# Current-session verification record

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

Object: `SGD with Vanishing Correlated Gradient Errors (technique notes)`, the exact `main.pdf` in this package.
SHA-256: `f02ce29f4b79672a3b23bc88c7d29279ef1316bddd2c8bff355b2e239cafc0ca`.

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 objective/noise assumptions, both main theorems, both examples, the normalized-weight lower bound and polynomial-gain attainment statements, and the complete `appendix_proofs.tex` covering their derivations. The public original is the supplied 19-page PDF; the current report is the organizing assistant's own check, not an independent reviewer and not a replacement for a later expert review. Experiments and external references were not rerun/reproved.

### Mathematical checks

1. Adapted bias: $b_t$ is measurable with the past and cannot be discarded as fresh mean-zero noise. The contraction argument uses cocoercivity and Young's inequality, giving a remainder $(3\eta_t/\mu)\|b_t\|^2+\sigma^2\eta_t^2$. With $\mathbb E\|b_t\|^2\le B/t$ this remainder is summable in expectation, hence almost surely by Tonelli. This supports the stated almost-supermartingale convergence route. The comparison constants $4D\sigma^2/\mu$ and $12B/\mu^2$ agree with the displayed scalar recursion.
2. PR remainders: the generalized MSE estimate is $O(t^{-\alpha}+t^{-\beta})$. Summation-by-parts boundary terms vanish if $\beta>2\alpha-1$ and $\alpha<1$, while the Hessian-Lipschitz nonlinear sum needs $\alpha,\beta>1/2$. The summary preserves the combined condition and the separate small cumulative-remainder condition.
3. Joint covariance: finite sum exchange gives the cumulative weights $K_{T,s}$. The current gradient-noise index can be shifted from $\xi_{t+1}$ to $\xi_t$ with two $L^2$-negligible endpoints. The joint CLT then has cross terms $C_0K^\top+KC_0^\top$. The proof's range factorization deals with singular joint covariance. The distributional covariance parameter $\Gamma$ is not automatically the limit of the actual cumulative covariance when the historical remainder is only $o_p(1)$.
4. Examples: exact finite-sum computations verify the running-mean identities. Direct integration gives $\int_0^1k_c=1$ and $\int_0^1k_c^2=2c/(2c-1)$ for $c>1/2$; the logarithmic $c=1$ case matches. Product/tail envelopes, not pointwise convergence alone, are needed for the limit. Single-step and downstream coefficients have different gain optima; the original conclusions are for fixed gain $c$.
5. Sharp bound: row normalization implies $\sum_sk_{T,s}=T$, so $\Gamma_T-Q=(Q/T)\sum_s(k_{T,s}-1)^2$. The downstream lower bound requires the normalized deterministic scalar estimator class, zero cross-covariance and a valid PR limit. It is not a universal lower bound over arbitrary algorithms.
6. Attainment: initialization $\gamma_1=1$ ensures exact row normalization. The scalar variance recursion gives $O(t^{-\rho})$; Abel summation gives the $L^2$ order $O(T^{(\rho-1)/2})$ of the cumulative difference, hence the mean-square column flatness rate. That rate itself implies the maximum-weight CLT condition. The extra downstream exponent restriction is $\rho>2\alpha-1$ in addition to $1/2<\rho<1$. The finite-weight script supports the identities and numerical behavior but does not replace the analytic asymptotic argument.

### Findings and limitations

No contradiction was found in the checked identities, exponent bookkeeping or the summary's restricted lower-bound claim. This is not a declaration that every line is correct. No historical review was found for this paper; no such review is represented as existing. The verification record consists of this actual current-session task/report and the executed supporting script/output. The original PDF is unchanged; the takeaways were derived from its corresponding source. Experiment reproduction, literature priority and a separate independent review remain outside scope.


### 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: exact running-mean identities and normalized-weight variance-gap identity, T=1,2,7,29.
PASS: exact recursive-kernel integral algebra at c=3/4,3/2,2,5; c=1 has log-square integral 2.
T=100: mean-square cumulative-weight gap=0.335780217, max-weight/sqrt(T)=0.359290116
T=1000: mean-square cumulative-weight gap=0.151419034, max-weight/sqrt(T)=0.114938441
T=10000: mean-square cumulative-weight gap=0.0736136618, max-weight/sqrt(T)=0.0363474912
T=100000: mean-square cumulative-weight gap=0.0383324989, max-weight/sqrt(T)=0.0114940859
PASS: polynomial-gain finite-T normalization and variance gaps, a=0.8, rho=0.75.
Scope: finite checks support identities but do not prove the asymptotic rate or reproduce the paper experiments.
```