At most one pair ±t of real logarithms of algebraic numbers makes t² + π² rational

Let $t_0, t_1$ be real numbers such that $e^{t_0}$ and $e^{t_1}$ are algebraic. If $t_0^{2} + \pi^{2}$ and $t_1^{2} + \pi^{2}$ are both rational, then

The number $u = t + i\pi$ is a logarithm of the negative algebraic number $-e^{t}$, and $|u|^{2} = t^{2} + \pi^{2}$. Diaz's conjecture predicts that $t^{2} + \pi^{2}$ is transcendental for every such $t$. The theorem shows that it can fail at a rational value for at most one pair $\pm t$.

Proof. Put $u = t_0 + i\pi$ and $v = t_1 + i\pi$. Both are non-zero, $e^{u} = -e^{t_0}$ and $e^{v} = -e^{t_1}$ are algebraic, and $u\bar u$, $v\bar v$ are the two rationals. Both are algebraic over $\mathbb{Q}(\pi)$, because $t_k^{2} = (t_k^{2} + \pi^{2}) - \pi^{2}$ and $i^{2} = -1$. DiazModulus.nongeneric_rational_modulus_orbit gives $v = qu$ or $v = q\bar u$ with $q$ rational. The imaginary parts give $q = 1$ in the first case and $q = -1$ in the second, and the real parts then give $t_1 = t_0$ or $t_1 = -t_0$.

Novelty. Not found in the sources read (among them Diaz 2004 and 2007, Roy–Waldschmidt 1995 and 1997, and Waldschmidt 1973, 1974 and 2005). It follows in a few lines from the four exponentials theorem in transcendence degree one (M. Waldschmidt, Solution du huitième problème de Schneider, J. Number Theory 5 (1973), Corollaire 4; Theorem 1 of D. Roy and M. Waldschmidt, Proc. Japan Acad. 71 (1995)), which is DiazModulus.log_pair_rigid_of_trdeg_one on this site.