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# Balanced reference selection for T353
The symmetry line $a=b$, $x=1/2$ is supplemented by asymmetric beta integrals at their distribution mean $x=a/(a+b)$ and by adjacent-shape pairs at the equal-split argument $x=1/2$ CITE{BalancedBeta}. Half-integer shapes are selected by chi-square degrees of freedom CITE{BalancedStatistics}; restricting the second shape to $1/2$, or to a one-unit offset, avoids a free parameter box. The adjacent-shape and low-complexity arcsine references are bounded editorial choices, not claims that every input has a separate application. These are unregularized integrals. Positive integer shapes at rational arguments are exact rational polynomial integrals; the mean is not claimed to be a median in asymmetric cases.
Coverage: it includes $a=b=n/2$, $x=1/2$ for $1\leq n\leq20$; the pairs $(n/2,1/2)$ and their reversals at their means for $2\leq n\leq12$; the pairs $(n/2,n/2+1)$ and their reversals at their means and at $x=1/2$ for $1\leq n\leq12$; and $a=b=1/2$ at $x=0,1/4,3/4,1$; all previous entries are retained.
## Mathematical sources
- DLMF 8.17: incomplete beta identities: https://dlmf.nist.gov/8.17
- DLMF 8.23: beta and chi-square applications: https://dlmf.nist.gov/8.23
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.