# Argument selection for T630

This is a build-stage justification, not the independent critique acceptance.
The governing instructions are archived in
`agents/runs/20261030T195647Z-build-skills/manifest.json`; both local and live
versions have SHA-256
`dedecbdca15a728b6e594bfa79f0d149896fca336080ac27b53f513e0b16cd2e`.
The same manifest is attached as skill-manifest.json.

The sole free parameter is the integer k>=1, the axisymmetric branch index.
Its role is `discrete_index`: the neutral perturbation has Legendre degree
2k+2 in Ansorg, Kleinwächter and Meinel (AKM), Section 2, equations (3)–(4).
The integral definition of the real oblate radial functions and the unique
positive root of equation (3) determine the spheroid. There is no independently
chosen eccentricity, density, size, rotation rate, coupling or probability.
The squared angular momentum is J^2 rho^(1/3)/(4 pi G M^(10/3)), exactly AKM (7).
The physical scales cancel. They are not additional table arguments.

The enumeration is every integer k=1,...,10, in increasing order. This finite
prefix is a practical, noncanonical cutoff, independent of numeral notation.
AKM investigated these ten branches and supplies a coarse angular-momentum
check for each in Table 1. The prefix includes the Dyson-ring branch, both parity
classes of higher branches, and later mathematical equilibria past the source's
axisymmetric dynamical-instability threshold. It does not claim stability.

Breadth was assessed before depth: this named quantity ranges over one neutral-mode
hierarchy. The prefix covers the distinct parity classes without restricting
coverage to the famous first branch. The Jacobi point and other non-axisymmetric
landmarks satisfy different perturbation conditions; continuation endpoints need
non-spheroidal surface calculations. They are outside this named object's scope.
The eccentricity, energy ratio, and angular velocity at these same configurations
are companion quantities, not additional rows or parameter choices here. All four
family proposals share k because they describe identical equilibria, not because
unrelated continuous functions received the same mesh.

There is no exploratory continuous reference group, decimal grid, Cartesian
product or quantile selection. The domain remains k>=1; `complete: no` and the
completeness note describe the finite prefix. The generator defaults to exactly
this selection and the stored entries have the same order. Ten values comfortably
fit the display: dry_run.py measured a longest ball representation of 145
characters and a block of 1.9 KB. Size is a check on this selection, not its reason.

Sources checked: AKM, MNRAS 339 (2003), 515–523,
DOI 10.1046/j.1365-8711.2003.06190.x, arXiv:astro-ph/0208267v1,
https://arxiv.org/html/astro-ph/0208267v1, Sections 1, 2 and 4, equations (3),
(4), (7), and Table 1. Live T627 and T628 establish the companion conventions;
T629 was also read as an in-progress draft. The original family issue #251 was
read before work. Its provenance remains in campaign evidence, not table prose.

A separate critique must inspect this justification, the live document and
all attached source dependencies, perform its own numerical check, and create
its own fingerprinted -arguments.json. This build creates no acceptance sidecar.