At a candidate u, with Roy's strong six exponentials theorem as hypothesis: u/(u² − a) ∉ ℒ̃ for every non-zero algebraic a with aā ≠ |u|⁴, and u/(u² − a)² ∉ ℒ̃ when aā = |u|⁴

Here $\mathcal{L}$ is the set of logarithms of algebraic numbers, and $\widetilde{\mathcal{L}}$ is the $\overline{\mathbb{Q}}$-vector space spanned by $1$ and $\mathcal{L}$. Roy's strong six exponentials theorem is carried as the hypothesis hSSE: if $x_1, x_2$ are $\overline{\mathbb{Q}}$-linearly independent and so are $y_1, y_2, y_3$, then one of the six products $x_iy_j$ is not in $\widetilde{\mathcal{L}}$.

Let $u$ be a candidate for Diaz's conjecture (DiazModulus.IsCandidate), so $\rho = u\bar u$ is algebraic, and let $a \in \overline{\mathbb{Q}}$, $a \neq 0$. Then $u/(u^2 - a) \notin \widetilde{\mathcal{L}}$ if $a\bar a \neq \rho^2$, and $u/(u^2 - a)^2 \notin \widetilde{\mathcal{L}}$ if $a\bar a = \rho^2$. It concerns candidates, so it is vacuous if Diaz's conjecture holds.

Proof. $\rho = |u|^2$ is algebraic, $u \notin \overline{\mathbb{Q}}$ by Hermite–Lindemann (DiazModulus.hermite_lindemann_holds), and $u, \bar u \in \mathcal{L}$ (DiazModulus.logAlg_conj_stable). If $z \in \widetilde{\mathcal{L}}$, then $\bar z \in \widetilde{\mathcal{L}}$ (DiazModulus.logAlgTilde_conj_stable), so $H_0 + \overline{\mathbb{Q}}z + \overline{\mathbb{Q}}\bar z \subseteq \widetilde{\mathcal{L}}$, and the configuration of DiazModulus.circle_point_conjugate_pair_extension_carries_two_by_three_configuration contradicts hSSE.

Novelty. Not asserted. The first part is one substitution in Diaz (2004, Th. 2) at $x = (u, \bar u)$, $y = 1/(u^2 - a)$, where $\bar u\,y = (\rho z - \bar u)/a$; it is also Diaz (2007, Cor. 4(4) and Th. 7(1)). The second is Diaz (2007, Th. 6(3)) at $(\lambda, s) = (u, 1/(u^2 - a))$, after one line.