For u ∉ Q̄ with |u| = s algebraic and g = a₁/(u − a) + a₂/(u + a), |a| ≠ s, a₁ ≠ a₂: span{1, u, u⁻¹, g, ḡ} is conjugation-stable, spanned by a 2×3 configuration, invisible to Theorem B

Let $u \in \mathbb{C} \setminus \overline{\mathbb{Q}}$ and let $s \in \overline{\mathbb{Q}}$ be real with $s^2 = u\bar u$ (so $\bar u = s^2/u$). Let $a, a_1, a_2 \in \overline{\mathbb{Q}}^\times$ with $a\bar a \neq s^2$ and $a_1 \neq a_2$, put $g = a_1/(u - a) + a_2/(u + a)$, and let $W$ be the $\overline{\mathbb{Q}}$-span of $1, u, u^{-1}, g, \bar g$. Then $W$ is stable under conjugation, a $2 \times 3$ configuration's six products span it, and for no $w \in W$ outside $H_0 = \overline{\mathbb{Q}} + \overline{\mathbb{Q}}u + \overline{\mathbb{Q}}\bar u$ does $H_0 + \overline{\mathbb{Q}}w$ carry one.

Conjugation maps the orbit $\{a, -a\}$ of $u \mapsto -u$ to $\{s^2/\bar a, -s^2/\bar a\}$, a different orbit when $|a| \neq s$: this is the swapped case of the conjugation-stable two-orbit spaces, whose fixed case is DiazModulus.conj_stable_two_orbit_space.

Proof. Put $b = s^2/\bar a$. Since $\bar u = s^2/u$, $\bar u - \bar a = -\bar a(u - b)/u$ and $\bar u + \bar a = \bar a(u + b)/u$, so $\bar g = (\bar a_2 - \bar a_1)/\bar a + \gamma/(u - b) + \delta/(u + b)$ with $\gamma = -\bar a_1 b/\bar a$ and $\delta = -\bar a_2 b/\bar a$, both non-zero. As $1 \in W$, $W$ is also the span of $1, u, u^{-1}, g$ and $\gamma/(u - b) + \delta/(u + b)$: a two-orbit space for the orbits $\pm a$ and $\pm b$. These differ, since $a^2 = b^2$ would give $(a\bar a - s^2)(a\bar a + s^2) = 0$, and $a\bar a + s^2 = |a|^2 + |u|^2 > 0$. With $a_1 \neq a_2$, DiazModulus.two_orbit_configuration_invisible_to_four_dimensional_extensions gives the spanning configuration and the invisibility. Conjugation maps $1, u, u^{-1}, g, \bar g$ into $W$ ($\bar u = s^2u^{-1}$, $\overline{u^{-1}} = u/s^2$), and induction on the span finishes.

Novelty. The configuration 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 $v = u$, $w = ug$ (credited here on 16 October 2026). That no four-dimensional extension of $H_0$ inside $W$ carries a configuration was not found in the sources read; routine.