Two distinct candidates whose difference is real or purely imaginary: a rational ratio of squared moduli, equal moduli, and v = ±ū are equivalent

Let $u \neq v$ be candidates for Diaz's conjecture with $v - u$ real or purely imaginary. The following are equivalent: (i) $|v|^2/|u|^2 \in \mathbb{Q}$; (ii) $|v| = |u|$; (iii) $v = \bar u$ if $v - u$ is purely imaginary, and $v = -\bar u$ if $v - u$ is real.

This is the equivalence in Theorem 2.3 of C. Perassi's unpublished manuscript on Diaz's modulus conjecture; its last assertion is Diaz.axis_triple_indep. It concerns candidates, so it is vacuous if Diaz's conjecture holds.

Proof. (ii) and (iii) are equivalent by plane geometry: equal real parts and equal moduli force opposite imaginary parts, and equal imaginary parts and equal moduli force opposite real parts. (ii) gives (i) with ratio $1$. For (i) ⇒ (ii), the condition on $v - u$ gives $v - \bar v = u - \bar u$ or $v + \bar v = u + \bar u$. With $\bar u = |u|^2/u$ and $\bar v = |v|^2/v$, it says that $v$ satisfies a quadratic equation over $\overline{\mathbb{Q}}(u)$, so $u$ and $v$ are algebraically dependent. By DiazModulus.candidate_pair_dichotomy, $v = cu$ or $v = c\bar u$ with $c \in \mathbb{Q}^\times$. Candidates lie on neither axis (Hermite–Lindemann), so $v = cu$ forces $c = 1$, which is excluded, and $v = c\bar u$ forces $c = \pm 1$, hence $|v| = |u|$.

Novelty. Not asserted. The step (i) ⇒ (ii) is Waldschmidt's Corollary 4 (1973, p. 192) at $(mu, \bar u, v)$; the rest is elementary.