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# Argument selection for T542
This note follows the coordinated skill archived in
`agents/runs/20261027T075514Z-build-skills/manifest.json`, copied to the attachment
`skill-manifest.json`. Both local and live versions have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
It is the builder's justification, not an independent critique or acceptance.
The single named quantity is the parity-twisted spectral zeta value
Z_N^P(n)=sum_{k>=0} (-1)^k E_{N,k}^{-n}. The positive, simple full-line
spectrum is indexed from its even ground state, so this is Neumann minus
Dirichlet. The mathematical parameter domain is N a positive integer and n
an integer satisfying n>(N+2)/(2N). Although the alternating series has a
wider convergence domain, this table deliberately uses the domain in which
both boundary traces converge absolutely, as does its companion full-trace
family. There is no finite-part prescription.
## Degree N: discrete_index
N labels the homogeneous oscillator H_N=-d^2/dx^2+|x|^N. Voros (2022),
Section 1.1, equations (2)-(5), explicitly treats N=1,2,... with this scale and
parity convention. It includes both polynomial even-degree wells and confining
odd absolute-power wells. It is not a continuous coupling renamed an index.
The selection is every degree 1<=N<=12. This is a noncanonical practical prefix,
not a claim of a special application at each degree. It spans the linear Airy
and harmonic controls, the cubic and quartic cases, the sextic case with a
closed parity second moment, and several progressively steeper wells of each
parity. Including both even and odd degrees gives broader coverage than
extending just the famous even quartic and sextic cases. Higher degrees are
mathematically admissible; twelve is a modest reference extent rather than a
storage or entry-count target. This integer prefix is independent of numeral
base. It is not an exploratory continuous-input grid.
## Inverse-power order n: discrete_index
n indexes inverse powers of one operator with a parity insertion. The operators
H_N^{-n} and the Taylor coefficients of the logarithmic determinant supply a
family-specific reason for integer moments; see Voros (2022), equation (6),
and Watkins (2012), equations (2.2) and (2.9). The varying order does not select
different named quantities.
We include n=2 at every degree 1<=N<=12, and n=1 at 3<=N<=12. Thus there are
22 entries, ordered by increasing N then increasing n. The pair (1,1) and
(2,1) fail absolute convergence and are excluded, even though parity
cancellation can define an alternating sum. Higher moments are meaningful,
but the first two have explicit Gamma and generalized hypergeometric formulas
across the whole selected degree family (Watkins (2.5a,b); Voros (1983),
Appendix C, (C.22), (C.28), (C.33)). The same appendix explains the additional
multiple-integral difficulty at higher orders. We prioritize these two useful
moment slices across degrees over depth at just one oscillator. We do not
form a Cartesian product and then regularize its missing cells.
## Fixed choices and interaction
The kinetic coefficient and potential coefficient are exactly one, without
factors of one half. This is the scale of the checked sources. For a potential
coefficient g>0, rescaling x=g^{-1/(N+2)}u gives
H_{N,g}=g^{2/(N+2)}H_{N,1} up to unitary equivalence, so a separate sampled g
would add a trivially rescaled continuum. We keep g=1 and do not claim it is
a selected reference family. The additive energy shift is fixed at zero,
where the inverse moments and logarithmic-determinant expansion apply; it is
not the radial lambda in Watkins's formulas. His alpha=0 and radial
lambda=+1/2 or -1/2 encode the Dirichlet and Neumann spectra respectively.
Their difference is the named output, not a boundary selector parameter.
All eigenvalues are positive, and integer inverse powers need no complex
branch choice. The 22-pair enumeration and completeness note state the same
selection. No auxiliary rational mesh, quantile, or reference_sample group
occurs.
## Sources checked
- A. Voros, arXiv:2206.14482v3, Section 1.1, (2)-(6), and Section 2.1.
- J. Watkins, DOI 10.1142/S140292511250026X, Sections 1 and 2; (1.1),
(1.2), (1.6), (2.2), (2.5a,b), (2.9), (2.13), and Example 2.2.
- A. Voros, Ann. IHP Phys. théor. 39 (1983), Appendix C, printed pages
327-330. The full and parity formulas (C.22), (C.28), and (C.33) were
inspected in the scanned pages, including their Gamma factors.
- NIST DLMF 9.2.1, 9.2.3, 9.2.4 for the Airy controls.
The numerical method is heuristic agreement, not certified quadrature or a
certified tail enclosure. Benchmark and accuracy evidence are separate from
the mathematical rationale for selecting these indices.