For u, w₁…w_m algebraically independent over a subfield K of ℂ with uū ∈ K, every p×q configuration over K (p, q ≥ 2) in K + Ku + Kū + Kw₁ + … + Kw_m is 2×2 and lies in K + Ku + Kū

A $p \times q$ configuration over a field $K$ in a $K$-subspace $V \subseteq \mathbb{C}$ is a pair $x_1, \dots, x_p$ and $y_1, \dots, y_q$ of complex numbers, each family linearly independent over $K$, with every product $x_iy_j$ in $V$; it is the input of Roy's strong six exponentials theorem when $p = 2$, $q = 3$, $K = \overline{\mathbb{Q}}$ and $V = \widetilde{\mathcal{L}}$.

Let $K$ be a subfield of $\mathbb{C}$, let $u, w_1, \dots, w_m$ be algebraically independent over $K$, suppose $\rho = u\bar u \in K$, and put $H_0 = K + Ku + K\bar u$. Every $p \times q$ configuration over $K$ with $p, q \ge 2$ in $H_0 + Kw_1 + \dots + Kw_m$ has $p = q = 2$ and all its products in $H_0$.

For $K = \overline{\mathbb{Q}}$ this contains DiazModulus.generic_circle_point_no_two_by_three_configuration.

Proof. The algebraic independence of $u, w_1, \dots, w_m$ over $K$ makes $u$ transcendental over $K$ and $w_1, \dots, w_m$ algebraically independent over $K(u)$. Since $H_0 \subseteq K(u)$, DiazModulus.rank_one_config_separation puts every product in $H_0$, and DiazModulus.circle_point_config_card_le_four gives $p + q \le 4$, so $p = q = 2$.

Novelty. Not found in the sources read; it is short: separation (DiazModulus.rank_one_config_separation) plus DiazModulus.circle_point_config_card_le_four. It extends the library's $\overline{\mathbb{Q}}$ statement to any base field.