Eight proof attacks on the Birch–Swinnerton-Dyer formula at (389.a1, 11), checked: the 11-adic and geometric data they compute, and where each stops

The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it. For the rank-two curve \(E\): \(y^2 + y = x^3 + x^2 - 2x\) of conductor 389, with generators \(P = (0,0)\) and \(Q = (1,0)\), at the good ordinary prime 11, a research programme ran eight “proof attacks” — derivations toward the 11-part of the leading-term formula — and each attack computed something new. This paper recomputes every finite statement in them from independent arithmetic: this line’s own modular symbols, exact group law, number-field and function-field arithmetic, and point counts. Everything recomputes.

The data, verified: the mod-11 cyclotomic p-adic L-function at conductor 121, all 110 summands, with \(\mu = 0\) and \(\lambda = 2\); the multiples \([16]P\) and \([16]Q\) to the last digit and their normalized formal logarithms \((4, 10) \bmod 11\); the local multiplier \(e_{11}/\theta = 16\) exactly in \(\mathbb Q(\alpha)\); the value \(L''(E,1)/2 = 0.7593165002884\) from a Mellin formula and this line’s own coefficients — the first time this line computed a number it had earlier taken as input; an explicit genus-three curve on \(E \times E\) with a Bézout certificate and a chain whose boundary is four times the exterior product of the two generators; the norms and pushforwards of its tangent function, and a closed correction showing that the constant they produce is not intrinsic; twisted L-values of \(E\) that tie this line’s mod-11 eigenline to archimedean values by one unit per sign; and the 11-adic logarithm of a circular unit in \(\mathbb Z[\zeta_8]\) to \(11^{10}\). Every attack states, in its own words, that it is not a proof and that the unit it needs is unknown; the finite data are what any \(p\)-adic approach at this point consumes.

Not claimed anywhere in this paper: the Birch–Swinnerton-Dyer conjecture or its 11-part for this curve; any of the attacks’ conditional theorems; the canonical scalar \(\mathfrak s_{11}\), the period \(C\) or the classes \(B_2\) that the attacks leave unknown; that \(\lambda = 2\) for the mod-11 Iwasawa invariant gives the order of vanishing in characteristic zero; and that the numerically vanishing twisted L-values of §7 are zero as algebraic numbers — they are identified with zero to \(10^{-15}\), with an independent mod-11 agreement.

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 — wrote the eight proof attacks this paper rests on, each a derivation with its own program and output, working from a handoff bundle of the programme.

Neo.K set the programme’s target — the 11-part of the leading-term formula for this curve — and its method, handed the attacks their inputs, and relayed them to this line.

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — recomputed every finite statement in the eight attacks from independent arithmetic, built the minus eigenline and the twisted L-values that tie it to archimedean values, computed the leading term the line had only taken as input, built and drilled the gates, withdrew its own wrong reading of Attack 09, and wrote this paper. The attacks are Aletheia’s; nothing in them is this line’s invention.

Priority, stated plainly. No result here is claimed as new mathematics. What is offered as this line’s own is the independent verification of the attacks’ data, the unit relations of §7, and the computation of \(L''(E,1)/2\) from the line’s own coefficients.