Calibrating a rank-two curve against a rank-zero one at the prime 11, reproduced: the calibrator 19.a1 rebuilt over the rationals, its plus period aligned to Néron's, and a relative regulator germ recomputed to degree 120

The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it directly. A research programme at the rank-two curve 389.a1 and the prime 11 wants to calibrate the leading coefficient of an 11-adic family against a curve of rank zero, so that common unknown scalars cancel. A separate local verification session built that calibration in two rounds: a rank-zero calibrator, 19.a1, with its modular symbols over the rationals and its 11-adic measures; and a relative regulator germ, the quotient \(Q(t)\) of four 11-adic series of the two curves, with a cup pairing that makes it invariant under rescaling the four eigenlines. This note reproduces both with independently written code. The calibrator’s modular-symbol space, its plus and minus eigenlines and five Hecke checks come out identical; every residue of its first layer is reproduced; and the alignment the session left undone is supplied on the archimedean side: modulo 11 the plus line is the Néron-normalised modular symbol with unit exactly 1, shown by twisted L-values that are exact integers, and the minus line agrees with the twisted L-values, with unit 2, at every discriminant tried. The four series are recomputed to \(t^{120}\) from the layer of level \(11^3\), all 1210 summand rows on each curve identical, and \(Q(t)\) has Weierstrass degree 2 and leading coefficient 7; the cup pairing gives \(J_E = 1\) and \(J_0 = 3\), and the invariance holds. What the session itself declines to claim — that \(Q\) is the regulator ratio, that its cup ratio is the adjoint period ratio — this note does not claim either.

Not claimed anywhere in this paper: the Birch–Swinnerton-Dyer conjecture; that \(Q(t)\) is the ratio of regulators, which rests on Loeffler and Rivero’s degeneration as the session cites it; that the cup ratio 3 is the adjoint period ratio, which the session itself says it is not; that the minus period is an 11-adic unit times the imaginary period, which the gate’s log does not record; the Kato, Coleman and adjoint alignments of the periods beyond the archimedean one; and the Beilinson–Flach classes the calibration is meant to compare, which neither the session nor this note constructs.

The three contributions behind this paper are different in kind, and the paper is worth less if they are blurred. They are given here in alphabetical order, the order the author list keeps as data; on the title page the three sit at a round table, with no first and no last.

Aletheia — model: ChatGPT, GPT series — is a standing author of the programme, by Neo.K’s decision, and wrote with Neo.K its documents on the cross-rank calibration that this construction serves. The calibration itself is not hers.

Neo.K set the programme’s target at \((389.\mathrm a1, 11)\) and the arrangement in which a separate session builds and this line verifies.

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — recomputed every object of the calibration with independent code, supplied the archimedean alignment of its plus period, built and drilled the gates, and wrote this note.

Input from outside the byline. The calibration — the choice of 19.a1, its modular symbols and measures, the four series, the quotient \(Q\), the cup pairing and its invariance — was built by a separate local verification session, a Claude Code session (model: Claude Opus 5), on its own branch, in two rounds; this note reproduces it and does not claim it. That session is credited here and is not an author, by the standing rule for local sessions. The theorems it cites are their authors’.

Priority, stated plainly. Nothing here is claimed as new mathematics. What is offered as this line’s own is an independent reproduction, and the archimedean alignment of the plus period that the construction left undone.