A five-dimensional Laurent hull span(1, v, v̄, z, z̄) without v² P(v) in it carries a 2×3 configuration only in the involution or real pencil normal form

Let $v \in \mathbb{C} \setminus \overline{\mathbb{Q}}$ with $v\bar v = 1$. Let $k \geq 0$, let $c_j \in \overline{\mathbb{Q}}$ for all $j \in \mathbb{Z}$, and put $z = \sum_{j=-k}^{k} c_jv^j$ and $W = \mathrm{span}_{\overline{\mathbb{Q}}}(1, v, \bar v, z, \bar z)$. Suppose $\dim_{\overline{\mathbb{Q}}} W = 5$, $W$ contains no $v^2P(v)$ with $P \in \overline{\mathbb{Q}}[X]$ non-zero, and $W$ carries a $2 \times 3$ configuration (free $x_0, x_1$ and free $y_0, y_1, y_2$ over $\overline{\mathbb{Q}}$ with all six products $x_iy_j$ in the space). Then $W$ is in one of the two classes

• (Q) $W = \mathrm{span}_{\overline{\mathbb{Q}}}(1, v, \bar v, \bar v^2 + ev^2, \beta\bar v^3 + \zeta v^2 + ev^3)$ 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) $W = \mathrm{span}_{\overline{\mathbb{Q}}}(1, v, \bar v, g, \bar g)$ 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 $\deg G \geq 2$.

Proof. The shifted polynomial model (DiazModulus.laurent_conj_hull_shifted_polynomial_model) writes $W$ as the image of $H_N + M$ under $\Phi : F \mapsto F(v)/v^N$, with $M$ two-sided. A configuration $x \otimes y$ in $W$ factors (DiazModulus.config_shifted_coprime_polynomial_factors) through coprime $a_0, a_1$ and free $b_0, b_1, b_2$ with $a_ib_j \in H_N + M$ and $x_1a_0(v) = x_0a_1(v)$; as $x$ is free the ratio is not constant, DiazModulus.two_by_three_configuration_ratio_degree_le_two bounds its degree by two, DiazModulus.coprime_pencil_of_degree_le_two_normal_form leaves four pencils, and DiazModulus.laurent_window_quadratic_pencils_carry_no_three_factors rules out $(1, X^2 + \alpha X)$ and $(X + \beta, X^2 + \gamma X)$. For $(1, X)$, DiazModulus.laurent_window_degree_one_pencil_factor_space gives $W = \mathrm{span}(1, v, \bar v, g, vg)$ with $g = \Phi(G_0)$; conjugation-stability of $W$ and the factor-space property give $\bar g \in \mathrm{span}(1, v, vg)$, and DiazModulus.laurent_degree_one_conj_stable_space_real_pencil_normal_form gives class (R), as $v^2 \notin W$ excludes $\deg G \leq 1$. For $(X, X^2 + \beta)$, DiazModulus.laurent_window_involution_pencil_normal_form puts $M$ in the involution normal form, $\Phi$ carries it to $\mathrm{span}(1, v, \bar v, \bar v^2 + ev^2, \beta\bar v^3 + \zeta v^2 + ev^3)$, and DiazModulus.laurent_involution_space_conj_stable_constraints gives the constraints of class (Q).

Novelty. Not found in the sources read. It is most of the "only if" half of Theorem E (DiazModulus.laurent_conj_hull_two_by_three_configuration_iff); given its children, routine.