Equal-modulus chords land in the norm-one group of the coefficient field

Source. Carlo Perassi's manuscript C. Perassi, Rigidity of logarithms with algebraic modulus — Around a conjecture of Diaz, unpublished manuscript, 15 August 2026, statement 53 of 57, Corollary cor:elliptic-chords (Elliptic chords: the norm-one orbit) of Appendix C, An elliptic outlook (source line 2591).

Statement, as formalised. Let k ⊆ ℂ be a conjugation-stable subfield, u ≠ 0, and v = γ u with γ ∈ k and ‖v‖ = ‖u‖. Then

1. γ conj γ = 1;
2. if k is fixed pointwise by conjugation (the non-CM case k = ℚ), then v = u or v = -u;
3. if k is not fixed pointwise (the CM case, k imaginary quadratic), then γ = δ / conj δ for some non-zero δ ∈ k — Hilbert's Theorem 90 for the quadratic extension k/k^{conj}.

Where this sits. The manuscript's corollary reads: for distinct u, v in the elliptic Diaz locus with |u| = |v|, one has |u−v| algebraic if and only if v = γu with γ ∈ k, N(γ) = 1. The "only if" is Theorem thm:elliptic-distance (statement 52), whose plane half is the separate node Diaz.elliptic_plane_rigidity and whose distance half rests on elliptic Baker; the corollary's own content, and all of it, is what happens to γ afterwards — and that is what this node records, with v = γ u as the hypothesis rather than the conclusion. Over ℚ the norm-one group is {±1} and one recovers the four-point orbit {±u, ±conj u} of the complex case (on this mission: Diaz.orbit_of_candidate); over an imaginary quadratic field, Hilbert 90 exhibits k^{(1)} as {δ/conj δ}, which is infinite.

What is not formalised. The manuscript closes the corollary with an analytic remark: k^{(1)} being an infinite subgroup of the unit circle, it is dense, so a single complex-multiplication candidate forces a dense set of candidates on its own circle — "elliptic Diaz is empty or dense on each centered circle". Neither the infinitude of k^{(1)} nor the density of an infinite subgroup of the circle is formalised here; this node stops at the algebraic dichotomy. That remark is unaffected by anything stated above and is not claimed.

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 This statement is not in C. Perassi, Rigidity of logarithms with algebraic modulus — Around a conjecture of Diaz, unpublished manuscript, 15 August 2026, the later version of the same note.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. Part 3 is Hilbert's Theorem 90 for a quadratic extension, which is classical; the proof given is the standard explicit one (δ = 1 + γ, with the trace-zero line covering γ = -1).