For u ∉ Q̄ and real algebraic s with s² = uū: the two-orbit space with |a| = |b| = s and β/α, δ/γ real is stable under complex conjugation

Let $u \in \mathbb{C} \setminus \overline{\mathbb{Q}}$, let $a, b \in \overline{\mathbb{Q}}^\times$ with $a^2 \neq b^2$ and $\alpha, \beta, \gamma, \delta \in \overline{\mathbb{Q}}^\times$, put $g_1 = \alpha/(u - a) + \beta/(u + a)$, $g_2 = \gamma/(u - b) + \delta/(u + b)$, and let $W$ be the $\overline{\mathbb{Q}}$-span of $1, u, u^{-1}, g_1, g_2$. Let $s \in \overline{\mathbb{Q}}$ be real with $s^2 = u\bar u$, suppose $a\bar a = b\bar b = s^2$ and that $\beta/\alpha$ and $\delta/\gamma$ are real. Then $W$ is stable under complex conjugation.

With DiazModulus.two_orbit_configuration_invisible_to_four_dimensional_extensions this gives a family of conjugation-stable spaces, invisible to Theorem B, whose configuration factor $(1, u^2)$ has degree two.

Proof. Substitute $\bar u = s^2/u$, $\bar a = s^2/a$, $\bar b = s^2/b$ and $\bar\beta = \beta\bar\alpha/\alpha$, $\bar\delta = \delta\bar\gamma/\gamma$. Then $\bar g_1 = a(\bar\beta - \bar\alpha)/s^2 - (a^2\bar\alpha/(s^2\alpha))\,g_1$ and similarly for $g_2$, while $\bar u = s^2u^{-1}$ and $\overline{u^{-1}} = s^{-2}u$; conjugation of \overline{\mathbb{Q}}-scalars stays in \overline{\mathbb{Q}}, and induction on the span finishes.

Novelty. Not found in the sources read; it is routine. With it, the configuration of DiazModulus.two_orbit_configuration_invisible_to_four_dimensional_extensions is printed: rescaled by $u^{\mp 1}$, it is the configuration $(v, \bar v) \otimes (1, w, \bar w)$ of Diaz (2007, Th. 7(1)) at $v = u$, for any $w$ with $\overline{\mathbb{Q}} + \overline{\mathbb{Q}}w + \overline{\mathbb{Q}}\bar w = u\,(\overline{\mathbb{Q}}u^{-1} + \overline{\mathbb{Q}}g_1 + \overline{\mathbb{Q}}g_2)$ (credited here on 16 October 2026).