back to table · edit · history · where entries came from · files · download
1768 bytes, as of the version from 2026-10-27 01:03 (current). Recorded here, not run.
# Argument selection for T353
The complement identity $I_x(a,b)+I_{1-x}(b,a)=1$ CITE{SelectionBetaSymmetry} fixes $x=1/2$ when $a=b$, so $B(1/2;a,a)=B(a,a)/2$. For independent chi-square variables with the same $n$ degrees of freedom, their ratio to their sum has beta parameters $a=b=n/2$ CITE{SelectionStatistics}; exchange symmetry selects its mean and median $1/2$, equivalently the unit median of the $F(n,n)$ ratio. This selects a symmetry line, not a Cartesian box of distribution parameters. The entries are unregularized beta integrals; rational values are retained exactly.
Coverage: it holds $B(1/2;n/2,n/2)$ for $1\leq n\leq10$, the equal-degree chi-square ratio symmetry values; the dimension bound is a finite reference prefix and no asymmetric parameter box is claimed.
## Primary mathematical evidence
- DLMF 8.17.4: Incomplete Beta Symmetry: https://dlmf.nist.gov/8.17.E4
- DLMF 8.23: Beta and Chi-square Distributions: https://dlmf.nist.gov/8.23
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 T353 --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.