A conjugation-aligned transcendental admits no algebraic multiple of algebraic norm

Source. Carlo Perassi's manuscript C. Perassi, Rigidity of logarithms with algebraic modulus — Around a conjecture of Diaz, unpublished manuscript, 15 August 2026, statement 51 of 57, Theorem thm:elliptic-axis of Appendix C, An elliptic outlook (source line 2533), part (c): in the complex multiplication case, the elliptic Diaz locus contains no elliptic logarithm of a torsion point.

Statement, as formalised. Let ω ∈ ℂ be non-algebraic with conj ω = λ ω for some algebraic λ, and let α be a non-zero algebraic number. Then (α ω) * conj (α ω) is not algebraic.

Why this is the manuscript's statement. The manuscript's torsion logarithms are the elements of $\Lambda \otimes \mathbb{Q}$, which in the complex-multiplication case is $k\omega$ for a period $\omega$; reality of the invariants $g_2,g_3$ gives $\bar\Lambda = \Lambda$ and hence $\bar\omega = \lambda\omega$ with $\lambda \in k$, and Schneider's theorem says a non-zero period is transcendental. The conclusion is that $u = \alpha\omega$ with $\alpha \in k^\times$ never satisfies $u\bar u \in \bar{\mathbb{Q}}$. Substituting the two inputs — "$\omega$ transcendental" and "$k \subseteq \bar{\mathbb{Q}}$" — for the objects they constrain gives exactly the formal statement, with λ and α arbitrary algebraic numbers rather than elements of the endomorphism field. As with part (b), the substitution loses nothing and needs no definition of the period lattice or of the elliptic logarithm set.

The proof. α ω is not algebraic (else ω = (αω)/α would be), and conj (α ω) = (conj α · λ / α) · (α ω) with the multiplier algebraic; the alignment lemma (Diaz.elliptic_axis_alignment, statement 51 (a),(b)) then rules out an algebraic norm.

Dropped from the successor manuscript. This statement is not in C. Perassi, Rigidity of logarithms with algebraic modulus — Around a conjecture of Diaz, unpublished manuscript, 15 August 2026, the later version of the same note. The reason is scope, and the manuscript says so itself (line 119): consequences of the same machinery that concern all logarithms rather than the Diaz locus, "and the transfers to elliptic and p-adic settings, are developed separately and are not needed here." The whole of the manuscript's Appendix C (An elliptic outlook) and Appendix D (A p-adic outlook) were removed as blocks; the appendix between them, the conjugation-degree framework, was kept and promoted to a body section. Nothing was withdrawn as wrong: the manuscript still asserts that the rank inequality rank(M) > ½ r_str(M) "also governs the elliptic and p-adic appendices". The appendices were moved out, not retracted.

Novelty. No novelty is claimed, either for the mathematics or for the formalisation. The manuscript itself presents these appendices as transfers of a complex argument to another setting. Possibly known; not checked against the literature.