Two independent candidates have a ℚ-independent conjugate quadruple

Source. Carlo Perassi's manuscript C. Perassi, Rigidity of logarithms with algebraic modulus — Around a conjecture of Diaz, unpublished manuscript, 15 August 2026, statement 41 of 57, Proposition prop:diaz-quadruple-indep (Independence certificate on the Diaz locus, source line 2125), in Appendix A, Beyond the Diaz locus: products of logarithms.

What the manuscript claims. If u and v are points of the Diaz locus that are algebraically independent over ℚ, then u, conj u, v, conj v are linearly independent over ℚ. The point of the proposition is that the linear-independence certificate demanded by the independence-upgrade lemma is automatic on the Diaz locus once the pair dichotomy has been applied.

Statement formalised. Let K be a subfield of ℂ (the intended K is ℚ̄), let u, v be non-zero with u * conj u ∈ K and v * conj v ∈ K — this is the Diaz condition — and suppose u is transcendental over K and v is transcendental over the hull K(u). Then any rational relation a u + b conj u + c v + d conj v = 0 has a = b = c = d = 0.

The two transcendence hypotheses are what the manuscript's proof actually uses; they follow from algebraic independence of u and v over ℚ together with K algebraic. Stating them directly keeps the node free of a ℚ̄-algebraicity side condition.

The mechanism. Substituting conj u = (u conj u)/u and conj v = (v conj v)/v and clearing denominators turns the relation into (c u) v² + (a u² + b (u conj u)) v + (d (v conj v) u) = 0, a quadratic in v with coefficients in K(u). Transcendence of v over K(u) kills all three: c = 0 and d = 0 at once, and a u² + b (u conj u) = 0 with a ≠ 0 would make u a root of X² + (b/a)(u conj u) over K, contradicting transcendence of u. So a = 0, and then b = 0.

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 This statement is not in C. Perassi, Rigidity of logarithms with algebraic modulus — Around a conjecture of Diaz, unpublished manuscript, 15 August 2026, the later version of the same note.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 it is stated in the manuscript itself: its introduction says that "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 A (products of logarithms) and both of its outlook appendices (elliptic, p-adic) were removed as blocks; the intervening appendix, the conjugation-degree framework, was kept and promoted to a body section. Nothing here was withdrawn as wrong, and no statement in the dropped blocks was replaced by a corrected version. It is worth recording because the argument is unconditional and short, and because a statement that survives only in a superseded draft is the kind that gets lost.

Novelty. No novelty is claimed. The manuscript presents this as bookkeeping that makes an existing lemma of Waldschmidt applicable, not as a result. Possibly known; not checked against the literature.