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1471 bytes, as of the version from 2026-10-27 01:00 (current). Recorded here, not run.
# Balanced reference selection for T51
The positive AGM is invariant under one arithmetic/geometric iteration and is related to a complete elliptic integral CITE{BalancedAGM}. The retained lemniscatic values are supplemented by a unit-normalized integer-input family $(1,n)$ and a quadratic-input family $(1,\sqrt d)$. These are bounded exploratory references for identifying means and elliptic integrals with simple coefficients, not claims of a published application for each pair. Perfect squares are omitted from the second family because their input pairs already occur in the first; no Cartesian box of both initial values is taken.
Coverage: it retains both lemniscatic pairs and adds $(1,n)$ for $1\leq n\leq16$ and $(1,\sqrt d)$ for nonsquare $2\leq d\leq15$; these practical simple-coefficient bounds are exploratory reference choices, and the diagonal anchor is exact.
## Mathematical sources
- DLMF 19.8: AGM iteration and elliptic integrals: https://dlmf.nist.gov/19.8
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.