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# Argument selection for T176
The endpoints $x=1,-1$ give zeta and alternating-zeta values. The dilogarithm reflection formula is symmetric at $x=1/2$ CITE{SelectionPolylog}. The trilogarithm at $1/2$ is the classical reducible value in CITE{formula-half}. Broadhurst's three-loop calculation uses $\operatorname{Li}_4(1/2)$ in equation (14), and identifies $\operatorname{Li}_2(1/4)$ as a weight-two primitive after equation (143) CITE{SelectionBroadhurst}. The argument $-1/2$ is its real distribution-identity partner: $\operatorname{Li}_2(1/4)=2\operatorname{Li}_2(1/2)+2\operatorname{Li}_2(-1/2)$. The classical Landen pair $1/3,1/9$ in CITE{formula-landen} is retained: $1/3$ is the transform of $-1/2$ under $z\mapsto z/(z-1)$; the reflection and distribution formulas in CITE{SelectionPolylog} give the displayed reduction. These named reductions and documented applications do not justify every rational argument of bounded denominator.
Coverage: it holds $\operatorname{Li}_s(1)$, $\operatorname{Li}_s(-1)$ and $\operatorname{Li}_s(1/2)$ for $s=2,3,4$, together with the dilogarithms at $1/4,-1/2,1/3,1/9$, for the reasons in CITE{argument-selection}; the orders are the first three non-elementary integer weights, including the weight-four diagram application, not an exhaustive rational-argument coverage.
## Primary mathematical evidence
- DLMF 25.12: Polylogarithms, reflection and distribution formulas: https://dlmf.nist.gov/25.12
- D. J. Broadhurst: Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity: https://arxiv.org/abs/hep-th/9803091
The mathematical deductions in the selection paragraph are distinguished from a source publishing this exact finite list. The range cutoff is a reference limit, not a theorem that every included dimension or order has a separate documented application.
## Reproduction and history
Run `python generate.py` to verify and `python checks.py T176 --live` for independent checks. Neither writes. The source uses exact rational arithmetic and certified Arb balls, with mpmath as a separate agreement check. Retained numeric strings and annotations are copied from, and verified against, the website export; the generator is not claimed to have authored those annotations. Removed sample identities and old attached sources remain in website revision history. No identity is reused for another argument.