Compiling the Fouquet–Wan hypotheses for a quadratic-twist family of 696.e1: one witness, a uniform certificate false at one prime, an obstruction that does not stop, and a twisting prime where H2 fails

The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it directly. A research programme proposes to prove its strong form for every quadratic twist \(E^{(q)}\) of the curve 696.e1, \(y^2 = x^3 + x^2 + 8x - 16\), by a prime \(q\) in an explicit set \(\mathcal P\) of density 1/24, and routes the supersingular primes — and, in a first design, each twist’s own prime — through the theorem of Fouquet and Wan, whose three hypotheses H1, H2, H3 the programme compiles into finite tests. This paper runs those tests. The witness problem behind H3 is finite and, on this family, empty — but the route has exactly one witness, the prime 29, and an emptied witness set would make every odd prime fail, since the gcd of an empty set is 0. Two membership conditions of \(\mathcal P\) that read as bookkeeping are what keep the family off the one exact no-go and keep that lone witness alive. The programme’s boxed uniform certificate for H3 is false at exactly one prime, \(p = 29\), because it drops the clause \(\ell \ne p\); the same argument restricted to good supersingular primes is true. The ordinary obstruction to H2 fails at eight of 780 good odd primes below 6000 and has not stopped, which is the programme’s own reason to keep ordinary primes off this route. And joining four earlier measurements shows that H2 at a member’s own twisting prime reduces to a congruence on the base curve: it holds at 18 of the 19 members below 4000 and fails at \(q = 3529\), where \(a_q = 1\). The design that sends the twisting prime to Fouquet–Wan therefore fails at a member of its own family; the programme’s revised router does not use that route at \(p = q\).

Not claimed anywhere in this paper: the Birch–Swinnerton-Dyer conjecture, or its strong form for any twist; the theorem of Fouquet and Wan, or that the programme’s compilation of its hypotheses is exact; the programme’s Lemma B, which its own text calls a candidate; that the ordinary obstruction set is infinite; that the revised router’s branches apply, since their theorems’ hypotheses are cited and not verified here; anything about the truth of strong BSD for \(E^{(3529)}\), where one route fails a hypothesis and the statement itself is untouched; and that the Manin constant is 1.

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 second-phase documents this paper reads: the witness lemmas, the twist-invariance bridge, the H2 and H3 compilers, the ordinary-obstruction analysis, the local isogeny criterion, the routers, the family schema and the implementation specification.

Neo.K set the programme’s target — strong BSD for an explicit infinite family, every prime routed to a theorem — and its method, that each hypothesis be compiled into a finite test before anything is claimed, and wrote the second phase with Aletheia.

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — computed the local data and the witness gcds, found the empty-set cases and the control curve by search, measured the membership conditions on both sides, refuted the uniform H3 certificate at \(p = 29\) and located why the supersingular form survives, measured the ordinary obstruction and exhausted the H2 equivalence, derived the mutual exclusivity of the kernel tests, joined the reduction of §6 and found member 3529, ran the compiler to where it stops, built and drilled the gates, and wrote this paper. The route, its lemmas and its documents are the programme’s; nothing in them is this line’s invention.

Priority, stated plainly. No result here is claimed as new mathematics; the equivalences used are standard representation theory, and the external theorems are their authors’. What is offered as this line’s own is the measurement of a compiled proof plan against its own family: one false boxed certificate and the quantifier that rescues it, two membership conditions shown load-bearing, a self-disagreement in the corpus and the document that resolves it, and a member at which the provisional design fails.