For u transcendental over a subfield K of ℂ and integers 4 ≤ k < l, span_K{u^s : s ∈ {0, ±1, ±k, ±l}} carries a 2×3 configuration iff l ∈ {k+1, k+2, 2k−1, 2k, 2k+1, 3k}

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 $\mathbb{C}$, $u$ transcendental over $K$, and $4 \le k < l$ integers. The span of $u^s$, $s \in \{0, \pm1, \pm k, \pm l\}$, carries a $2 \times 3$ configuration over $K$ if and only if $l \in \{k + 1, k + 2, 2k - 1, 2k, 2k + 1, 3k\}$.

So at a candidate, Roy's strong six exponentials theorem applied through this hull excludes exactly these pairs $u^k, u^l \in \widetilde{\mathcal{L}}$ (DiazModulus.candidate_power_pair_not_both_mem_logAlgTilde) and says nothing about the others.

Proof. Compose DiazModulus.laurent_hull_config_iff ($p = 2$, $q = 3$), DiazModulus.two_sumset_iff_difference_count and DiazModulus.power_pair_difference_count_iff.

Novelty. Not found in the sources read; it is short. The exclusions themselves are Diaz (2007) by substitution; what is added is that the list is complete for this hull.