Over the whole rational orbit the quantisation bound says exactly that Re u is non-zero

Statement. Let $u$ lie on a line $\Im u = k\pi$ with $k\in\mathbb{Z}\setminus\{0\}$ --- the real branch $\mathcal{D}_{\mathbb{R}}$ of the manuscript, where $e^{u}$ is real. Assert the conclusion of Diaz.order_quantisation at every point $qu$ of the rational orbit and for every admissible order $m$:

That whole family is equivalent to the single condition $\Re u\neq 0$.

No algebraicity hypothesis is used in either direction. So on this branch the quantisation bound, asserted everywhere it can be asserted, is exactly as strong as a hypothesis the configuration already grants, and no stronger.

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Perassi's, in his manuscript C. Perassi, Rigidity of logarithms with algebraic modulus — Around a conjecture of Diaz, unpublished manuscript, 15 August 2026, §Polar coordinates and the discreteness of the period.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 two halves are Theorem Quantisation of the real branch (thm:real-quantisation), which gives $u\bar u>\pi^{2}$ on $\mathcal{D}_{\mathbb{R}}$ and observes that $qu$ stays in $\mathcal{D}_{\mathbb{R}}$ only when $qk\in\mathbb{Z}$, and Corollary Effective lower bound in the torsion branch (cor:degree-quantisation), published here as Diaz.order_quantisation. What this node adds is the equivalence --- that the family over the whole orbit collapses to $\Re u\neq 0$, hence contributes nothing beyond it. That converse direction is possibly new and is in any case small; it is possibly folklore, and it was not found stated in the sources consulted.

Why it is on the board. It is a negative result and it is meant as one. Anyone reaching for the quantisation group to constrain the real branch --- the natural first move, since it is the only published family that speaks about exactly this configuration --- can read off here that the attempt cannot succeed, without re-running the search.

Proof. ($\Rightarrow$) If $\Re u = 0$, take $q = 1/k$; then $q u$ has real part $0$ and imaginary part $\pi$, the order-$1$ hypothesis holds by Diaz.exp_ratio_pow_eq_one_iff, and the conclusion reads $\pi^{2}<\pi^{2}$. ($\Leftarrow$) Given $\Re u\neq 0$, unwrap the hypothesis with Diaz.exp_ratio_pow_eq_one_iff to $m\,\Im(qu)=n\pi$; here $\Im(qu)=qk\pi\neq 0$ forces $n\neq 0$, so $|n|\ge 1$ and $\Im(qu)^{2}\ge\pi^{2}/m^{2}$, while $\Re(qu)^{2}=q^{2}(\Re u)^{2}>0$ supplies the strictness.