# Current-session verification record

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

Object: `Complex-Corrected Stochastic Gradient Descent for Noisy Observations`, the exact `main.pdf` in this package.
SHA-256: `3e222f197266ca0283b9618f4eb4d4cc1818c0e90a916d724c3981404da6eb02`.

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

Read the original Sections 3–4, the proofs of exact correction, relative/higher moments and convergence, and the localization and uniform-averaging endpoint argument in Appendix A. The supplied AI edition was used only as a locator; key claims in the new takeaways were compared with the original LaTeX and the readable original PDF. This is a selected-proof and summary-consistency audit, not a full verification of every theorem or numerical experiment.

### Mathematical checks

1. Exact correction: for $D=U+iV$, equal independent Gaussian covariances give rotational invariance without invertibility of $\Omega$. Applying the one-variable entire mean-value identity to $s\mapsto g^{\mathbb C}(\theta;z+sD)$ is valid after the stated derivative envelope supplies integrability. This verifies the conditional centering mechanism used in Lemma 3.4.
2. Moment transfer: centering of $\Delta_t$ conditional on both the past and $Z_t$ kills its cross term with the clean-score noise. The bound with $K_a$ and $M(\theta)^2\le A+B\|\nabla F(\theta)\|^2$ yields the stated deterministic relative-variance constants. The amplitude hypothesis must not be removed. Subtracting $g(\theta;0)$ before estimating higher moments is consistent with the original derivation.
3. Optimization assumptions: convex and nonconvex small-step restrictions, the distinction between step-weighted and uniform averages, and all three strongly convex endpoint MSE rates were preserved. The CLTs retain local Hessian regularity and covariance continuity, not just strong convexity.
4. Endpoint averaging: the manuscript localizes using $m>1/(2c)$ below $\lambda_{\min}(H)$ and proves a square-integrable cumulative kernel. The summary retains the $c\lambda_{\min}(H)>1/2$ condition. In the scalar case $cH=1$, $K_c(u)=-c\log u$ gives twice the interior PR covariance; automatically substituting the interior formula at the endpoint would be wrong.
5. Response and extension boundaries: the response is held fixed during input complexification; joint-data moments are separate assumptions. Non-Gaussian conjugate identities require integrability. Trainable sine-network experiments are not labeled as theorems.

### Changes and limitations

The supplied Markdown had malformed covariance symbols and a damaged array separator. The new takeaways were written from the original formulas; no original PDF or LaTeX file was edited. No new correctness, priority, optimality or experiment-reproduction claim is made. The full last-iterate CLT, all example-specific calculations, external literature and experiment outputs were not independently reverified in this check.

The supplementary `supplied-review.pdf` is the unchanged user-provided historical 6 Astra report. Its text describes `main(6).pdf`; local repository notes assert correspondence to the current manuscript, while the PDF itself cannot establish that correspondence by hash. Its original prompt and actual review date are unavailable. It is supplementary evidence, not the qualifying dated current-session report and not a newly reproduced 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: E[(U+iV)^n]=0, independent standard Gaussians, n=1,...,14 (exact moments).
PASS: observation/simulation decomposition at v_obs=3, v_sim=5, n=4, m=7 gives 13/14.
PASS: scalar cH=1 endpoint covariance = 3/2; interior covariance = 3/4.
Scope: these checks do not establish the general CLTs or rerun experiments.
```