Diaz's conjecture (Qr2) in transcendence degree one: for logarithms ℓ₀, ℓ₁ off both axes, a real or purely imaginary ℓ₁/ℓ₀ is rational

Let $\ell_0, \ell_1$ be logarithms of algebraic numbers, neither of them real or purely imaginary, such that $\ell_0, \bar\ell_0, \ell_1, \bar\ell_1$ generate a $\mathbb{Q}$-algebra of transcendence degree at most $1$. If $\ell_1/\ell_0$ is real or purely imaginary, then $\ell_1/\ell_0 \in \mathbb{Q}$; in particular it is never purely imaginary.

This is Diaz's conjecture (Qr2) (2007, p. 376) under the hypothesis on the transcendence degree. Diaz states (Qr2) as a consequence of the four exponentials conjecture; here that conjecture is replaced by the theorem that holds in transcendence degree one.

Proof. Put $\lambda = \bar\ell_0$ and $\mu = \ell_1$, both non-zero logarithms of algebraic numbers. Then $\lambda\mu = |\ell_0|^2\,\ell_1/\ell_0$, so $\lambda\mu$ is real, or purely imaginary, exactly when $\ell_1/\ell_0$ is. In the real case, DiazModulus.log_mul_real_trichotomy_of_trdeg_one leaves only $\mu \in \mathbb{Q}\bar\lambda = \mathbb{Q}\ell_0$, since its other two cases put $\ell_0$ on an axis. In the imaginary case, DiazModulus.log_mul_imaginary_mixed_of_trdeg_one would put $\bar\ell_0$, hence $\ell_0$, on an axis.

Novelty. Not asserted. The statement is the transcendence-degree-one case of Diaz's conjecture (Qr2) (2007, p. 376), and the proof is Diaz's own argument for (Qr2) from the four exponentials conjecture (p. 377), with that conjecture replaced by the theorem of Waldschmidt (1973) and Brownawell (1974) in transcendence degree one, which Diaz quotes on p. 386. That combination was not found in the sources read.