# Balanced reference selection for T207

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 twenty extremal heights at $x=\sqrt{2n}$, $1\leq n\leq20$; the selection is not exhaustive over the real domain. In addition, it includes the paired axis-tangent coordinates through $n=8$, four diagonal-tangent arguments, the four unit-quadratic references and simple arc lengths $1/2,1,2,3,4$, as described in CITE{balanced-cornu}, CITE{balanced-quadratic} and CITE{balanced-simple}. The exploratory families are identification references with practical finite bounds, not claims of documented applications for every input. All prior entries are retained.

Cornu's spiral has coordinates $(C(x),S(x))$, arc length $x$, and tangent angle $\pi x^2/2$ CITE{BalancedCornu}. The reference set includes both coordinates at the axis-tangent arguments $\sqrt{2n}$ and $\sqrt{2n-1}$ through $n=8$, retaining the longer existing extrema prefix. The four diagonal-tangent arguments $\sqrt{(2n-1)/2}$, $1\leq n\leq4$, have tangent slope $\pm1$.

The inputs $x=n\sqrt{2/\pi}$, $1\leq n\leq4$, correspond to the unit-phase integrals with $\sin(t^2)$ or $\cos(t^2)$ and simple integer upper limits. This bounded exploratory family makes the normalization change explicit CITE{BalancedFresnelDefinitions}.

The additional arc lengths $x=1/2,1,2,3,4$, where not already included as tangent points, are a short exploratory reference for coordinates of Cornu's spiral. They are not an interpolation mesh.

## Normalization formulas

$\int_0^n\sin(t^2)\,dt=\sqrt{\pi/2}\,S(n\sqrt{2/\pi})$.

## Primary references

- [DLMF 7.20(ii): Cornu's spiral](https://dlmf.nist.gov/7.20#ii)
- [DLMF 7.2(iii): Fresnel normalization](https://dlmf.nist.gov/7.2#iii)

## 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.