Laurent hulls: for u transcendental over a subfield K of ℂ and a finite S ⊆ ℤ, span_K{u^s : s ∈ S} carries a p×q configuration iff S contains a sumset A + B with |A| = p, |B| = q

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 $\mathbb{C}$ and $u$ transcendental over $K$. For a set $S$ of integers write $V_S = \operatorname{span}_K\{u^s : s \in S\}$ (the Laurent hull of $S$). For a finite $S$ and $p, q \ge 1$, $V_S$ carries a $p \times q$ configuration over $K$ if and only if there are $A, B \subseteq \mathbb{Z}$ with $|A| = p$, $|B| = q$ and $A + B \subseteq S$.

At a candidate $u$, $u^k \in \widetilde{\mathcal{L}}$ brings $u^{-k} = \rho^{-k}\,\overline{u^k} \in \widetilde{\mathcal{L}}$, so hypotheses on powers of $u$ put Laurent hulls inside $\widetilde{\mathcal{L}}$; this criterion says exactly when Roy's strong six exponentials theorem can refute them through those hulls. The case $S = \{0, \pm 1, \pm k\}$ is DiazModulus.power_hull_strong_six_exp_configuration_iff.

Proof. If $A + B \subseteq S$, take $x_i = u^{a_i}$, $y_j = u^{b_j}$; distinct integer powers of $u$ are independent. Conversely, choose $N$ with $S + N \subseteq \mathbb{N}$ and polynomials $P_{ij}$ supported on $S + N$ with $P_{ij}(u) = u^Nx_iy_j$. The spans $V$ of the $P_{i1}$ and $W$ of the $P_{1j}$ have dimensions $p$ and $q$, hence (DiazModulus.polynomial_submodule_trailing_degrees_card) exactly $p$ and $q$ orders at $0$. For $f \in V$, $g \in W$, $fg = P_{11}h$ with $h$ a combination of the $P_{ij}$ (both sides agree at $u$), so $\operatorname{ord} f + \operatorname{ord} g - \operatorname{ord} P_{11} - N \in S$; shifting the order set of $V$ gives $A$, and the order set of $W$ is $B$.

Novelty. Not found in the sources read. The nearest printed results are the progression lemma of Fischler (2001, Lemma 6.1) and its repetition in Diaz (2007, Th. 7(2)); the library had the single family $S = \{0, \pm1, \pm k\}$.