At a candidate u, under Roy's strong six exponentials theorem, a rational function z of u whose conjugation hull meets Theorem F's criterion is not in ℒ̃

Here $\mathcal{L}$ is the set of logarithms of algebraic numbers (the $z \in \mathbb{C}$ with $e^z$ algebraic), and $\widetilde{\mathcal{L}}$ is the $\overline{\mathbb{Q}}$-vector space spanned by $1$ and $\mathcal{L}$; spans and linear independence are over $\overline{\mathbb{Q}}$, the algebraic numbers. Roy's strong six exponentials theorem is carried as the hypothesis hSSE: if $x_1, x_2 \in \mathbb{C}$ are linearly independent and so are $y_1, y_2, y_3 \in \mathbb{C}$, 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 $z$ lie in $\overline{\mathbb{Q}}(u)$, the subfield of $\mathbb{C}$ generated by $\overline{\mathbb{Q}}$ and $u$, and let $W_z = \mathrm{span}(1, u, \bar u, z, \bar z)$, the conjugation hull of $z$. Suppose $W_z$ meets Theorem F's criterion, that is, one of the following holds:

• $\dim W_z = 4$, and $z \in \mathrm{span}(1, u, \bar u, (u - a)^{-1})$ for some $a \in \overline{\mathbb{Q}}$ with $a\bar a = u\bar u$;
• $\dim W_z = 5$, and $W_z$ satisfies (D1), (D2a) or (D2b) below.

Here, with $W = W_z$:

• (D1) there are linearly independent $t_0, t_1, t_2 \in \mathbb{C}$ with $t_j \in W$ and $ut_j \in W$ for every $j$;
• (D2a, two-orbit) $W = \mathrm{span}(1, u, u^{-1}, Q'(u)S_1(u)/(uP'(u)), P'(u)S_2(u)/(uQ'(u)))$ for some $P', Q', S_1, S_2 \in \overline{\mathbb{Q}}[X]$ with $P', Q'$ coprime, $\max(\deg P', \deg Q') = 2$ and $\deg S_1, \deg S_2 \leq 2$;
• (D2b, cyclic) $W = \mathrm{span}(1, u, u^{-1}, Q'(u)S_1(u)/(uP'(u)), Q'(u)(Q'(u)S_1(u) + P'(u)S_2(u))/(uP'(u)^2))$ for some such $P', Q', S_1, S_2$.

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

Proof. By Hermite–Lindemann (DiazModulus.hermite_lindemann_holds, 3.2), $u \notin \overline{\mathbb{Q}}$, since $u \neq 0$ and $e^u$ is algebraic; and $u\bar u = |u|^2$ is algebraic, as $|u|$ is. So Theorem F (DiazModulus.rational_conj_hull_two_by_three_configuration_iff, 10.146) applies to $z \in \overline{\mathbb{Q}}(u)$, and its “if” direction puts a $2 \times 3$ configuration in $W_z$. If $z \in \widetilde{\mathcal{L}}$, then $W_z \subseteq \widetilde{\mathcal{L}}$: $1 \in \widetilde{\mathcal{L}}$, $u \in \mathcal{L}$ 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. Not found in the sources read as a family; routine. Its content is the “if” direction of Theorem F (DiazModulus.rational_conj_hull_two_by_three_configuration_iff): with a configuration in $W_z$ as the hypothesis instead of Theorem F's criterion, the conclusion holds for every $z \in \mathbb{C}$, and Theorem F is not needed. Each instance is a substitution in Roy's strong six exponentials theorem (Roy 1992, Cor. 2), as in DiazModulus.candidate_laurent_conj_hull_class_not_mem_logAlgTilde (10.106) for Theorem E's classes. Special cases are printed configurations: in dimension 4, Diaz's Théorème 7(1) (2007) at $v = 1/(u - a)$, with the exclusion in its Corollaire 5(4); for polynomials of degree 2 and 3 in $u$, Théorème 7(1); and Theorem E's class (Q) is a substitution in Diaz's Théorème 2 (2004) and Théorème 4(1) (2007). It concerns candidates, so it is vacuous if Diaz's conjecture holds.