# Argument selection for T351

For a chi-square variable $Y$ with $n$ degrees of freedom, $\Pr(Y\geq y)=\Gamma(n/2,y/2)/\Gamma(n/2)$ CITE{SelectionStatistics}. Since $Y$ is a sum of $n$ squared standard normals, its mean is $n$; evaluation at the mean therefore gives $a=x=n/2$. Independently, $x=a$ is the transition $\lambda=x/a=1$, $\eta=0$ of the uniform incomplete-gamma expansion CITE{SelectionTransition}, studied explicitly in CITE{SelectionTransitionPaper}. The entries are unregularized numerators, not probabilities and not medians. The upper bound on $n$ specifies a finite dimensional prefix, not a conventional significance level.

Coverage: it holds $\Gamma(n/2,n/2)$ for $1\leq n\leq10$, the first ten chi-square dimensions at their mean and the exact incomplete-gamma transition $x=a$; no off-diagonal sampling grid is included.

## Primary mathematical evidence

- DLMF 8.23: Gamma and Chi-square Distributions: https://dlmf.nist.gov/8.23
- DLMF 8.12.1-4: The transition x=a: https://dlmf.nist.gov/8.12
- Nemes and Olde Daalhuis: Asymptotic expansions for the incomplete gamma function in the transition regions: https://arxiv.org/abs/1803.07841
- DLMF 8.4: Incomplete Gamma Special Values: https://dlmf.nist.gov/8.4

The mathematical deductions in the selection paragraph are distinguished from a source publishing this exact finite list. The range cutoff is a reference limit, not a theorem that every included dimension or order has a separate documented application.

## Reproduction and history

Run `python generate.py` to verify and `python checks.py T351 --live` for independent checks. Neither writes. The source uses exact rational arithmetic and certified Arb balls, with mpmath as a separate agreement check. Retained numeric strings and annotations are copied from, and verified against, the website export; the generator is not claimed to have authored those annotations. Removed sample identities and old attached sources remain in website revision history. No identity is reused for another argument.