Two candidates with a rational ratio of squared moduli: one is a rational multiple of the other or of its conjugate exactly when they are algebraically dependent

Let $u, v$ be candidates for Diaz's conjecture (DiazModulus.IsCandidate) with $|v|^2 = m\,|u|^2$ for some $m \in \mathbb{Q}$. Then $v \in \mathbb{Q}^\times u \cup \mathbb{Q}^\times\bar u$ if and only if $u$ and $v$ are algebraically dependent over $\mathbb{Q}$. So exactly one of two things holds: $v \in \mathbb{Q}^\times u \cup \mathbb{Q}^\times\bar u$, or $u$ and $v$ are algebraically independent.

This is Theorem 2.5 of C. Perassi's unpublished manuscript on Diaz's modulus conjecture, where it was derived from Roy and Waldschmidt (1997); here it follows from the four exponentials theorem in transcendence degree one. It concerns candidates, so it is vacuous if Diaz's conjecture holds.

Proof. If $v \in \mathbb{Q}^\times u \cup \mathbb{Q}^\times\bar u$, then $u$ and $v$ are dependent (Diaz.pair_dichotomy_exclusive). Conversely, if they are dependent, they generate an algebra of transcendence degree at most one ($u$ is transcendental by Hermite–Lindemann), and so do $u, v, \bar u, \bar v$, since $\bar u = |u|^2/u$ and $\bar v = |v|^2/v$ with algebraic numerators. Then DiazModulus.log_pair_rigid_of_trdeg_one gives $v \in \mathbb{Q}u \cup \mathbb{Q}\bar u$, with a non-zero factor since $v \neq 0$.

Novelty. Not asserted. It is Waldschmidt's Corollary 4 (1973, p. 192) at $(mu, \bar u, v)$.