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# Argument selection for T178
The signatures $r=2,3,4,6$ are the classical and alternative-base theories described in CITE{BBG1995}. For each signature, $x=1/2$ is fixed by exchanging the complementary arguments $x$ and $1-x$, and Gauss's second summation theorem gives the gamma quotient in CITE{formula-half} CITE{SelectionHalf}. This selects the half argument without appealing to a small denominator. Other singular moduli require their own arithmetic specification and are not replaced by a rational mesh.
Coverage: it holds the complementary-symmetry value $F_r(1/2)$ for each signature $r=2,3,4,6$; this is not a catalogue of all singular or arithmetic moduli.
## Primary mathematical evidence
- DLMF 15.4.28: Gauss's second summation theorem: https://dlmf.nist.gov/15.4.E28
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 T178 --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.