Paper against code: how a strong-BSD twist census's algorithm changed across three commits, and why one of its branches could never fire

The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it. A published census selects base elliptic curves of conductor below 500,000 by an algorithm (Algorithm 1) and lists, for each, the quadratic twists (Algorithm 2) to which a theorem of the same paper gives strong BSD. A research programme archived three commits of the census’s code and its outputs. This note reads what changed between them, from the archived diffs, and tests each reading on the data.

Algorithm 1’s isogeny condition changed three times. At the commit that generated the twist output it reads set(bad_primes) and set([3,5,7]), which in Python evaluates to {3,5,7} on every curve with a bad prime — a strict rule, whatever was meant; the next commit made it conditional on the conductor; the current one restored the strict rule and added a filter \(a_3 \ne \pm 3\). The intermediate rule makes a prediction — every isogeny curve in its base escapes its trigger — and 1355 of 1355 do. Each of the three archived artefacts matches the rule that produced it. On the twist side, a deleted predicate explains why the census’s “expand” branch never fired: 0 of the 36687 kept curves fail it at any prime up to 997, so deleting it could add nothing. The corpus’s own replay of the change, a 2×2 of 268697 twist pairs, is recomputed here to the pair. And the programme’s specifications for a full reproduction are held against what was actually produced: the descent that would complete it has not been run by anyone, and every document says so.

Not claimed anywhere in this paper: the Birch–Swinnerton-Dyer conjecture; the census’s theorem, or strong BSD for any twist; that the census’s 36687 curves are correct beyond the two filters recomputed — the rank, descent and 2-adic BSD filters were not re-run, here or anywhere in the programme; and anything about intent, such as whether and was meant to be &.

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 census-reproduction documents this paper rests on: the paper-against-code audit, the one-commit analyses, the replay of the deleted predicate, the reproduction specifications, and the agent-experiment specification.

Neo.K set the programme’s target and its method — that a census be reproduced as exact artefacts with its gaps stated — and wrote its documents with Aletheia.

墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — read the three rules from the archived diffs and tested the prediction they make, recomputed every figure of the four commit documents and the replay to the pair, found why the expand branch could not fire, held the specifications against the package and against this line, audited the freeze conditions, built and drilled the gates, and wrote this paper. The census and its code are Banwait and Huang’s; the documents are the programme’s; nothing in either 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 mechanism of §2 and its test, and the reason of §4.