For u algebraically independent of π on a circle of algebraic radius, no 2×2 configuration with Q̄-independent pairs lies over Q̄u + Q̄ū + Q̄iπ

Algebraic coefficients do not help without a constant.

Let $u \neq 0$ with $|u|^{2}$ algebraic, and suppose that $u$ and $i\pi$ are algebraically independent over $\overline{\mathbb{Q}}$. Then there are no $x_1, x_2$ and $y_1, y_2$, each pair linearly independent over $\overline{\mathbb{Q}}$, with all four products $x_i y_j$ in the $\overline{\mathbb{Q}}$-span of $u$, $\bar u$, $i\pi$.

So the homogeneous version of the strong four exponentials conjecture, with $\overline{\mathbb{Q}}$ in place of $\mathbb{Q}$ but no constant term, has no instance on the logarithms a generic candidate certifies. With rational coefficients this is DiazModulus.generic_conj_pair_four_exp_barrier. The configuration that does detect a candidate, $\begin{pmatrix}1 & \bar u\\ u & \rho\end{pmatrix}$, needs the constant, and DiazModulus.generic_period_never_enters shows that $i\pi$ cannot enter it. That homogeneity is what separates the provable from the open is classical; see the remark following Théorème 0.2 in Roy–Waldschmidt (1997), for quadrics in transcendence degree one.

Novelty. None claimed.