Theorem D: if u is transcendental over the field of ℒ̃ and uū is algebraic, then ℒ + ℚu + ℚū satisfies conjugation, Baker, Hermite–Lindemann, Gel'fond–Schneider, Roy, six and four exponentials

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 $\mathcal{L}$. Let $F$ be the subfield of $\mathbb{C}$ generated by $\widetilde{\mathcal{L}}$, that is, by $\overline{\mathbb{Q}}$ and $\mathcal{L}$. Let $u \in \mathbb{C}$ be transcendental over $F$, with $u\bar u$ algebraic, and put $\mathcal{L}_u = \mathrm{span}_{\mathbb{Q}}(\mathcal{L}) + \mathrm{span}_{\mathbb{Q}}(u, \bar u)$; as $\mathcal{L}$ is a $\mathbb{Q}$-vector space, this is $\mathcal{L} + \mathbb{Q}u + \mathbb{Q}\bar u$. Let $\widetilde{\mathcal{L}}_u = \mathrm{span}_{\overline{\mathbb{Q}}}(\{1\} \cup \mathcal{L}_u)$. Then $\mathcal{L} \subseteq \mathcal{L}_u$, $u \in \mathcal{L}_u$, $u \notin \overline{\mathbb{Q}}$, $u \neq 0$ and $|u|$ is algebraic, and $\mathcal{L}_u$ has the following properties:

• (conjugation) $\bar z \in \mathcal{L}_u$ for every $z \in \mathcal{L}_u$;
• (Baker) if $n \geq 0$, $l_1, \dots, l_n \in \mathcal{L}_u$ are linearly independent over $\mathbb{Q}$, and $b_0, b_1, \dots, b_n$ are algebraic numbers, not all zero, then $b_0 + \sum_{i=1}^{n} b_il_i \neq 0$;
• (Hermite–Lindemann) every algebraic $z \in \mathcal{L}_u$ is $0$;
• (Gel'fond–Schneider) if $l_0 \neq 0$ and $\beta l_0$ both lie in $\mathcal{L}_u$ and $\beta$ is algebraic, then $\beta \in \mathbb{Q}$;
• (Roy, relative to $\widetilde{\mathcal{L}}$) if no $x_1, x_2$, linearly independent over $\overline{\mathbb{Q}}$, and $y_1, y_2, y_3$, linearly independent over $\overline{\mathbb{Q}}$, have all six products $x_iy_j$ in $\widetilde{\mathcal{L}}$, then no such families have all six products in $\widetilde{\mathcal{L}}_u$;
• (six exponentials) no $x_1, x_2 \in \mathbb{C}$, linearly independent over $\mathbb{Q}$, and $y_1, y_2, y_3 \in \mathbb{C}$, linearly independent over $\mathbb{Q}$, have all six products $x_iy_j$ in $\mathcal{L}_u$;
• (four exponentials in transcendence degree one) if $\ell_{11}, \ell_{12}, \ell_{21}, \ell_{22} \in \mathcal{L}_u$ are non-zero, $\ell_{11}\ell_{22} = \ell_{12}\ell_{21}$, and the $\mathbb{Q}$-algebra they generate has transcendence degree at most $1$ over $\mathbb{Q}$, then there are $a, b \in \mathbb{Q}$, not both zero, with $a\ell_{11} + b\ell_{21} = a\ell_{12} + b\ell_{22} = 0$ (the rows of the matrix $(\ell_{ij})$ are linearly dependent over $\mathbb{Q}$), or with $a\ell_{11} + b\ell_{12} = a\ell_{21} + b\ell_{22} = 0$ (its columns are).

The Roy clause is an implication: Roy's strong six exponentials theorem for $\widetilde{\mathcal{L}}$ (Roy 1992, Cor. 2) is its hypothesis, not a hypothesis of the node, so the statement is unconditional. By DiazModulus.mem_logAlg_adjoin_circle_point_iff, $\widetilde{\mathcal{L}}_u = \widetilde{\mathcal{L}} + \overline{\mathbb{Q}}u + \overline{\mathbb{Q}}\bar u$. So $u$ is a non-zero element of $\mathcal{L}_u$ with algebraic modulus, and $e^u$ is not involved; DiazModulus.theorem_D_barrier draws the consequence.

Proof. $\mathcal{L} \subseteq \mathcal{L}_u$ and $u \in \mathcal{L}_u$ by construction; $u \notin \overline{\mathbb{Q}}$ because $\overline{\mathbb{Q}} \subseteq \widetilde{\mathcal{L}} \subseteq F$ and $u$ is transcendental over $F$, so $u \neq 0$; and $|u|$ is algebraic because $|u|^2 = u\bar u$. Conjugation follows from the membership criterion DiazModulus.mem_logAlg_adjoin_circle_point_iff: $\overline{c + pu + q\bar u} = \bar c + qu + p\bar u$ with $\bar c \in \mathcal{L}$ (DiazModulus.logAlg_conj_stable). Baker, Hermite–Lindemann and Gel'fond–Schneider are DiazModulus.logAlg_adjoin_circle_point_baker_linear_forms and DiazModulus.logAlg_adjoin_circle_point_hermite_lindemann_gelfond_schneider, six exponentials is DiazModulus.logAlg_adjoin_circle_point_six_exponentials, and four exponentials is DiazModulus.logAlg_adjoin_circle_point_four_exponentials_trdeg_one. For the Roy clause, the second half of DiazModulus.mem_logAlg_adjoin_circle_point_iff rewrites $\widetilde{\mathcal{L}}_u$ as $\widetilde{\mathcal{L}} + \overline{\mathbb{Q}}u + \overline{\mathbb{Q}}\bar u$, and DiazModulus.logAlgTilde_adjoin_circle_point_no_two_by_three_configuration turns Roy's theorem for $\widetilde{\mathcal{L}}$, the hypothesis of the clause, into Roy's theorem for that space.

Novelty. Not found in the sources read; routine. With $\mathcal{L}_u$ in place of $\mathcal{L}$, each clause restates a classical theorem: Baker (1966), Hermite–Lindemann, Gel'fond–Schneider (1934), the six exponentials theorem (Lang 1966, Ramachandra 1968), the four exponentials theorem in transcendence degree one (Waldschmidt 1973, Brownawell 1974), and, as an implication, Roy's strong six exponentials theorem (Roy 1992, Cor. 2).