For a Laurent polynomial z in u/s with a five-dimensional conjugation hull, Theorem F's shapes D1, D2a, D2b hold exactly when Theorem E's classes (P), (Q), (R) do

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 $s \in \overline{\mathbb{Q}}$ with $s^2 = u\bar u$, and put $v = u/s$. Let $k \geq 0$, let $c_j \in \overline{\mathbb{Q}}$ for $j \in \mathbb{Z}$, put $z = \sum_{j=-k}^{k} c_jv^j$, and let $W = \mathrm{span}(1, u, \bar u, z, \bar z)$, the conjugation hull of $z$. Suppose $\dim W = 5$. Then $W$ has one of Theorem F's shapes (D1), (D2a), (D2b) if and only if it is in one of Theorem E's classes (P), (Q), (R).

Theorem F's shapes:

• (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 $P', Q', S_1, S_2$.

Theorem E's classes:

• (P) $\alpha u^2 + \beta u^3 \in W$ for some $\alpha, \beta \in \overline{\mathbb{Q}}$, not both zero;
• (Q) $W = \mathrm{span}(1, u, \bar u, \bar v^2 + ev^2, \beta\bar v^3 + \zeta v^2 + ev^3)$ for some $e, \beta, \zeta \in \overline{\mathbb{Q}}$ with $e\bar e = 1$, $\beta\bar\beta = 1$ and $\zeta = -\beta e^2\bar\zeta$;
• (R) $W = \mathrm{span}(1, u, \bar u, g, \bar g)$ with $g = G(v) + \kappa\,\overline{vG(v)}$, for some $\kappa \in \overline{\mathbb{Q}}$ with $\kappa\bar\kappa = 1$ and some $G \in \overline{\mathbb{Q}}[X]$ with $G(0) = 0$ and $\deg G \geq 2$.

Theorem E (DiazModulus.laurent_conj_hull_two_by_three_configuration_iff) and Theorem F (DiazModulus.rational_conj_hull_two_by_three_configuration_iff) both say when $W$ carries a $2 \times 3$ configuration; this node says that, for Laurent $z$, their criteria agree.

Proof. Put $\rho = u\bar u = s^2$, which is algebraic as $s$ is. Since $s \in \overline{\mathbb{Q}}$, $v = u/s$ lies in the field $\overline{\mathbb{Q}}(u)$, and so do its integer powers and $z$. Theorem E (DiazModulus.laurent_conj_hull_two_by_three_configuration_iff, 10.98) says that $W$ carries a $2 \times 3$ configuration if and only if $\dim W = 5$ and $W$ is of class (P), (Q) or (R). Theorem F (DiazModulus.rational_conj_hull_two_by_three_configuration_iff, 10.146), which applies because $\rho$ is algebraic and $z \in \overline{\mathbb{Q}}(u)$, says that $W$ carries one if and only if either $\dim W = 4$ and $z \in \mathrm{span}(1, u, \bar u, (u - a)^{-1})$ for some $a \in \overline{\mathbb{Q}}$ with $a\bar a = \rho$, or $\dim W = 5$ and $W$ has shape (D1), (D2a) or (D2b). As $\dim W = 5$, the dimension-four alternative is excluded. So (D1), (D2a) or (D2b) gives a configuration by Theorem F, hence (P), (Q) or (R) by Theorem E; and (P), (Q) or (R) gives a configuration by Theorem E, hence (D1), (D2a) or (D2b) by Theorem F. The proof uses Theorem E: it does not derive Theorem E from Theorem F.

Novelty. Not found in the sources read; routine. A corollary of two equivalences: Theorem E (DiazModulus.laurent_conj_hull_two_by_three_configuration_iff, 10.98) and Theorem F (DiazModulus.rational_conj_hull_two_by_three_configuration_iff, 10.146) both say when the hull carries a $2 \times 3$ configuration, and Theorem F asks Theorem E's question for every $z$ rational in $u$. That their criteria agree on Laurent hulls was earlier checked by exact computation only; here it is proved. The proof goes through Theorem E: it does not derive Theorem E from Theorem F, and Theorem E's explicit criteria still do not follow from Theorem F. The explicit maps from the classes to the shapes are DiazModulus.laurent_polynomial_class_conj_hull_two_orbit_form ((P) to (D2a)), DiazModulus.laurent_involution_class_space_cyclic_form ((Q) to (D2b)) and DiazModulus.laurent_real_pencil_class_space_u_multiplier_triple ((R) to (D1)).