Transcendence of the modulus of a generic conjugate pair of logarithms

Let $u \in \mathbb{C}$ be such that both $e^{u}$ and $e^{\overline{u}}$ are algebraic, so that $u$ and $\overline{u}$ are simultaneously logarithms of algebraic numbers. Assume the pair is generic: neither $\mathrm{Re}\,u$ nor $\mathrm{Im}\,u$ is algebraic, and neither is their ratio $\mathrm{Re}\,u/\mathrm{Im}\,u$. Then

is transcendental.

The content is a statement about a quadratic relation between two logarithms. Writing $\lambda_1 = u$ and $\lambda_2 = \overline{u}$, the assertion is that the product $\lambda_1 \lambda_2$ escapes $\overline{\mathbb{Q}}$ whenever the pair is generic — and products of logarithms are exactly what the linear theory of logarithms, Baker's theorem included, does not reach.

Relation to the four exponentials circle. The statement follows from the strong four exponentials conjecture: applying it to $x = (1, \lambda_1)$ and $y = (1, \lambda_2)$ places all four products $1$, $\lambda_2$, $\lambda_1$, $\lambda_1\lambda_2$ in $\widetilde{\mathcal{L}}$ and forces $u \in \overline{\mathbb{Q}}$. It is, however, strictly weaker than that consequence: the three transcendence hypotheses above together imply $u \notin \overline{\mathbb{Q}}$, while the converse fails, since $u \notin \overline{\mathbb{Q}}$ leaves open the case of an algebraic ratio. The interest of isolating it is that the whole modulus conjecture currently rests on the strong four exponentials conjecture, and this node asks for strictly less.

Formalization note. The three genericity hypotheses are stated over $\mathbb{Q}$; since $\overline{\mathbb{Q}} \cap \mathbb{R}$ is what is meant, and algebraicity over $\mathbb{Q}$ and over $\overline{\mathbb{Q}}$ agree for these quantities, no generality is lost. ‖u‖ is coerced through ℝ into ℂ to match the ambient conventions of the mission's other nodes.