The multiplier set of a point of one conjugate plane is exactly the opposite plane

Source. Carlo Perassi's manuscript C. Perassi, Rigidity of logarithms with algebraic modulus — Around a conjecture of Diaz, unpublished manuscript, 15 August 2026, statement 57 of 57, Theorem thm:padic-multiplier (p-adic multiplier bound and conjugate planes) of Appendix D, A p-adic outlook (source line 2793) — its second half, the identification of the multiplier set.

Statement, as formalised. Let F be a field, A ⊆ F a subfield, and V ⊆ F a set containing 1, closed under addition and under multiplication by A. Let u, v ∈ V with u v ∈ A and v ∉ A, and let x = a + b u with a, b ∈ A. Assume the multiplier bound

any three elements of M_x = {y ∈ V : x y ∈ V} are A-linearly dependent.

Then M_x = A ⊕ A v.

Why this is the manuscript's statement. There F = ℂ_p, A = ℚ̄, V = Λ_p = ℚ̄ + span_ℚ̄ ℒ_p with ℒ_p = {log_p α : α ∈ ℚ̄^×} the Iwasawa logarithms of algebraic numbers, u ∈ 𝒟_p a point of the p-adic Diaz locus and v = σ u its image under the Galois involution of the fixed quadratic extension K/ℚ_p; u v = N(u) ∈ ℚ̄ is the norm, the ultrametric replacement for the archimedean modulus. U_+ = ℚ̄ ⊕ ℚ̄ u and U_- = ℚ̄ ⊕ ℚ̄ σu are the two conjugate planes, and the claim is M_x = U_- for every x ∈ U_+ ∖ ℚ̄.

The involution itself never enters the argument: only that v is a second element of V with u v ∈ A. So σ is dropped from the statement and v left free. Similarly Λ_p appears only through the three closure properties actually used, and ℂ_p — which cannot be built here without a permanent definition node — only as an arbitrary field. Nothing is lost: the intended instance satisfies every hypothesis.

What is a hypothesis and why. Two inputs are cited rather than proved.

• The multiplier bound dim_A M_x ≤ 2, which in the manuscript comes from the p-adic strong six exponentials theorem of Waldschmidt and Roy (Corollary 2.2.2 of Maksoud). Its elementary half is the separate node Diaz.rank_one_six_exponentials; the transcendence half is unavailable, so the bound is the hypothesis hbound, stated concretely as "any three elements of M_x are A-dependent".
• v ∉ A, which in the manuscript is Mahler's p-adic Hermite–Lindemann theorem ℒ_p ∩ ℚ̄ = {0}, and is what makes dim_A U_- = 2 rather than 1.

What is proved is the assembly: the inclusion U_- ⊆ M_x is the identity (a + bu)(C + Dv) = (aC + bD·uv) + (bC)u + (aD)v, which the closure properties keep inside V; the reverse inclusion applies the bound to the triple 1, v, y and uses v ∉ A to force the coefficient of y to be non-zero. Over ℂ with v = conj u the same identity is already on this mission as Diaz.conj_planes_mul, and the independence of 1, u as Diaz.conj_planes_inter; this node is the ultrametric form of both, with the multiplier bound added so that the inclusion becomes an equality.

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. The manuscript's own remark rem:padic-prior records that no p-adic analogue of Diaz's question was found in the sources consulted, and calls that a negative search result rather than a proof of novelty.