At a candidate u, under Roy's strong six exponentials theorem, a polynomial in u and ū whose hull is in the polynomial, involution or real pencil class is not in ℒ̃

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: $u \neq 0$, with $|u|$ and $e^u$ algebraic), let $s \in \overline{\mathbb{Q}}$ with $s^2 = u\bar u$, and put $v = u/s$. Let $k \geq 0$, let $c_j \in \overline{\mathbb{Q}}$ for all $j \in \mathbb{Z}$, and put $z = \sum_{j=-k}^{k} c_jv^j$ and $W = \mathrm{span}_{\overline{\mathbb{Q}}}(1, u, \bar u, z, \bar z)$. If $W$ is in one of the classes

• (P) $\alpha u^2 + \beta u^3 \in W$ for some $\alpha, \beta \in \overline{\mathbb{Q}}$, not both zero;
• (Q) $W = \mathrm{span}_{\overline{\mathbb{Q}}}(1, u, \bar u, \bar v^2 + ev^2, \beta\bar v^3 + \zeta v^2 + ev^3)$ for some $e, \beta, \zeta \in \overline{\mathbb{Q}}$ with $e\bar e = 1$, $\beta\bar\beta = 1$ and $\zeta = -\beta e^2\bar\zeta$;
• (R) $W = \mathrm{span}_{\overline{\mathbb{Q}}}(1, u, \bar u, g, \bar g)$ with $g = G(v) + \kappa\,\overline{vG(v)}$, for some $\kappa \in \overline{\mathbb{Q}}$ with $\kappa\bar\kappa = 1$ and some $G \in \overline{\mathbb{Q}}[X]$ with $G(0) = 0$ and $\deg G \geq 2$,

then $z \notin \widetilde{\mathcal{L}}$.

Proof. By Hermite–Lindemann (DiazModulus.hermite_lindemann_holds), $u \notin \overline{\mathbb{Q}}$. The class forces $\dim W = 5$ (DiazModulus.laurent_conj_hull_class_finrank_eq_five), so Theorem E (DiazModulus.laurent_conj_hull_two_by_three_configuration_iff) puts a configuration in $W$. If $z \in \widetilde{\mathcal{L}}$, then $W \subseteq \widetilde{\mathcal{L}}$: $1 \in \widetilde{\mathcal{L}}$, $u \in \mathcal{L}$ since $e^u$ is algebraic, $\bar u \in \mathcal{L}$ (DiazModulus.logAlg_conj_stable) and $\bar z \in \widetilde{\mathcal{L}}$ (DiazModulus.logAlgTilde_conj_stable). The configuration then lies in $\widetilde{\mathcal{L}}$, against hSSE.

Novelty. Each instance is a substitution in Roy's strong six exponentials theorem (Roy 1992, Cor. 2); for class (Q) it is also one in Diaz (2004, Th. 2). Not found in the sources read as a family; routine. It concerns candidates, so it is vacuous if Diaz's conjecture holds.