Separation: for subfields K ≤ F of ℂ, a K-space V₀ ⊆ F and w₁…w_m algebraically independent over F, every p×q configuration over K (p, q ≥ 2) in V₀ + Kw₁ + … + Kw_m lies in V₀

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 $K \subseteq F$ be subfields of $\mathbb{C}$, $V_0 \subseteq F$ a $K$-subspace, and $w_1, \dots, w_m$ algebraically independent over $F$. Every $p \times q$ configuration over $K$ with $p, q \ge 2$ in $V_0 + Kw_1 + \dots + Kw_m$ has all its products in $V_0$.

With $K = \overline{\mathbb{Q}}$, $F = \overline{\mathbb{Q}}(u)$ and $V_0 = \overline{\mathbb{Q}} + \overline{\mathbb{Q}}u + \overline{\mathbb{Q}}\bar u$ this contains the separation steps of DiazModulus.generic_period_never_enters ($w = i\pi$, $2 \times 2$) and of DiazModulus.generic_circle_point_no_two_by_three_configuration ($2 \times 3$).

Proof. Induction on $m$. For $m + 1$, let $F'$ be the field generated over $F$ by $w_1, \dots, w_m$; then $w_{m+1}$ is transcendental over $F'$, and $V_0' = V_0 + Kw_1 + \dots + Kw_m \subseteq F'$. Every entry $(i, j)$ lies in a $2 \times 2$ block (rows $i \neq i'$, columns $j \neq j'$, using $p, q \ge 2$), which is a $2 \times 2$ configuration in $V_0' + Kw_{m+1}$; DiazModulus.rank_one_config_separation_two puts $x_iy_j$ in $V_0'$, and the induction hypothesis puts it in $V_0$.

Novelty. Not found in the sources read. It generalises the library's separation steps, which are over $\overline{\mathbb{Q}}$ with $V_0 = H_0$, to any base field, any $V_0 \subseteq F$, any number of generic numbers and any $p, q \ge 2$.