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2153 bytes, as of the version from 2026-10-27 01:09 (current). Recorded here, not run.
# Balanced reference selection for T197
Run `python generate.py` for complete offline reproduction or `python generate.py --independent` for the independent checks. Install `numberdb`, `python-flint`, and `mpmath`. Neither command writes to the website.
it holds the origin and the one- through six-standard-deviation references $x=n/\sqrt2$, $1\leq n\leq6$, under CITE{argument-selection}; the selection is not exhaustive over the real domain. In addition, it includes the simple inputs $-1,-1/2,1/2,1,3/2,2,3,4$ and $\sqrt n$ for $1\leq n\leq6$, counting equivalent arguments only once. The exploratory families are identification references with practical finite bounds, not claims of documented applications for every input. All prior entries are retained.
The simple inputs $x=-1,-1/2,1/2,1,3/2,2,3,4$ supplement the conventional standard-deviation references. They are a bounded exploratory Gaussian-integral reference, including negative-side reflection examples and moderate positive tails, not quantile cutoffs CITE{BalancedGaussianDefinitions}.
Integer quadratic coefficients in the truncated Gaussian integral $\int_0^1e^{-nt^2}\,dt$ select $x=\sqrt n$. The exploratory range $1\leq n\leq6$ is finite and noncanonical; coincident arguments such as $\sqrt2=2/\sqrt2$ are represented only once CITE{BalancedGaussianDefinitions}.
## Normalization formulas
$\int_0^1e^{-nt^2}\,dt=\frac{\sqrt\pi}{2\sqrt n}\operatorname{erf}(\sqrt n)$, $n>0$.
## Primary references
- [DLMF 7.2(i): error function and complement](https://dlmf.nist.gov/7.2#i)
## Provenance
The current generate.py reproduces the complete attached selection, including the retained entries. Their original written values and entry comments are preserved rather than reassigned to a new source. The selection manifest records the predecessor revision and source hashes; earlier generators and checks remain historical provenance, not the enumeration of this extension.
The source manifest identifies the predecessor revision and records whether the preparation used a fresh export or a stale cached repaired readback. It is not authorization to overwrite a later revision.