← BSD Conjecture · LAYER 3 · III · Crystallization
An independent recomputation of a published computational census at full scale — discriminants, conductors by Tate's algorithm, the isogeny filter decided in both directions through modular curves, every twist list rebuilt, the census's own accounting measured term by term — with every disagreement explained and the directions its own checks could not see
I · Precursor (five sub-lines: Phase 0, P5, Phase 1, Phase 2 and the stress test) → II · Computation and Reproduction (107 runs, RUN-001 to RUN-107) → III · Crystallization (this page, the formal paper).
dd2ecf7ec3ac5534…bf108f3d60730eb4…The PDF is built from the Markdown source by tools/build.py of the crystallization repository, amral-crystallization, at commit 8953b08e8f5f2fcab5a87d101acac208577dc948. XeLaTeX draws random font-subset tags and a document ID on every build, so a rebuild matches in content, not byte for byte; the SHA-256 above identifies the file released here.
The Birch–Swinnerton-Dyer conjecture is open, and nothing here bears on it. Banwait and Huang’s census selects semistable base elliptic curves of conductor below 500,000 and lists, for each, the quadratic twists to which a theorem of theirs gives strong BSD. A research programme reproduced the census as an exact artefact census, and graded its own headline count as taken from the official output and consistency-audited, with only 2 curves and their twists recomputed. This paper closes that gap for all of them, with independent code and exact integer arithmetic throughout.
Recomputed from the a-invariants alone: 40749 discriminants and 135787 valuations; 40749 conductors by Tate’s algorithm — every bad prime multiplicative, so the census reaches no non-semistable curve; the trace of Frobenius at 3 on every removed curve; and the 3-, 5- and 7-isogeny filter decided in both directions through rational points on \(X_0(n)\) — over the whole base, the kept curves included, which the census’s own checks could not see. Then every one of the 36687 twist lists, 247391 pairs, both branches, exactly; the twist conditions on every pair (soundness) and on every candidate up to the stated bound (completeness); and the census’s global accounting identity with all eight of its terms measured rather than back-solved. Zero disagreements wherever the census and this line measure the same thing. Along the way: the base is two populations split exactly by 2-torsion, which are the theorem’s two branches; a docstring states a condition its own code does not compute, and the output follows the code; a bucket labelled \(a_3 = 1\) holds two kinds of number; and one of this line’s own checks was vacuous on this data, and is reported as such.
The paper covers a full independent recomputation of the Banwait–Huang strong-BSD twist census, with the second layer's RUN-004 to RUN-008, RUN-012, RUN-016, RUN-028, RUN-029, RUN-055 and RUN-061 to RUN-063 and RUN-066: discriminants, 135787 valuations and Tate's-algorithm conductors of the 40749 base curves; the trace of Frobenius at 3 on the removed curves; the 3-, 5- and 7-isogeny filter decided both ways through X0(n), including the kept direction the census's own checks cannot see; all 36687 twist lists and 247391 pairs; the condition map, completeness, the eight-term accounting identity and the diff classified line by line. The base list's provenance conditions (optimality, analytic rank 0, BSD(E,2)) and the curve database's a-invariants are premises (INHERITED); the census's rank, descent and 2-adic filters have never been re-run (UNMEASURED). The Birch–Swinnerton-Dyer conjecture itself is OPEN; the paper claims strong BSD for no twist.
Not claimed anywhere in this paper: the Birch–Swinnerton-Dyer conjecture, or strong BSD for any twist; the census’s theorem; that the base’s selection conditions are the right ones — this paper verifies that they were applied, not that they should be; and the base list itself as an independent scan of the elliptic curves below conductor 500,000, which was not done.
| Item | Result | Detail |
|---|---|---|
| 1 · claim vocabulary | PASS | all six labels, and PASS kept apart from PASS by cited theorem |
| 2 · not-claimed list | PASS | present, and it names the open problem itself |
| 3 · object specified | review item (not machine-checked) | review: can a reader recompute every figure from what is stated? |
| 4 · numbers trace to logs | PASS | 50/50 traced — 50 to recomputed evidence, 0 only to a quotation of the programme (7 small integers set aside) |
| 5 · instrument can fail | PASS | 692 defects planted, 687 caught by the check named for each; 179 controls; 218 checks |
| 6 · reproduction | PASS | section present; running it is a review step |
| 7 · contributions | PASS | section present; 3 author(s), listed alphabetically, each named in it |
| 8 · attachments | PASS | 109 gate logs; 18 input manifest(s) with SHA-256; public projection linked |
| rule · 4 nothing local | PASS | nothing machine-local or credential-shaped |
Evidence pinned at agent/bsd-verification-zhuiheng @ 6f9c8cc10139ee05f235bf9a22085479c3d2966a · bsd-verification-zhuiheng/
This paper pins its evidence at research-tree commit 6f9c8cc (RUN-109, 2026-10-24), from which the second layer's pages are regenerated, so every figure in it can be seen there, on the pages of the runs it cites; the corrections this version carries are recorded on the pages of RUN-108 and RUN-109.
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 reads: the condition map, the accounting report, the small-fixture analyses, the adversarial corpus and the mirror.
Neo.K set the programme’s target — an exact artefact census of a published result, with its evidence grade stated — and wrote its documents with Aletheia.
墜衡 (Zhuì Héng) — the independent verification line; model: Claude, Opus series — recomputed every column of the census from the a-invariants with independent code, decided the isogeny filter in both directions through modular curves, rebuilt every twist list, measured the accounting identity term by term, took the diff apart, built and drilled the gates, reported its own vacuous check, and wrote this paper. The census and its code are Banwait and Huang’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 full-scale independent recomputation and the measurements of where the census’s checks, and this line’s, could not see.
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