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# Balanced reference selection for T515
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 $a_n$ for $1\leq n\leq20$, with $a_n$ defined in CITE{airy-arguments}; these are the arguments in CITE{brownian-application}; these are finite prefixes of intrinsic indexed families, not a complete real-argument table. In addition, it includes the simple real arguments $\pm1/2,\pm1,\pm2,\pm4,\pm8$ and unit-cubic arguments $k/3^{1/3}$ for $1\leq|k|\leq3$. 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 $1\leq|k|\leq3$ is an exploratory simple-coefficient reference choice, not a source-prescribed cutoff CITE{BalancedCubicIntegrals}.
## Normalization formulas
$\int_0^\infty e^{-u^3+ku}\,du=\pi3^{-1/3}\operatorname{Hi}(k/3^{1/3})$.
## Primary references
- [DLMF 9.2: Airy differential equation](https://dlmf.nist.gov/9.2)
- [DLMF: defining cubic integral representations](https://dlmf.nist.gov/9.12)
## 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.