Theorem F: for z rational in u with |u|² algebraic, which conjugation hulls span(1, u, ū, z, z̄) carry a 2×3 configuration

Let $u \in \mathbb{C}$ be transcendental, that is, $u \notin \overline{\mathbb{Q}}$ (the algebraic numbers); spans and linear independence are over $\overline{\mathbb{Q}}$. Suppose $u\bar u$ is algebraic, let $z$ lie in $\overline{\mathbb{Q}}(u)$, the subfield of $\mathbb{C}$ generated by $\overline{\mathbb{Q}}$ and $u$, and let $W_z = \mathrm{span}(1, u, \bar u, z, \bar z)$, the conjugation hull of $z$. Then $W_z$ 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 $x_iy_j \in W_z$) if and only if one of the following holds:

• $\dim W_z = 4$, and $z \in \mathrm{span}(1, u, \bar u, (u - a)^{-1})$ for some $a \in \overline{\mathbb{Q}}$ with $a\bar a = u\bar u$;
• $\dim W_z = 5$, and $W_z$ satisfies (D1), (D2a) or (D2b) below.

Here, with $W = W_z$:

• (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. $W_z$ is spanned by five vectors and contains $\mathrm{span}(1, u, \bar u)$, of dimension $3$ (a relation $a + bu + c\bar u = 0$ times $u$ is $c\rho + au + bu^2 = 0$ with $\rho = u\bar u \neq 0$ algebraic), so $\dim W_z \in \{3, 4, 5\}$. In dimension $3$, $W_z = \mathrm{span}(1, u, \bar u)$, and a $2 \times 3$ configuration there contradicts DiazModulus.circle_point_config_card_le_four ($2 + 3 > 4$). In dimension $4$ the equivalence is DiazModulus.conj_hull_finrank_four_two_by_three_configuration_iff. In dimension $5$, DiazModulus.rational_conj_hull_le_adjoin gives $u^{-1} \in W_z$ and $W_z \subseteq \overline{\mathbb{Q}}(u)$, so DiazModulus.rational_space_two_by_three_configuration_iff applies to $W = W_z$. Both directions follow.

Novelty. Not found in the sources read: the “only if” half, the classification. 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. 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). For the conjugation-stable families they are printed configurations: Diaz's Théorème 7(1) (2007) and Théorème 2 (2004); in dimension 4, Théorème 7(1) at $v = 1/(u - a)$.