# Balanced reference selection for T202

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 first twenty extremal heights specified in CITE{argument-selection}; the selection is not exhaustive over the real domain. In addition, it includes the simple real arguments $\pm1/2,\pm1,\pm2,\pm4,\pm8$ and unit-cubic arguments $k/3^{1/3}$ for $1\leq|k|\leq2$. The exploratory families are identification references with practical finite bounds, not claims of documented applications for every input. All prior entries are retained.

The exploratory reference arguments $x=\pm1/2,\pm1,\pm2,\pm4,\pm8$ give a sparse set of simple real inputs on both sides of the Airy turning point. They complement root-indexed values with near-origin, moderate and larger arguments. This is a bounded identification reference, not a claim that each point has a documented application CITE{BalancedAiryEquation}.

The change of variable $t=3^{1/3}u$ in the defining cubic integrals gives $x=k/3^{1/3}$ when the cubic coefficient is one and the linear coefficient is the integer $k$. The finite range $1\leq|k|\leq2$ is an exploratory simple-coefficient reference choice, not a source-prescribed cutoff CITE{BalancedCubicIntegrals}.

## Normalization formulas

$\int_0^\infty[e^{-u^3+ku}+\sin(u^3+ku)]\,du=\pi3^{-1/3}\operatorname{Bi}(k/3^{1/3})$.

## Primary references

- [DLMF 9.2: Airy differential equation](https://dlmf.nist.gov/9.2)
- [DLMF: defining cubic integral representations](https://dlmf.nist.gov/9.5)

## 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.