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# Balanced reference selection for T250
The earlier trigamma and tetragamma arguments are retained. Higher integer derivative orders are included on the small arithmetic slices $x=1,1/2,1/3,2/3,1/4,3/4$, selected by recurrence, reflection and multiplication identities CITE{BalancedPolygamma}. At $x=1$ and $1/2$ the values are zeta multiples. Orders through 12 are a practical reference prefix, not a claim of completeness in derivative order; higher orders do not inherit the full lower-order argument mesh.
Coverage: it retains all earlier arguments at orders 1 and 2, and adds orders $3\leq n\leq12$ at $x=1,1/2,1/3,2/3,1/4,3/4$; the higher-order arithmetic slices broaden order coverage without multiplying the entire earlier argument range.
## Mathematical sources
- DLMF 5.15: polygamma special values and multiplication: https://dlmf.nist.gov/5.15
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.