Grading evidence for the Birch–Swinnerton-Dyer conjecture curve by curve: an eleven-rung ladder, an audited rejection, and a frontier metric faithful only in exact arithmetic

The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it. This paper is about how evidence for it should be stated. A research programme on the conjecture opens with framework documents, and the first of them refuses to let a curve carry a bare true or false: it defines an eleven-rung certificate ladder, from the curve’s identity (C0) through numerical and rigorous analytic rank (C2, C3), algebraic bounds, weak and single-prime strong BSD, finiteness of Ш, full strong BSD, to a family theorem (C10), so that each curve and prime is placed at the rung some certificate has actually closed. Checked against the programme’s 85 curated documents, the ladder’s vocabulary is used cleanly — no document names a rung the framework does not define — but every summary of it renders six rungs, dropping C1 to C5, and with them the one boundary the framework forbids crossing: a number a computation produced against a number a proof produced.

Three further results. The programme’s audit rejecting its own author’s “lattice-point rank convergence” route holds on all four objections, nothing in the corpus claims the route’s salvage conditions met, and a later document refutes the route outright with \(f_a(z) = z^2 + az\); checked here in exact rationals, that refutation has a second half — the zero count in a fixed neighbourhood does converge while the order at the point does not — which says the route fails at infinite precision, not for lack of it. The programme’s frontier quantity \(\mathfrak B_k(s) = \sum_{i \in H_k} i^{-s}\) satisfies all four of its stated properties in the rationals, one hypothesis is unnecessary, and its design goal — never to swallow a single uncertified curve — fails in double precision at a computable index, 208064 for \(s = 3\). And the programme’s three routing maps, read after sixty-four rounds of this line, are scored by what can be computed about them.

Not claimed anywhere in this paper: the Birch–Swinnerton-Dyer conjecture, or that any curve reaches any rung; that the lattice-point route’s salvage conditions are unmet — only that no document in the corpus claims them met; that the abbreviated ladder misled any reader; and a replacement weight for the frontier quantity, which is a research decision rather than an arithmetic fact.

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 framework documents this paper rests on: the certificate ladder, the audit of the lattice-point route, the frontier quantity, and the closure map, route matrix and agent briefs.

Neo.K set the programme’s target and its method — that evidence be stated at its actual strength, rung by rung — proposed the lattice-point idea that the programme then audited and rejected, and wrote the framework with Aletheia.

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — measured the ladder’s use across the corpus, checked the audit in all three directions and recomputed the refutation in exact arithmetic, added its neighbourhood half, verified the frontier quantity’s claims exactly and located where a float computation of it goes blind, scored the maps against this line’s own rounds, built and drilled the gates, and wrote this paper. The framework is the programme’s; nothing in it 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 measurement of how a framework for evidence is used and summarized, and two sharpenings: the neighbourhood half of the refutation, and the index at which the frontier quantity’s faithfulness stops being computable.