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# Argument selection for T540
Policy: the archived manifest is
`agents/runs/20261027T061240Z-build-skills/manifest.json` in this source snapshot.
Both archived versions, `skill-local.md` and `skill-live.md`, were read and match
SHA-256 `e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The same digest was checked against the canonical snapshot skill. The attached
`skill-evidence.json` records the paths and digests. This is the builder's
justification, not an independent argument-selection acceptance.
## Quantity and conventions
One number is stored for each degree: the zeta-regularized Neumann determinant
D_N^+(0) for -d²/dx²+x^N on [0,infinity), y'(0)=0, square-integrable at infinity.
Both coefficients are one, matching Voros, Exercises in exact quantization,
Appendix A, equation (125). The zero is a fixed additive spectral shift, not
a sampled energy or an eigenvalue. Zeta regularization is equation (7), with
analytic continuation before differentiation at s=0. The spectrum is positive,
so its complex powers use the real logarithm of each positive eigenvalue.
For odd N the corresponding full-line problem uses |x|^N. Its even-parity
sector gives this Neumann spectrum. No radial coupling or further variable is
sampled. Dirichlet determinants concern a different spectrum and belong to
T539, not to another selector in this table.
## Sole parameter N: discrete_index
The mathematical domain is every positive integer N. This is the documented
integer-degree homogeneous-oscillator family, not a grid of a continuously
varying coupling. The checked sources are Voros, math-ph/0005029v2, Appendix A,
equations (125), (135), (136), (148), (160), and Voros, 2206.14482v3, sections
1.1 and 3.1. They cover both even polynomial wells and odd absolute-power wells,
and identify the linear, harmonic and quartic cases. They do not select N=12
as a canonical endpoint.
Default enumeration is exactly N=1,2,...,12 in increasing order. This practical,
noncanonical prefix includes the two distinct solvable controls, cubic and
quartic wells, and several further steeper wells of both parities. It offers
breadth across the discrete family before a long listing of high degrees.
No claim is made that each selected degree has its own famous application.
Evaluation is cheap and produces uniformly short real entries, so size does
not force the cutoff. The reason to stop is a bounded low-degree reference,
not a target count or an assertion of mathematical completeness. Extending
past 12 is possible but is not needed to represent another kind of degree.
Neither odd degrees nor less familiar intermediate degrees have been pruned.
There is no reference_sample group, rational grid, auxiliary root index,
Cartesian product, or excluded statistical quantity. The finite prefix is a
bound on one genuine discrete family. The parameter declaration states N>=1;
the completeness note, program and generator state the selected prefix.
## Independent review still required
The critic must examine this rationale and the numerical checks independently,
then fingerprint the live selection and at least `generate.py`, which alone
controls enumeration and values. `checks.py` is a separate verification program;
`argument-selection.md` records the rationale. The builder has not written an
acceptance sidecar and has not offered the draft for review.