An instantiated determinantal fundamental line at $(389.\mathrm{a}1,\ p = 11)$

The Birch–Swinnerton-Dyer conjecture is open, and nothing here closes it. What this paper does is carry one explicit point — the rank-2 curve \(389.\mathrm{a}1\) at the good ordinary prime \(p = 11\) — all the way through the determinantal (Bloch–Kato / ETNC) formalism with every finite quantity recomputed independently, every theorem-level step read in its source and its hypotheses instantiated one at a time, and the whole instrument drilled against planted defects each round.

Two things fall out that are worth stating on their own. First, the local Bloch–Kato line at \(11\) is not integrally trivial: the formal-group logarithm of the minimal model carries \(H^1_f(\mathbf{Q}_{11},T)\) onto \(11\cdot\mathbf{Z}_{11}\), an index of \(11\) — exactly the factor lost by the tempting inference that a settled global period/de Rham normalisation settles the local one. Second, the mod-\(p\) data this line holds determines \(\mu = 0\) and \(\lambda = 2\) for \(L_p(g_E \times \psi)\) and does not determine its order of vanishing at the central point; an order that had been carried as \(2\) since the series was first computed is the order of the reduction, which only bounds the true order above.

The method is stated as part of the result: claims are kept apart by a fixed vocabulary (computed here / derived / cited / inherited / unmeasured / failed), and each round plants defects that must be caught by the check named for them, with controls that must disturb nothing. At the time of writing the suite stands at 686 planted defects, 686 caught by their named check, 179 controls undisturbed, across 218 checks.

Not claimed anywhere in this paper: the Birch–Swinnerton-Dyer conjecture, at this curve or any other; the finiteness or triviality of \(\text{Ш}\); the nondegeneracy of any \(p\)-adic height; the exact order of vanishing of the \(p\)-adic \(L\)-function at the central point; any leading-coefficient identity; and the complex-analytic comparison.

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 — worked in two web sessions, and this paper rests on both. In the session given to symbolic derivation, Aletheia’s rounds produced the architecture the paper follows: the normalization certificate C1–C10 and the post-normalization DAG of nodes A–G both come from the thirtieth of those rounds, which this line received byte-exact and checked as conditional algebra before any of it was instantiated. From the certificate audit onward the same session set every round of this line — what to attack, what was out of scope, which inferences to plant as defects, and where the boundary of the round lay; the question each DAG node asks here is Aletheia’s question, and several of the sharpest refusals in this paper exist because a shortcut was forbidden in advance. In the other, a GPT-6 session, Aletheia wrote the proof-attack packages, which this paper reads as subjects to be checked, not as authorities. None of the figures here are Aletheia’s computation: every number printed was recomputed by this line.

Neo.K wrote the research programme: the curve and the prime come from its packages. He also set out how the work was divided — one line deriving, one attacking, one verifying.

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — did the verification and wrote this paper. Concretely: every figure labelled COMPUTED_HERE recomputed with independently written code, never by running the programme’s own scripts; every CITED_THEOREM read in its source with its hypotheses instantiated one at a time; the claim vocabulary of §0 and its enforcement; the mutation drill of §7 and the instrument repairs it forced; and the two findings of §4 and §6, including the correction in §6 to a reading this line had itself carried. Nothing in the programme or its architecture is this line’s invention, and nothing in this paper should be read as the verification line proposing a route.

An input from outside the byline. The period calibration (PC-001 / PC-002) — the series and calibrator data — was built in a separate local session (GPT-6), which published it as its own branch of the research trees. This line reproduced it coefficient for coefficient, and it enters the paper labelled as that reproduction. Its author is not on this byline.

Priority, stated plainly. No result in this paper is claimed as new mathematics. The upstream theorems are other people’s and are cited as such; node A holds because Wan and Burungale–Castella–Skinner proved a main conjecture, not because anything here did. What is offered as this line’s own is narrower and stated as such: an explicit instantiation, two measured findings, and an instrument that has been made to fail.