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1733 bytes, as of the version from 2026-10-27 01:02 (current). Recorded here, not run.
# Balanced reference selection for T176
Integer orders give a weight-indexed polylogarithm family, with zeta and alternating-zeta endpoints CITE{BalancedPolylog}. The half-argument values, their negative partners, and the low-weight reflection and Landen partners broaden recognition beyond the few reducible constants. Half-integer orders occur in Fermi-Dirac and Bose-Einstein integrals CITE{BalancedPolylog}; their selected fugacity arguments here are bounded reference choices, not a claim of a specific experiment or quantile. The negative arguments use the real principal continuation; the singular endpoint $x=1$ is included only for $s>1$. The finite order cutoffs and the extra simple low-weight arguments are explicitly exploratory, and are not a denominator-complete rational mesh.
Coverage: it preserves the previous 13 values and includes integer weights $2\leq s\leq12$ at $x=1,-1,1/2,-1/2$; weights $2,3,4$ also have $x=1/3,2/3,1/4,3/4,-1/3,-1/4,-2,-3$; orders $s=1/2,3/2,5/2,7/2$ have $x=-1,-1/2,1/2$ and, for $s>1$, $x=1$; the zero-argument anchor at weight 2 is exact, and the earlier dilogarithm at $1/9$ is retained.
## Mathematical sources
- DLMF 25.12: polylogarithms and Fermi/Bose integrals: https://dlmf.nist.gov/25.12
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.