# Argument selection for T51

The pairs $(1,\sqrt2)$ and $(1/\sqrt2,1)$ are the lemniscatic and unit-normalized complementary-modulus initial pairs. The first gives the reciprocal of Gauss's constant; homogeneity relates the second to it. The integral representation of the AGM CITE{SelectionAGM} and the beta integral CITE{SelectionBeta} give CITE{formula-lemniscatic-agm}. The square roots are exact arguments, not rounded decimal inputs. These are two distinguished normalizations, not a sampling of integer pairs.

Coverage: it holds the lemniscatic pairs $(1,\sqrt2)$ and $(1/\sqrt2,1)$; arbitrary integer pairs and their homothetic copies are not enumerated.

## Primary mathematical evidence

- DLMF 19.8(i): Gauss's Arithmetic-Geometric Mean: https://dlmf.nist.gov/19.8#i
- DLMF 5.12: Beta Function: https://dlmf.nist.gov/5.12

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 T51 --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.