A rational multiple of a failure of the one-relation form is a failure only for the multiplier plus or minus one

Statement. Suppose $t_{1}\neq 0$, $e^{t_{1}}$ is algebraic, and both $t_{1}^{2}+\pi^{2}$ and $t_{2}^{2}+\pi^{2}$ are algebraic, with $t_{2}=r\,t_{1}$ for a rational $r$. Then $r=\pm 1$.

So the set where the relation "$t\neq 0$, $e^{t}$ algebraic $\Rightarrow$ $t^{2}+\pi^{2}$ transcendental" fails is rigid under $\mathbb{Q}^{\times}$: no rational multiple of a failure other than $\pm$ itself is a failure. Combined with Diaz.two_failures_give_algebraic_log_product, distinct failure classes $\{b,b^{-1}\}$ are $\mathbb{Q}$-linearly independent and each pair of them produces two real logarithms of algebraic numbers with non-zero algebraic product.

Source and attribution. All the mathematics of this section is Carlo Perassi's, in his manuscript C. Perassi, Rigidity of logarithms with algebraic modulus — Around a conjecture of Diaz, unpublished manuscript, 15 August 2026, §Polar coordinates and the discreteness of the period. No novelty is claimed. The relation whose failures are being constrained is Corollary The torsion branch is one relation (cor:torsion-one-relation), $(\log b)^{2}+\pi^{2} \in\bar{\mathbb{Q}}$ for some real algebraic $b>0$, $b\neq 1$. The rigidity statement itself is possibly new and is small --- it is one application of Hermite--Lindemann; it is possibly folklore, and it was not found stated in the sources consulted.

Proof. If $r\neq\pm 1$ then $1-r^{2}\neq 0$, and $(1-r^{2})t_{1}^{2}=(t_{1}^{2}+\pi^{2})-(t_{2}^{2}+\pi^{2})$ is algebraic; dividing by the non-zero rational $1-r^{2}$ makes $t_{1}^{2}$ algebraic, hence $t_{1}$ algebraic. But $t_{1}\neq 0$ is algebraic with $e^{t_{1}}$ algebraic, which DiazModulus.hermite_lindemann_holds forbids.