Balanced jets: $u^j\bar u^{\,k}\alpha$ is algebraic exactly when $j=k$

Balanced jets, and only balanced jets, are algebraic.

Let $K \subseteq \mathbb{C}$ be a subfield, $u \neq 0$ transcendental over $K$ with $u\bar u \in K$, and let $\alpha \in K$ be non-zero. Then for $j, k \geq 0$

Why. Since $\rho = u\bar u \in K$ we may write $\bar u = \rho/u$, so $u^{j}\bar u^{\,k}\alpha = (\rho^{k}\alpha)\, u^{\,j-k}$ with $\rho^k\alpha \in K^\times$. The claim reduces to $u^{\,j-k} \in K \iff j = k$, which is the sparsity statement for integer powers of a transcendental element.

Role. This is the arithmetic content of the manuscript's Balanced lattice jets. For a candidate $u$ with $\alpha = e^{u}$, the function $F(z,w) = \exp(uz + \bar u w)$ has

and this node says that this jet is algebraic exactly on the diagonal $j = k$ — so that $\Delta = \partial_z\partial_w$ acts as multiplication by $\rho$ and every balanced jet on $\mathbb{Z}^2$ stays in the fixed number field $\mathbb{Q}(\alpha, \bar\alpha, \rho)$. The Lean statement is the algebraic dichotomy itself, stated for an arbitrary subfield and without any differential calculus, so it holds unconditionally; the manuscript's use of it is the case $K = \overline{\mathbb{Q}}$, with the algebraic factor $\alpha^{m}\bar\alpha^{\,n}$ of the jet absorbed into the single parameter $\alpha$.

Source: Carlo Perassi's private manuscript on Diaz's modulus conjecture, section The precise open boundary, proposition Balanced lattice jets. No novelty is claimed.