For a candidate u and a logarithm μ outside ℚu ∪ ℚū, the numbers u, μ and e^{uū/μ} generate transcendence degree at least two

Let $u$ be a candidate for Diaz's conjecture and $\mu$ a logarithm of an algebraic number with $\mu \notin \mathbb{Q}u \cup \mathbb{Q}\bar u$. Then $u, \mu, e^{u\bar u/\mu}$ generate a $\mathbb{Q}$-algebra of transcendence degree at least $2$.

This is Theorem 3.9 of C. Perassi's unpublished manuscript on Diaz's modulus conjecture. When $\mu$ is algebraic over $\mathbb{Q}(u)$ it gives the transcendence of $e^{|u|^2/\mu}$, which is DiazModulus.candidate_norm_div_log_not_log. On the two excluded rays the transcendence degree is one. It concerns candidates, so it is vacuous if Diaz's conjecture holds.

Proof. Apply Waldschmidt's theorem (DiazModulus.two_algebraically_independent_of_exp_column) to $x = (u, \mu)$ and $y = (\bar u/\mu, 1)$. Both pairs are linearly independent over $\mathbb{Q}$ because $\mu \notin \mathbb{Q}u \cup \mathbb{Q}\bar u$, and the column $y_2 = 1$ gives the algebraic numbers $e^u$ and $e^\mu$. So two of $u, \mu, \bar u/\mu, 1, e^{u\bar u/\mu}, e^u, e^{\bar u}, e^\mu$ are algebraically independent. All eight are algebraic over $\mathbb{Q}(u, \mu, e^{u\bar u/\mu})$, since $\bar u = |u|^2/u$ with $|u|^2$ algebraic.

Novelty. Not asserted. It is Waldschmidt's Corollary 4 (1973, p. 192) at $(u, \bar u, \mu)$; see also Waldschmidt, LN 402, Exercise 7.4.d (p. 223), and Roy and Waldschmidt (1997, Cor. 1.2(c), p. 757).