# Argument selection for T541

The quantity is the full trace Z_N(n) of the nth inverse power of
H_N = -d²/dx² + |x|^N on L²(R), with eigenvalues counted once. It is one named
quantity with two integer arguments, not a selector between full and parity
twisted traces. The latter belongs to a separate proposal.

## Degree N: discrete_index

Positive integer degrees label the homogeneous oscillator family. Watkins,
*Spectral zeta functions of a 1D Schrödinger problem*, equations (1.1), (1.2),
(2.13), and the paragraph defining the full potential |x|^(2M), establish the
family and its full-line meaning with M=N/2. Voros, *Exercises in exact
quantization*, Section 2 and Appendix A.1, explicitly treats the integer-degree
family |q|^N, including odd degrees.

The selection is every degree 1 through 12. This noncanonical practical prefix
covers the linear and harmonic controls, cubic and quartic wells, and several
steeper wells of each parity. The degree is a genuine discrete family index;
this is not a continuous coupling disguised as an integer. No particular
application is claimed for each degree. Extending this single prefix is less
valuable here than covering both of the first two convergent inverse moments.
The cutoff is editorial and computational, not a mathematical endpoint.

## Inverse-power order n: discrete_index

The domain consists of positive integers n>(N+2)/(2N), the absolute convergence
condition obtained from the eigenvalue asymptotic in Watkins Section 1.
Integer orders are inverse-operator moments and coefficients of a logarithmic
spectral determinant (Watkins (2.9)). We select orders one and two because
Watkins (2.5a,b) gives explicit Gamma and generalized hypergeometric formulas
for that entire slice. Higher orders require further analysis or multiple
integrals. This is a meaningful shallow slice across degrees, rather than a
Cartesian product formed to fill a table.

The exact selection is {(N,2): 1<=N<=12} union {(N,1): 3<=N<=12}.
There are 22 entries, ordered by increasing N then n. The two omitted first
moments diverge; no finite part or regularized value is substituted.

## Fixed choices

Both kinetic and potential coefficients are one, consistent with Watkins
(1.1) and Voros Appendix A. The potential is |x|^N for odd as well as even N;
it is not x^N on the full line for odd N. The additive energy shift is fixed at
zero, rather than sampled. The whole full-line spectrum is used without a
parity insertion. Half-line Dirichlet and Neumann spectra are intermediate
computations, whose union is the full spectrum. The ground-state index is
zero and even parity. All eigenvalues are positive and counted once, so the
integer inverse powers entail no branch ambiguity.

Watkins uses a radial parameter lambda=+/-1/2 to select these two boundary
conditions; that symbol is not an energy shift or a new table parameter.
Neither a reference_sample nor a mixed parameter group is present. These
classification statements remain subject to the controller's independent
argument-selection critique.

## Scope and evidence

The generator enumeration and the table completeness note state the same
selection. Measurements and independent formula checks are recorded in
numerical-evidence.md; those establish numerical reliability, not argument
relevance. The 100-digit aim is achievable with 150/200-digit evaluations and
varied asymptotic truncations; it does not motivate the selection itself.

The coordinated skill manifest is
agents/runs/20261027T064642Z-build-skills/manifest.json. Both archived versions
were read and match SHA-256
e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.

Checked sources:
- https://arxiv.org/abs/1110.2004 (version 2), Sections 1 and 2.
- https://arxiv.org/abs/math-ph/0005029 (version 2), Section 2 and Appendix A.1.
- https://www.numdam.org/item/AIHPA_1983__39_3_211_0/, Appendix C,
  equations (C.22), (C.23), and (C.28), for independent full-trace formulas.
- https://dlmf.nist.gov/9.2, Airy equation and initial values for the linear case.