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# Balanced reference selection for T351
For a chi-square variable with $n$ degrees of freedom, $\Gamma(n/2,x)/\Gamma(n/2)$ is its upper-tail probability at threshold $2x$ CITE{BalancedStatistics}. The half-integer shape family is extended beyond the transition $x=a$ to the unit scale and half/double mean thresholds. These are explicit reference comparisons, not quantiles or conventional significance levels. The shape $a=1/2$ also includes integer standard-normal-deviation thresholds $x=k^2/2$ CITE{BalancedGamma}. The values remain unregularized integrals. No free Cartesian grid of shape and threshold is used.
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 $\Gamma(1,0)=1$ is included; the dimensional and threshold bounds delimit reference families rather than a quantile table.
## 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.