Under the strong six exponentials theorem, π² and π³, π² and 1/π, 1/π and 1/π² are pairwise not both 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 $\widetilde{\mathcal{L}}$.

1. $\pi^2$ and $\pi^3$ are not both in $\widetilde{\mathcal{L}}$.
2. $\pi^2$ and $1/\pi$ are not both in $\widetilde{\mathcal{L}}$.
3. $1/\pi$ and $1/\pi^2$ are not both in $\widetilde{\mathcal{L}}$.
4. If $e^{\gamma/(\pi i)}$ is algebraic for some algebraic $\gamma \neq 0$, then $e^{\beta\pi^2}$ and $e^{\beta/\pi^2}$ are transcendental for every algebraic $\beta \neq 0$.

Part 4 concerns the statement (S) of the Diaz mission, that $e^{\gamma/(\pi i)}$ is transcendental for every algebraic $\gamma \neq 0$ (DiazModulus.recip_pi_not_log, Open). Under the strong six exponentials theorem, an exception to (S) would make $e^{\pi^2}$ transcendental. Part 1 is the $\widetilde{\mathcal{L}}$ strengthening of the unconditional DiazModulus.exp_pi_sq_or_exp_i_pi_cube_transcendental.

Proof. Parts 1–3 are Conséquence 3 of Corollaire 2 (P) and the two "in particular" clauses of Corollaire 1 (PQ) 3) and 4) at $\lambda = i\pi$ (DiazModulus.diaz_2007_cor2_P_consequences, DiazModulus.diaz_2007_cor1_PQ). Here $i\pi \in \mathcal{L}$ and is transcendental by Hermite–Lindemann. For part 4, $\gamma/(\pi i) \in \mathcal{L}$ gives $1/\pi \in \widetilde{\mathcal{L}}$, and parts 2 and 3 exclude $\pi^2$ and $1/\pi^2$.

Novelty. None: Diaz (2007) draws parts 1–3 himself at $\lambda = i\pi$, as statements on $\exp(\pi^2)$, $\exp(\pi^3)$, $\exp(1/\pi)$ and $\exp(1/\pi^2)$ (pp. 380–381). Part 4 restates parts 2 and 3. The contribution of this node is the formal proof.