Adding the period to a point of the period plane returns to the locus only in two cases

Statement. Let $\Im u = k\pi$ with $k\in\mathbb{Z}\setminus\{0\}$ and $\lVert u\rVert^{2}\in\bar{\mathbb{Q}}$. For rationals $a,b,c$,

In the first case the point equals $(a+b)\bar u$, in the second $(a+b)u$. So on this branch the period $2\pi i$ contributes nothing new: adding any rational multiple of it to any point of the plane $\mathbb{Q}u\oplus\mathbb{Q}\bar u$ lands back on the locus only when the result is already a rational multiple of $u$ or of $\bar u$.

The reason this covers the whole space $V_u=\mathbb{Q}u+\mathbb{Q}\bar u+\mathbb{Q}\,2\pi i$ at once is that here $2\pi i=(u-\bar u)/k$ already lies in the plane spanned by $u$ and $\bar u$ --- which is exactly what fails off the torsion branch.

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Perassi's, in his manuscript C. Perassi, Rigidity of logarithms with algebraic modulus — Around a conjecture of Diaz, unpublished manuscript, 15 August 2026, §Polar coordinates and the discreteness of the period.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. This is part (a) of Theorem Period-plane classification (thm:period-plane) --- $V_u=\mathbb{Q}u\oplus\mathbb{Q}\bar u$ and $V_u\cap\mathcal{D}=\mathbb{Q}^{\times}u\,\dot\cup\,\mathbb{Q}^{\times}\bar u$ when $\Im u\in\pi\mathbb{Q}$ --- formalised in the range $\Im u\in\pi\mathbb{Z}$ and in the coordinates $(a,b,c)$ of that theorem's own proof. Of that theorem only the norm identity eq:pp-norm had been published, as Diaz.period_plane_norm. Nothing here is new.

Proof. Since $u-\bar u=2ik\pi$, the point $a u + b\bar u + 2\pi ci$ equals $(a+c/k)\,u+(b-c/k)\,\bar u$. Apply Diaz.plane_normSq_algebraic_iff to that pair of coefficients: the modulus is algebraic exactly when $a+c/k=0$ or $b-c/k=0$, that is $c=-ak$ or $c=bk$.