An algebraic norm forbids any algebraic alignment with the conjugate

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 Carlo Perassi's manuscript C. Perassi, Rigidity of logarithms with algebraic modulus — Around a conjecture of Diaz, unpublished manuscript, 15 August 2026, statement 51 of 57, Theorem thm:elliptic-axis (Elliptic axis and alignment lemma) of Appendix C, An elliptic outlook (source line 2533), parts (a) and (b).

Statement, as formalised. Let u ∈ ℂ be non-algebraic with u * conj u algebraic. Then conj u ≠ u and conj u ≠ -u, and more generally conj u ≠ γ u for every algebraic γ.

Why this is the manuscript's statement. In the manuscript u ranges over the elliptic Diaz locus

where $\mathcal{L}_E$ is the set of elliptic logarithms of algebraic points of a Weierstrass curve with real algebraic invariants, and $k = \operatorname{End}(E)\otimes\mathbb{Q}$ is either $\mathbb{Q}$ or an imaginary quadratic field. Part (a) says $\mathcal{D}_E \cap (\mathbb{R}\cup i\mathbb{R}) = \varnothing$; part (b) says $u$ and $\bar u$ are linearly independent over $k$.

The only property of $\mathcal{L}_E$ that either proof uses is Schneider's theorem in the form $\mathcal{L}_E \cap \bar{\mathbb{Q}} = \{0\}$, i.e. that a non-zero element of the locus is not algebraic; and the only property of $k$ used is $k \subseteq \bar{\mathbb{Q}}$. So the statement is recorded here with those two facts substituted for the objects they constrain: ¬ IsAlgebraic ℚ u in place of u ∈ ℒ_E \ {0}, and an arbitrary algebraic γ in place of γ ∈ k. Nothing about elliptic functions survives the substitution, and nothing is lost: the conclusion for arbitrary algebraic γ is formally stronger than the conclusion for γ ∈ k.

Being an identity of this shape, the same statement covers the archimedean and the ultrametric case: with a general involution in place of complex conjugation it is the p-adic axis lemma (statement 55), whose formalisation is already on the platform as Diaz.involution_alignment. That node fixes σ c = c and concludes c = ±1; this one lets the multiplier be any algebraic number and concludes there is none.

No new definition. ℒ_E, the endomorphism field and the elliptic Diaz locus are all absent from the formal statement. Introducing them would have meant a permanent definition node for the elliptic apparatus; the abstraction above carries the theorem without one.

The proof. If conj u = γ u with γ algebraic then γ ≠ 0 (else u = 0), and u * conj u = γ u², so u² = (u conj u)/γ is algebraic, hence so is u — contradiction. Parts (a) and (b) of the manuscript's (a),(b) are the cases γ = 1 and γ = -1.

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. The argument is three lines of field arithmetic over the transcendence input; possibly known, and in any case not checked against the literature.