$G_u/(2r)$ is a rank-one projection: $\det G_u=0$, $\operatorname{tr}G_u=2r$, $P_u^2=P_u$

The real form of the candidate matrix is $2r$ times a rank-one projection.

Write $u = x + iy$ with $x, y$ real, let $r \neq 0$, and assume the candidate relation $u \bar u = r^2$. Put

Then $\det G_u = 0$, $\operatorname{tr} G_u = 2r$, and $P_u := G_u/(2r)$ is idempotent: $P_u^2 = P_u$.

Why. The hypothesis $u\bar u = r^2$ says exactly $x^2 + y^2 = r^2$, whence $\det G_u = r^2 - x^2 - y^2 = 0$; the trace is immediate; and a $2\times2$ matrix with zero determinant satisfies $G^2 = (\operatorname{tr}G)\,G$, so $P_u^2 = P_u$.

Role. This is the substance of the manuscript's Real projection normal form: the obstruction attached to a Diaz candidate can be presented as a genuine real rank-one orthogonal projection whose entries lie in the augmented logarithm space $\widetilde{\mathcal L}$ — because $x = (u+\bar u)/2$ and $iy = (u - \bar u)/2$ are logarithms and $r$ is algebraic. The manuscript then observes that the projection has no vanishing non-zero algebraic matrix coefficient and that its spectral slopes lie outside $\widetilde{\mathcal L}$, which is what turns "every real rank-one orthogonal projection over $\widetilde{\mathcal L}$ has an algebraic image or kernel line" into a sufficient target for Diaz's conjecture. The Lean statement isolates the projection property itself, which is unconditional.

Source: Carlo Perassi's private manuscript on Diaz's modulus conjecture, section The precise open boundary, corollary Real projection normal form. Elementary; no novelty is claimed.