# Argument selection for T352

For a chi-square variable $Y$ with $n$ degrees of freedom, $\Pr(Y\leq y)=\gamma(n/2,y/2)/\Gamma(n/2)$ CITE{SelectionStatistics}. The mean of the sum of $n$ squared standard normals is $n$, selecting $a=x=n/2$. The same argument is the intrinsic transition $\lambda=x/a=1$, $\eta=0$ in CITE{SelectionTransition}, treated in CITE{SelectionTransitionPaper}. The unregularized values are probability numerators, not probabilities; the mean is not asserted to be the median.

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 $x=a$; the dimensional cutoff is a reference prefix, not a replacement rational grid.

## 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 T352 --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.