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# Argument selection: Airy zeta values (T531)
## Quantity and normalization
Let 0 > a_1 > a_2 > ... be the zeros of the conventional Airy function Ai.
The one stored quantity is Z_D(n) = sum_{k>=1} (-a_k)^(-n). Its operator is
H = -d^2/dx^2 + x on [0,infinity), with u(0)=0 and decay at infinity.
The kinetic coefficient, potential slope and additive shift are fixed at
1, 1 and 0. No coupling, root cutoff, scale or boundary-condition selector
is a free parameter. Each eigenvalue is counted once. Negative-root sums
are (-1)^n Z_D(n), not a second set of entries.
## Every free parameter
The only parameter is n, type Z, the integer inverse-power order n >= 2.
This is a discrete index of traces of powers H^(-n), not a grid of samples
of a continuously varying zeta argument. These same values are coefficients
of log(Ai(z)/Ai(0)) after its linear term. The spectral growth
-a_k ~ [(3*pi/2)(k-1/4)]^(2/3) forces n > 3/2 for an unregularized sum;
therefore 2 is the first convergent integer order. No analytic-continuation
values are selected.
The enumeration is every integer 2 <= n <= 20, in increasing order. Twenty
is an editorial initial-segment endpoint, not a singularity or a distinguished
integer. A small complete prefix supplies the low moments, determinant
coefficients and progressively sharper ground-energy bounds, while avoiding
an unmotivated expansion in the number of entries. The source exhibits the
use of higher integer moments for ground-state estimates (Crandall, equations
1.14-1.15 and 4.18-4.19). There is no continuous-argument grid and no omitted
interior order. The default generator, Programs and completeness note use
exactly this selection. Larger orders belong to the same mathematical family.
## Checked evidence
- R. E. Crandall, *On the quantum zeta function*, J. Phys. A 29 (1996),
6795-6816, Section 1 (inverse powers and ground-energy bounds), Section 4,
especially 4.20 and 4.22 (Airy-zero sums and convergence).
https://www.reed.edu/physics/faculty/crandall/papers/a62111.pdf
DOI: 10.1088/0305-4470/29/21/014.
His operator in 4.12-4.16 has eigenvalues (-a_k)/2; those formulas must be
divided by 2^n to match this table. Formula 4.20 is already in root units.
- A. Voros, *Airy function (exact WKB results for potentials of odd degree)*,
J. Phys. A 32 (1999), 1301-1311, arXiv:math-ph/9811001v2,
equations 5, 9, 29, 31, 42.2, 43.1 and 43.3. The odd parity sector Z^-
is the half-line Dirichlet spectrum in this normalization.
https://arxiv.org/pdf/math-ph/9811001
- NIST DLMF 9.9(i) and 9.9(iv) fix the signs, order and growth of Ai zeros;
9.2.1, 9.2.3 and 9.2.4 fix the differential equation and initial constants.
https://dlmf.nist.gov/9.9 and https://dlmf.nist.gov/9.2
## Instruction provenance and independent review
Read manifest: agents/runs/20261026T224836Z-build-skills/manifest.json.
Both archived skill versions, the canonical source skill and the freshly
fetched live /skill have SHA-256
b4d51ded0d52cfe3dcc0859de0845c3879b0c2363273ba7e809a7e06ded40b18.
This is the builder's justification. A separate critic must assess n and bind
its own sidecar to the live selection and generate.py. No builder acceptance
or offer is asserted by this file.