Strong six exponentials configurations in the span of u^{±k}, u^{±1}, 1 exist exactly for k = 2 and k = 3

Let $u$ be transcendental and $k \geq 1$. There exist $x_1, x_2$, linearly independent over $\overline{\mathbb{Q}}$, and $y_1, y_2, y_3$, linearly independent over $\overline{\mathbb{Q}}$, with all six products $x_iy_j$ in

if and only if $k = 2$ or $k = 3$.

Why it matters for Diaz's conjecture. At a candidate $u$ ($|u|$ and $e^u$ algebraic) the space $\widetilde{\mathcal{L}}$ contains $1$, $u$ and $\bar u = |u|^2/u$, and if it contained $u^k$ it would contain $\bar u^k = |u|^{2k}u^{-k}$ too. Roy's strong six exponentials theorem refutes $u^k \in \widetilde{\mathcal{L}}$ through numbers built from $u$ alone exactly when a configuration as above exists. So it excludes $u^2$ (DiazModulus.candidate_multiplier_module) and $u^3$ (DiazModulus.candidate_cube_and_axis_multiple_not_mem_logAlgTilde), and by this route nothing further: not $u^4$, nor any higher power. Configurations that also use logarithms unrelated to $u$ are not covered.

Proof. For $k = 2$ take $x = (1, u)$, $y = (u^{-1}, 1, u)$; for $k = 3$ take $x = (u, u^{-1})$, $y = (1, u^{-2}, u^2)$ (Diaz's configuration for his Corollaire 5(2), up to the factor $|u|^2$). For $k = 1$ and $k \geq 4$, multiply by $u^k$: since $u$ is transcendental, the products become polynomials supported on $E = \{0, k-1, k, k+1, 2k\}$, and the rank-one relations become polynomial identities. The orders at $0$ of the three independent polynomials of the first row are three distinct exponents $t$ with $t$ and $t + \delta$ both in $E$, for a fixed $\delta \neq 0$. But for $k = 1$ and for $k \geq 4$ at most two such $t$ exist for every $\delta \neq 0$.

Novelty. Not found in the sources read (Diaz 1997, 2004, 2007; Waldschmidt's Variations, Further variations and The role of complex conjugation; Roy–Waldschmidt 1995, and 1997 §§0 and 7; Diophantine Approximation on Linear Algebraic Groups, §§11.5–11.6 and chapter 12). Diaz notes that his list of conjectures contains no general one for $\lambda^n$ (2007, p. 376), and that inside the strong six exponentials theorem one cannot hope to go very far (p. 390); this node is a precise instance of the second remark.