# Current-session verification record

Model: Codex (GPT-6; current session). Date: 2026-10-17 (UTC).

Object: `Hodge Classes on Abelian Sixfolds: Constructions and a Conditional Proof`, the exact `main.pdf` in this package.
SHA-256: `efcac47add715c1cf605d6827d6484f514405b7cfe67ecdcbb386ccbd67cf88e`.

This is the organizing assistant's own limited check in the current upload session. It is not an independent peer review, a re-run of the old reviewer, or a claim that every proof, external theorem or experiment has been verified. The following is the complete task input and report for this limited check; it does not purport to reconstruct the missing historical review conversation.

## User task (verbatim)

hi 我想使用 MathHub 整理、上传和管理我的数学结果。

请帮我下载并安装 MathHub 命令行工具（用于连接网站）和 MathHub 技能（用于整理结果、上传和管理作品），并帮我登录账号；如果我还没有账号，请协助注册。然后询问我：“你有哪些数学作品想要上传？请告诉我这些文件在电脑中的位置。”

请先阅读 https://mathhub.eggtea.org/docs，按我的操作系统选择命令行工具，并从 https://mathhub.eggtea.org/downloads/SKILL.md 获取技能。安装前检查来源、安装内容与权限；如你的环境要求额外批准，请遵循该要求。

请与我确认要处理的文件、署名、许可证及公开发布范围，然后在授权范围内自动整理、验证、上传和管理。遵循 work.yaml、原文、简短的核心洞见与验证材料的格式；不得编造结果或审阅，也不要上传密码或私密材料。需要登录时请让我安全完成。

## Subsequent user input relevant to this report (verbatim)

> 需要注册 以及我们的四篇论文在 归档里面的 paper里面

> 统一 是的 许可都可以

Public authorship, provided in response to the author question:

> eggtea

The assistant selected CC BY 4.0 and prepared only the original, takeaways and verification materials. Account secrets and local absolute paths are excluded.

## Assistant report

### Scope and inspected material

This is a conditional-claim and method-map audit, not a full mathematical review of the 66-page manuscript. Read the supplied method map and scope ledger, the original polarization-adjustment proofs (`p:oddscale`, `p:quartic-scale`, `p:CMsplit`), the stabilization theorem statement and recovery construction (`p:all-sixfold-weil`), and the exact residual N2/N3 external contracts and selected opening construction steps (`p:N2auxiliary`, `p:N3KS`). The full residual spin/Clifford proofs and all twenty-one rows were not independently verified.

### Mathematical checks

1. Odd rank: scaling one rank-$r$ factor by $c=-d>0$ gives $d'=c^rd=-d^{r+1}$ for odd $r$. This is a negative rational square, representing the required norm class; positivity of the scaling is essential. The summary does not confuse invariance of the Weil space with a general assertion that changing a polarization creates cycles.
2. Real quadratic: from $D=(4d)^3N(b)$ and $N(a)=-dN(b)$, the new determinant is $D'=-(8d^3N(b)^2)^2$. This identity matches the source. Geometric applicability still requires the Rosati compatibility, positive choice of $a$ and the cited split-sixfold theorem.
3. Trace splitting: degree-six interpolation yields the five trace vanishings and the degree-five normalization. This makes the span of $1,\theta,\theta^2$ isotropic for the stipulated trace form because product degrees are at most four. The argument also needs a primitive element with the correct alternating signs, ensuring a positive rational change of polarization. A finite rational-root script only checks the interpolation identity, not the entire CM argument.
4. Stabilization: adding the auxiliary surface changes signature to $(4,4)$ and determinant to $d^2$. Recovery is by an actual algebraic cup/push correspondence; codimension is $4+1-2=3$. Algebraicity on the eightfold remains a separate external assumption.
5. Residual models: N2 uses dimension thirteen, signatures $(2,11)$ and Witt index at least two, then a specified four-slot Aux tensor and CM transfer. N3 uses the stated complete Clifford KS correspondence. Ordinary KS or an abstract Hodge isomorphism does not supply either algebraic cycle automatically. The summary keeps the sixfold/all-powers distinction and $\dim A^n=6n$.

### External status check

The official OpenAI mathematics release history was read on 2026-10-17:
https://github.com/openai/math/blob/main/history.md

Its 2026-10-07 entry records withdrawal of the split-eightfold, Kuga–Satake and K3-product manuscripts. These are treated here as stipulated assertions, not established theorems. The CM assertion remains unverified in this session. The new abstract and takeaways explicitly make the full twenty-one-row conclusion conditional on the exact external contracts; the original PDF, including its historical wording, is unchanged.

### Changes and limitations

The supplied companion combined a method map and the full manuscript. The new takeaways retain the method map, add explicit setup and external contracts, and omit the duplicate full proof text. This report does not certify spin descent, all invariant computations, every cycle construction, external Hodge assertions or overall correctness. Executed algebraic specializations are reproducible supporting checks, not proofs of algebraic-cycle existence.

`supplied-review.pdf` is the unchanged user-provided historical conditional report. Its PDF says the reviewer is unnamed and the review date is unavailable; later repository notes attribute it to 6 Astra and assert correspondence. Those are source declarations, not independently authenticated provenance. Missing historical prompt/date information is not invented. The supplied report is supplementary, not a new independent review.


### Executed supporting check

Command: `python3 verification/check.py` from the package root. All checks completed with exit code 0 on 2026-10-17. Source: `check.py`; complete stdout/stderr: `check-output.txt`. The captured output is:

```text
PASS: odd-rank determinant adjustment at three rational determinants and ranks 1,3,5.
PASS: real-quadratic trace determinant adjustment at six exact rational specializations.
PASS: degree-six interpolation trace identity at six distinct rational roots, m=0,...,5.
PASS: eightfold Weil contraction codimension 4+1-2=3.
Scope: no external Hodge/KS/Aux assertion, spin descent, or general cycle-algebraicity theorem is certified.
```