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# Balanced reference selection for T352
For a chi-square variable with $n$ degrees of freedom, $\gamma(n/2,x)/\Gamma(n/2)$ is its lower-tail probability at threshold $2x$ CITE{BalancedStatistics}. The selected half-integer shape slices compare unit scale, half mean, mean and double mean; only the mean is the intrinsic transition $x=a$, and the other ratios are exploratory reference comparisons. At shape $a=1/2$, $x=k^2/2$ gives the Gaussian integer-standard-deviation family CITE{BalancedGamma}. These are unregularized numerators, not probabilities or quantiles.
Coverage: it includes $a=n/2$ for $1\leq n\leq20$ at $x=a/2,a,2a,1$, with duplicate pairs merged; $a=1/2$ also has $x=k^2/2$ for $1\leq k\leq6$; the exact endpoint $\gamma(1,0)=0$ is included; these bounded families expand both shape and threshold coverage without a Cartesian sampling box.
## Mathematical sources
- DLMF 8.23: gamma and chi-square applications: https://dlmf.nist.gov/8.23
- DLMF 8.4: incomplete gamma special values: https://dlmf.nist.gov/8.4
The sources establish definitions and relationships, not our exact finite exploratory list. All previous values, identities, and manual row annotations are retained. The current generator recomputes the entire selection from mathematical inputs and never uses saved decimal values as arguments. Older attached helpers are historical; generate.py now uses balanced_values.py and balanced-selection.json. Exact arithmetic and independent checks are described in balanced-evidence.json.