The two conjugate planes meet only in the base field

Let $K \subseteq \mathbb{C}$ be a subfield, $u$ transcendental over $K$ with $u\bar u \in K$, and $A,B,C,D \in K$. If $A + Bu = C + D\bar u$, then $B = D = 0$ and $A = C$.

Equivalently: $(K \oplus Ku) \cap (K \oplus K\bar u) = K$, so every non-algebraic element of $U_+ \cup U_-$ has a unique chirality.

Where this sits. This is the first sentence of the proof of Corollary 3.4 (cor:chiral-multiplication, Chiral multiplication law) of the manuscript: "The intersection $U_+ \cap U_- = \bar{\mathbb{Q}}$ by the independence of $1, u, \bar u$, so every nonalgebraic element has a unique chirality."

Proof. The relation says $(A-C) + Bu + (-D)\bar u = 0$. Multiplying by $u$ and using $\bar u = \rho/u$ with $\rho = u\bar u \in K$ turns it into $Bu^{2} + (A-C)u + (-D\rho) = 0$, a polynomial relation for $u$ over $K$. Transcendence forces that polynomial to be zero, so $B = 0$ and $A = C$; the original relation then reads $D\bar u = 0$, and $\bar u \neq 0$.

This is a corollary of the independence of $1, u, \bar u$ over $K$, which is already on the mission as Diaz.indep_three; it is recorded separately because Corollary 3.4 is stated in terms of the two planes rather than of a linear relation, and because the chirality statement is what the product filter of Theorem 3.5 consumes.

What is deliberately not claimed. The equivalence of Corollary 3.4 — that $xy \in \widetilde{\mathcal{L}}$ exactly when $x$ and $y$ lie in opposite planes — needs the saturation of Theorem 3.3 and hence Roy's strong six exponentials theorem, and is not asserted.

Elementary; possibly known, not checked against the literature.

Source. Carlo Perassi, Rigidity of logarithms with algebraic modulus — Around a conjecture of Diaz (private manuscript, 15 August 2026). The mathematics is his; this node only records one step of it in Lean, and claims no novelty of its own.