A six-exponentials rank bound admits no rank-one witness

Source. Carlo Perassi's manuscript C. Perassi, Rigidity of logarithms with algebraic modulus — Around a conjecture of Diaz, unpublished manuscript, 15 August 2026: the common engine of statement 54, Theorem thm:elliptic-multipliers (Elliptic common multipliers, complex multiplication case, Appendix C, source line 2623), and of the first half of statement 57, Theorem thm:padic-multiplier (p-adic multiplier bound and conjugate planes, Appendix D, source line 2793).

Statement, as formalised. Let k ⊆ F be fields and V ⊆ F a set. Suppose the six-exponentials-type hypothesis

every 2 × 3 matrix over F whose entries lie in V, whose rows are k-linearly independent and whose columns are k-linearly independent, has F-linearly independent rows (i.e. rank two).

Then there are no a, b ∈ F with ![a,b] k-independent and no y : Fin 3 → F k-independent such that all six products a y_j, b y_j lie in V.

Why this is both statements. Each of the two theorems produces a matrix that is an outer product and derives a contradiction from the fact that a six-exponentials theorem forces its rank to be two.

• Statement 54: V = ℒ_E, k the endomorphism field, a = u, b = conj u for u in the elliptic Diaz locus, and y_1,y_2,y_3 three k-independent common multipliers, i.e. elements of K^E_u = {y : uy ∈ ℒ_E and conj u · y ∈ ℒ_E}. Rows are k-independent by statement 51 (b) — on this mission, Diaz.elliptic_axis_alignment. The conclusion dim_k K^E_u ≤ 2 is exactly the non-existence of such a triple. The rank-two input is the elliptic six exponentials corollary.
• Statement 57, first half: F = ℂ_p, V = Λ_p = ℚ̄ + span_ℚ̄ ℒ_p, k = ℚ̄, a = 1, b = x for x ∉ ℚ̄, and y_1,y_2,y_3 three ℚ̄-independent elements of M_x = {y ∈ Λ_p : xy ∈ Λ_p}. Rows ![1,x] are ℚ̄-independent precisely because x ∉ ℚ̄. The conclusion dim_ℚ̄ M_x ≤ 2 is again the non-existence of such a triple. Here the rank-two input is the p-adic strong six exponentials theorem of Waldschmidt and Roy, in the form of Corollary 2.2.2 of Maksoud.

Both deep inputs are unavailable — neither the elliptic nor the p-adic six exponentials theorem is in Mathlib — so the rank bound is carried as the explicit hypothesis hsix, and the citation boundary is visible in the statement. What is actually proved is the elementary half: the matrix (a y_j ; b y_j) has F-dependent rows, since b · (a y) − a · (b y) = 0 and (b, −a) ≠ 0, while its rows and columns are k-independent whenever ![a,b] and y are. It is that clash, not the transcendence, that both proofs turn on.

No new definition. ℒ_E and Λ_p appear only as the arbitrary set V; the elliptic apparatus and ℂ_p are absent.

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 rank-one observation is elementary linear algebra; possibly known — most likely folklore — not checked against the literature.