At a candidate u, with Roy's strong six exponentials theorem as hypothesis: for 4 ≤ k < l with l ∈ {k+1, k+2, 2k−1, 2k, 2k+1, 3k}, u^k and u^l are 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 $\widetilde{\mathcal{L}}$.

Let $u$ be a candidate for Diaz's conjecture (DiazModulus.IsCandidate) and $4 \le k < l$ with $l \in \{k + 1, k + 2, 2k - 1, 2k, 2k + 1, 3k\}$. Then $u^k$ and $u^l$ are not both in $\widetilde{\mathcal{L}}$. For example, no two consecutive powers $u^k, u^{k+1}$ with $k \ge 4$ are both in $\widetilde{\mathcal{L}}$. It concerns candidates, so it is vacuous if Diaz's conjecture holds.

Proof. $\rho = |u|^2$ is algebraic and $u$ is transcendental (DiazModulus.hermite_lindemann_holds), so powers of $u$ with distinct integer exponents are $\overline{\mathbb{Q}}$-independent. By conjugation (DiazModulus.logAlgTilde_conj_stable), $u^n \in \widetilde{\mathcal{L}}$ gives $u^{-n} = \rho^{-n}\,\overline{u^n} \in \widetilde{\mathcal{L}}$, so $u^s \in \widetilde{\mathcal{L}}$ for $s \in \{0, \pm1, \pm k, \pm l\}$. In each case $x = (u^a)_{a \in A}$, $y = (u^b)_{b \in B}$ is a $2 \times 3$ configuration in $\widetilde{\mathcal{L}}$, against hSSE: $l = k + 1$: $A = \{0, 1\}$, $B = \{-1, 0, k\}$; $l = k + 2$: $\{0, 2\}$, $\{-1, k, -k - 2\}$; $l = 2k - 1$: $\{0, k - 1\}$, $\{1, -k, k\}$; $l = 2k$: $\{0, k\}$, $\{0, -k, k\}$; $l = 2k + 1$: $\{0, k + 1\}$, $\{-1, -k, k\}$; $l = 3k$: $\{0, 2k\}$, $\{-k, k, -3k\}$.

Novelty. Not asserted: each case is one substitution in Diaz (2007): $l = k + 1$ in Cor. 2(P)(1); $l = k + 2$, $2k - 1$, $2k + 1$ in Th. 7(1); $l = 2k$ and $3k$ in Cor. 5(1) and 5(2), with $\lambda = u^k$ and $\bar u^k = \rho^k u^{-k}$. Diaz does not state them at a candidate.