# Argument selection

Integer orders $\nu=m$ label cylindrical Fourier modes $e^{im\varphi}$. Positive half-integer orders $\nu=\ell+1/2$ correspond to spherical harmonic radial orders through the standard relation with spherical Bessel functions. These are structural order families, not a rational order mesh. The prefixes $m,\ell=0,\ldots,5$ and six positive critical arguments per order are explicit finite lookup depths, not canonical cutoffs. Each argument $c_k$ is the unique zero of the derivative of the tabulated cylindrical Bessel function in its stated rational interval. The entries are extremal heights, not zero locations. The labels enumerate the isolating intervals; a global root-count proof is not claimed. At order zero the zero of $J_0^{\prime}$ at the origin is excluded. For half-integer order these are extrema of $J_\nu$ or $Y_\nu$, not of the differently scaled spherical radial functions.

## Limitations

No application-specific demand is asserted for every individual height. Both order families and the stationary arguments have intrinsic justification; the finite lookup depths are stated choices. This certificate proves isolation within each defining interval, not a global ordinal count.

## Primary references

- https://dlmf.nist.gov/10.73
- https://dlmf.nist.gov/10.47
- https://dlmf.nist.gov/10.21
- https://dlmf.nist.gov/10.6
- https://dlmf.nist.gov/10.51

The certified argument definitions are in `selection.json`. Run `python generate.py` to verify; run `python checks.py T187` for the independent checks. Neither command writes. Earlier attachments describe historical grids and remain available for provenance.