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# Balanced reference selection for T178
The signatures $r=2,3,4,6$ retain their alternative-base elliptic meaning. The half argument has the gamma quotient from Gauss's second summation theorem CITE{BalancedHypergeometric}. Complementary pairs $(x,1-x)$ with $x=1/4,1/3,1/9$ provide a bounded low-complexity reference set on both sides of the complementary centre; this list is our exploratory choice and is not asserted to consist of singular moduli. The origin supplies the exact hypergeometric normalization.
Coverage: it holds $F_r(x)$ for each signature $r=2,3,4,6$ at $x=0,1/2,1/4,3/4,1/3,2/3,1/9,8/9$; the complementary pairs are exploratory references, not a catalogue of arithmetic moduli, and the origin is exact.
## Mathematical sources
- DLMF 15.4: hypergeometric special cases: https://dlmf.nist.gov/15.4
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.