The real-generic case is equivalent to one relation: t² + π² is transcendental

Statement. The left-hand side is, verbatim, the statement of the open node DiazModulus.diaz_of_exp_real_generic: the real-generic case of Diaz's conjecture, where $e^{u}$ is real and $\neq 1$ and neither $\Re u$ nor $\Im u$ vanishes. This node says it is equivalent to a single relation in one real variable:

Equivalently: the principal logarithm of a negative algebraic number of modulus $\neq 1$ has transcendental modulus, since $(\log b)^{2}+\pi^{2}=\lVert\operatorname{Log}(-b)\rVert^{2}$. In particular the smallest instance, "is $\sqrt{(\log 2)^{2}+\pi^{2}}$ algebraic?", is not a sample of the case --- the case has one real parameter $b$ and nothing else.

Two things are folded into this. First, the integer $k$ with $\Im u=k\pi$ carries no arithmetic: if $t^{2}+k^{2}\pi^{2}$ were algebraic then $t/|k|$ is again a non-zero real logarithm of an algebraic number --- $e^{t/|k|}$ is a real $|k|$-th root of $e^{t}$ --- and $(t/|k|)^{2}+\pi^{2}$ is algebraic too. Second, "$e^{u}$ real" is "$\Im u\in\pi\mathbb{Z}$", which is where $\pi^{2}$ enters.

This is an equivalence, not a decomposition. The right-hand side is the case restated, not a weaker statement; it reduces nothing and creates no research work of its own. It is published because the description is the useful object: it names what would have to be proved.

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Perassi's, in his manuscript C. Perassi, Rigidity of logarithms with algebraic modulus — Around a conjecture of Diaz, unpublished manuscript, 15 August 2026, §Polar coordinates and the discreteness of the period.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 normal form is the closing clause of Theorem Torsion dichotomy (thm:torsion-dichotomy) --- $u_{0}:=u/s=\ell+i\pi$, $u_{0}\overline{u_{0}}=\ell^{2}+\pi^{2}$ --- together with Corollary The torsion branch is one relation (cor:torsion-one-relation) and Theorem Polar normal form of the conjecture (thm:polar-form), all stated there for the larger torsion branch $\Im u\in\pi\mathbb{Q}$ and hence subsuming the integer case here. None of the three is on the board: Diaz.torsion_dichotomy publishes only the (ii)$\Leftrightarrow$(iii) equivalence. Nothing in the mathematics is new; what this node adds is only the machine-checked chain from the platform's own statement of the case to it, and that is small.

Where the difficulty sits. The manuscript's Remark Where this sits (rem:polar-scope) identifies the obstruction as Waldschmidt's own open question: the remark following Theorem 15.30 of Diophantine Approximation on Linear Algebraic Groups, that the homogeneous rational quadratic theorem is available in transcendence degree one and "it would be interesting to extend this statement to nonhomogeneous quadratic polynomials", the model case being the transcendence of $e^{\lambda^{2}}$, with $e^{\pi^{2}}$ still open --- cited there to p. 593. The board shows the same wall in miniature: Diaz.salem_quartic_relations settles the homogeneous form $At^{2}+Bts+Cs^{2}=0$ for $t$ real, $s$ purely imaginary and rational $A,B,C$; this node is the inhomogeneous instance $(A,B,C)=(1,0,-1)$ with right-hand side an algebraic number instead of $0$. Caveat: that citation is taken from the manuscript and was not verified against the book, which is not held locally.

Proof. No transcendence input is used; the content is the change of variables $u=t+ik\pi$ and a descent in $k$. Forward: given the case, the witness $u=t+i\pi$ has $\lVert u\rVert^{2}=t^{2}+\pi^{2}$, real exponential $-e^{t}$ which is algebraic and $\neq 1$, and non-zero real and imaginary parts. Backward: $e^{u}$ real forces $\sin(\Im u)=0$, so $\Im u=k\pi$ with $k\neq 0$; then $e^{\Re u}=\pm e^{u}$ is algebraic and $\lVert u\rVert^{2}=(\Re u)^{2}+k^{2}\pi^{2}$ is algebraic; put $s=\Re u/|k|$, whose exponential is algebraic by Diaz.exp_ratMul_isAlgebraic at the rational $1/|k|$, and $s^{2}+\pi^{2}=\lVert u\rVert^{2}/k^{2}$ is algebraic --- contradicting the one relation at $s$.