An explicit quadratic-twist family of the non-semistable curve 696.e1: a density of 1/24 by three routes, an arithmetic certificate, the isogeny and image questions settled, and a derived theorem candidate checked against its own router

The Birch–Swinnerton-Dyer conjecture is open, and nothing here proves any case of it. Banwait and Huang proved strong BSD for infinite families of quadratic twists of semistable curves. A research programme proposes a family for a curve their method does not reach: the twists \(E^{(q)}\) of 696.e1, which has additive reduction at 2, by the primes \(q\) of an explicit set \(\mathcal P\), and it states, as a derived theorem candidate, strong BSD for every member, each prime of each twist routed to a published theorem. This paper recomputes everything about the family that can be computed. The anchor has conductor 696, analytic rank 0 and \(L(E,1)/\Omega = 1\) to one unit in the last place. The density of \(\mathcal P\) is \(1/24\), not the \(1/48\) its three conditions seem to multiply to, by three routes: an exact Galois computation, a Chebotarev derivation in which every field-theoretic step is computed, and a count to \(2\cdot 10^7\); one of the three conditions is implied by the other two, and that redundancy is the factor of 2. A certificate of twenty quantities is rebuilt from the a-invariants alone; all twelve of Mazur’s prime isogeny degrees are excluded; the mod-\(\ell\) image is surjective at 38 primes, and at \(\ell = 3\) provably beyond this method. The candidate is stated against the programme’s revised router, which sends every prime to exactly one theorem — checked as a partition — and not against its first design, which a companion paper shows fails at the member \(q = 3529\).

Not claimed anywhere in this paper: the Birch–Swinnerton-Dyer conjecture, or strong BSD for any member of the family; that the four theorems of the router apply to the primes routed to them — the partition is checked, the applicability is cited; that \(L(E^{(q)}, 1) \ne 0\) for every member, which is computed for the two smallest; that \(\#\text{Ш}(E^{(313)}) = 49\) independently of the BSD formula, which is the analytic order read from a measured ratio; the curve’s optimality and its Manin constant; and the result’s novelty, which the programme defers to a search this line has not made.

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, with Neo.K, the programme’s documents on the family: the set, its density note, the certificate’s specification, the routers, the provisional and revised statements, and the referee lists.

Neo.K set the programme’s target — strong BSD beyond the semistable reach of the published method, for an explicit infinite family — and wrote its documents with Aletheia.

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — recomputed the anchor, the density by three routes and the redundancy behind it, the two smallest members, the certificate from the a-invariants, Mazur’s twelve degrees and the images at 38 primes, the barrier, and the router’s partition; built and drilled the gates; and wrote this paper. The family and the candidate are the programme’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 a complete recomputation of the family’s computable content, the redundancy that explains its density, and the statement of its candidate against the router that survives.