# Argument selection for T539

The table holds one quantity, the zeta-regularized Dirichlet determinant at zero
additive energy shift for the homogeneous oscillator. It does not select a
boundary condition, an inverse-power moment, or another quantity per row.

## Policy and source consulted

This assessment uses both matching archived skill copies identified in
`agents/runs/20261027T052752Z-build-skills/manifest.json`. Each has SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The family proposal is numberdb-data issue 218, read through queue.py before
screening. This builder assessment does not replace the independent critic's
argument review or its required fingerprint sidecar.

The checked primary source is André Voros, *Exercises in exact quantization*,
J. Phys. A 33 (2000), 7423–7450, corrected arXiv:math-ph/0005029v2,
https://arxiv.org/abs/math-ph/0005029 . The half-line boundary conditions are
specified in the abstract and section 1.2. Appendix A defines the homogeneous
family; equations (135)–(136) give its zero-energy solution and determinants.
Appendix A.2.2 covers the linear/Airy member, and A.2.3 the harmonic member.

## Every free parameter

There is exactly one: `N`, type `Z`, with domain N >= 1. Its category is
`discrete_index`. It is the integer degree of a documented family of monomial
half-line potentials. Even degrees also give polynomial full-line wells; odd
degrees give the corresponding |x|^N wells on the full line. This is a discrete
oscillator family, not a continuous coupling sampled on an integer mesh.

The selection is every N from 1 through 12, in increasing order. The upper
bound is a noncanonical practical reference prefix. It includes the linear
and harmonic analytic controls, cubic and quartic members, and several steeper
wells of each parity. Its purpose is to identify determinants arising from
low-degree homogeneous spectral problems. All members need not have separately
famous applications. There is no arbitrary decimal grid and no Cartesian
product. The selection is independent of how the integers are written.

## Breadth and finite extent

The breadth review keeps odd as well as even degrees and keeps a contiguous
prefix rather than selecting only the two exactly solvable spectral controls.
There are six degrees of each parity. Extending much further is computationally
cheap, but no additional family-specific identification purpose was established
for a long listing of high degrees. Twelve is an editorial cutoff, not a
convergence or precision barrier, and no numerical size target selected it.
The measured written lengths and precision margins are in numerical-evidence.md.
The mathematical domain remains all positive integer degrees; the finite prefix
is stated only in enumeration and the completeness note.

No reference_sample or mixed group is hidden within this selection. There are
no independently sampled rational arguments. The derived rational nu=1/(N+2)
is an evaluation variable fixed by N, not an additional parameter.

## Fixed conventions

The potential coefficient and kinetic coefficient are 1, matching the source's
operator -d^2/dx^2+x^N. In particular no kinetic factor 1/2 is inserted. These
choices fix the spectral scale. The Dirichlet boundary y(0)=0 is part of the
named object; Neumann spectra require their own table. Square-integrability
at infinity selects the decaying solution. The energy shift is fixed at 0:
the quantity is the constant term of the spectral determinant, not a grid of
shifted determinants. Eigenvalues are positive and counted once. The spectral
zeta starts as a convergent sum for Re(s)>(N+2)/(2N), then is continued to 0;
the determinant uses its s derivative, not an energy derivative or the
unregularized eigenvalue product. Full-line odd parity agrees with the
half-line Dirichlet condition for the even potential |x|^N.

## Independent review boundary

The default enumeration, table parameter declaration, completeness note and
stored identities must be checked by the separate critic. The sole generator
source controlling the selection is generate.py; checks.py checks it but does
not supply generator inputs. The critic should obtain the current fingerprint
from the live draft and attachments and use its own NUMBERDB_RUN_ID. This file
is the builder's justification, not an independent acceptance.