For u ∉ Q̄ with uū = ρ algebraic and a ∈ Q̄ non-zero with aā = ρ², the conjugation-stable space Q̄ + Q̄u + Q̄ū + Q̄·u/(u² − a) carries no 2×3 configuration

Let $u \in \mathbb{C} \setminus \overline{\mathbb{Q}}$ with $\rho = u\bar u$ algebraic, and let $a \in \overline{\mathbb{Q}}$, $a \neq 0$, with $a\bar a = \rho^2$. Then there are no $x_1, x_2$, independent over $\overline{\mathbb{Q}}$, and $y_1, y_2, y_3$, independent over $\overline{\mathbb{Q}}$, with all six products $x_iy_j$ in $\overline{\mathbb{Q}} + \overline{\mathbb{Q}}u + \overline{\mathbb{Q}}\bar u + \overline{\mathbb{Q}}\,u/(u^2 - a)$.

Here $z = u/(u^2 - a)$ has $\bar z = -(\rho/\bar a)z$, so the space is stable under conjugation and a hypothesis $z \in \widetilde{\mathcal{L}}$ adds nothing to it; when $a\bar a \neq \rho^2$ the five-dimensional $H_0 + \overline{\mathbb{Q}}z + \overline{\mathbb{Q}}\bar z$ does carry a configuration (DiazModulus.circle_point_conjugate_pair_extension_carries_two_by_three_configuration). So $\bar z$ is what lets the strong six exponentials theorem see $u/(u^2 - a)$. This is also where the hypotheses of Diaz (2004, Th. 2) and Diaz (2007, Cor. 4(4), Th. 7(1)) fail for this family: $(1, U, \bar U)$ with $U = 1/(u^2 - a)$ is $\overline{\mathbb{Q}}$-linearly dependent.

Proof. Put $z_2 = u/(u^2 - a)^2$. Since $\rho^2/\bar a = a$, $\bar z_2 = (\rho/\bar a^2)\,u^3/(u^2 - a)^2$, and $w = u/(u^2 - a) = (\bar a^2/\rho)\bar z_2 - a z_2$ lies in $W_2 = H_0 + \overline{\mathbb{Q}}z_2 + \overline{\mathbb{Q}}\bar z_2$. Also $w \notin H_0$: if $w = c_0 + c_1u + c_2\bar u$, multiplying by $u(u^2 - a)$ gives a polynomial $c_1X^4 + c_0X^3 + (c_2\rho - c_1a - 1)X^2 - c_0aX - c_2\rho a$ vanishing at the transcendental $u$; its coefficients in degrees $4$, $0$ and $2$ give $c_1 = 0$, $c_2 = 0$ and $-1 = 0$. So DiazModulus.circle_point_conjugate_pair_configuration_invisible_to_four_dimensional_extensions, for $z_2$, says that $H_0 + \overline{\mathbb{Q}}w$ carries no configuration.

Novelty. Not found in the sources read; a short consequence of Theorem B (DiazModulus.circle_point_extension_two_by_three_configuration_iff).