Normalizing to the unit circle is not an arithmetic invariance

Source. Remark 1.4, Normalization is not arithmetic invariance, in Carlo Perassi's manuscript Rigidity of logarithms with algebraic modulus — around a conjecture of Diaz, Section 1, The conjecture and its elementary normal form: "Writing $u = rv$ with $|v| = 1$ changes the exponential condition to $e^{rv} \in \overline{\mathbb{Q}}$; it does not imply $v \in \mathcal{L}$. Hence one cannot assume $\rho = 1$ without loss of arithmetic information." The mathematics is Carlo Perassi's; no novelty is claimed. The witness below is the obvious one and the observation is possibly known — it has not been checked against the literature.

Statement (conditional). Assume Hermite–Lindemann in the form given as the hypothesis HL: for every non-zero $z \in \mathbb{C}$ algebraic over $\mathbb{Q}$, $e^{z}$ is transcendental over $\mathbb{Q}$. Then there exists $u \neq 0$ with $e^{u}$ algebraic — that is, $u \in \mathcal{L}$ — such that $e^{u/|u|}$ is transcendental; so $u/|u| \notin \mathcal{L}$.

Witness. $u = 2\pi i$. Then $e^{u} = 1$ is algebraic, $|u| = 2\pi$, and $u/|u| = i$, whose exponential is transcendental by HL applied to the non-zero algebraic number $i$.

Reading. The projection $u \mapsto u/|u|$ onto the unit circle destroys membership in $\mathcal{L}$, which is the whole arithmetic content of the hypothesis. So the parameter $\rho = u\bar u$ in the conic normal form $XY = \rho$ cannot be scaled away: the family of conics is not a single conic in disguise, and a proof of Diaz's conjecture may not assume $\rho = 1$.

On the hypothesis. Hermite–Lindemann is a theorem (Hermite 1873, Lindemann 1882) but is not in the platform's Mathlib at this revision — only the analytic half, NumberTheory.Transcendental.Lindemann.AnalyticalPart, is present — so it is carried as an explicit hypothesis, following the convention already used on this mission by DiazModulus.diaz_on_axes_of_hermite_lindemann. It can be discharged against DiazModulus.hermite_lindemann_holds. Only the instance at $z = i$ is used.