The four exponentials conjecture has no instance on the logarithms certified by a generic candidate

A barrier: the four exponentials conjecture sees nothing homogeneous.

Let $u \neq 0$ with $|u|^{2}$ algebraic, algebraically independent of $i\pi$ over $\overline{\mathbb{Q}}$. Let $M$ be a $2\times2$ matrix whose entries are rational linear combinations of $u$, $\bar u$ and $i\pi$. If $\det M = 0$, then the rows of $M$ are $\mathbb{Q}$-linearly dependent or its columns are.

For a candidate $u$ of Diaz's conjecture, the numbers $u$, $\bar u$ and $i\pi$ are the logarithms of algebraic numbers that the candidate certifies. The statement therefore says that no instance of the four exponentials conjecture ($\mathbb{Q}$-independent rows and columns, products in $\mathcal{L}$) can be built from them: on this data the conjecture is a theorem, by pure algebra. What does detect a candidate is a configuration with a constant entry, as in DiazModulus.diaz_of_sfe. For candidates algebraic over $\mathbb{Q}(\pi)$, see DiazModulus.candidate_nongeneric_four_exp_barrier. The statement does not use $e^{u}$.

Novelty. Not asserted.