}$ (the algebraic numbers); spans and linear independence are over $\overline{\mathbb{Q}}$. Let $W \subseteq \mathbb{C}$ be a $\overline{\mathbb{Q}}$-vector subspace of dimension $5$ with $1, u, u^{-1} \in W$, that is, $H_0 \subseteq W$ for $H_0 = \mathrm{span}(1, u, u^{-1})$. Suppose every element of $W$ lies in $\overline{\mathbb{Q}}(u)$, the subfield of $\mathbb{C}$ generated by $\overline{\mathbb{Q}}$ and $u$. Then $W$ carries a $2 \times 3$ configuration (linearly independent $x_0, x_1 \in \mathbb{C}$ and linearly independent $y_0, y_1, y_2 \in \mathbb{C}$ with every $x_iy_j \in W$) if and only if one of the following holds:

• (D1) there are linearly independent $t_0, t_1, t_2 \in \mathbb{C}$ with $t_j \in W$ and $ut_j \in W$ for every $j$;
• (D2a, two-orbit) $W = \mathrm{span}(1, u, u^{-1}, Q'(u)S_1(u)/(uP'(u)), P'(u)S_2(u)/(uQ'(u)))$ for some $P', Q', S_1, S_2 \in \overline{\mathbb{Q}}[X]$ with $P', Q'$ coprime, $\max(\deg P', \deg Q') = 2$ and $\deg S_1, \deg S_2 \leq 2$;
• (D2b, cyclic) $W = \mathrm{span}(1, u, u^{-1}, Q'(u)S_1(u)/(uP'(u)), Q'(u)(Q'(u)S_1(u) + P'(u)S_2(u))/(uP'(u)^2))$ for some such $P', Q', S_1, S_2$.

Proof. ($\Rightarrow$) DiazModulus.rational_space_configuration_coprime_ratio writes $x_1q(u) = x_0p(u)$ with $p, q$ coprime and $\max(\deg p, \deg q) \geq 1$, and DiazModulus.two_by_three_configuration_ratio_degree_le_two bounds that degree by $2$. Degree $1$: DiazModulus.mobius_ratio_configuration_u_multiplier_triple gives (D1). Degree $2$: Theorem NF (DiazModulus.degree_two_configuration_normal_form) gives (D1), (D2a) or (D2b). ($\Leftarrow$) Under (D1), $(1, u) \otimes t$ is a configuration, $1, u$ being independent as $u \notin \overline{\mathbb{Q}}$. Under (D2a) or (D2b), $W$ is the canonical span, of dimension $5$, and DiazModulus.quadratic_two_orbit_space_carries_two_by_three_configuration, resp. DiazModulus.quadratic_cyclic_space_carries_two_by_three_configuration, give the configuration $(1, P'(u)/Q'(u)) \otimes y$.

Novelty. Not found in the sources read: the “only if” half, the classification. The configurations themselves (the “if” half) give, at a candidate, exclusions that are substitutions in Roy's strong six exponentials theorem (Roy 1992, Cor. 2). Applied to the conjugation hulls of rational $z$, it gives the five-dimensional case of Theorem F (DiazModulus.rational_conj_hull_two_by_three_configuration_iff). Theorem F asks the question of Theorem E of the mission's research note (the Laurent case, formal as DiazModulus.laurent_conj_hull_two_by_three_configuration_iff) for every $z$ rational in $u$, and builds on Theorem B (DiazModulus.circle_point_extension_two_by_three_configuration_iff, 10.6), which holds for every complex $z$. For Laurent $z$, that its answer gives back Theorem E's classes (P), (Q) and (R) was checked by exact computation on random data, not proved; Theorem E's explicit criteria do not follow from it.