# Argument selection for T628

The single stored quantity is beta_k = T/|W|, the rotational kinetic energy
relative to the magnitude of negative gravitational potential energy at the
kth axisymmetric bifurcation of a homogeneous incompressible Newtonian
Maclaurin spheroid. The only free parameter is the integer k >= 1. It fixes
the neutral perturbation's Legendre degree 2k+2 and thereby the unique
positive root xi_k of AKM equation (3). Neither eccentricity, density, size,
nor rotation is an independently sampled input. The positive energy ratio
is used, not T/W. All dimensional factors cancel.

Category: discrete_index. The default enumeration is exactly k=1,...,10,
in increasing order. This is an initial segment of the discrete neutral-mode
hierarchy, not a sampling mesh in eccentricity or an arbitrary coupling
attached to a root index.

Checked source: Ansorg, Kleinwächter and Meinel, MNRAS 339 (2003), 515–523,
DOI 10.1046/j.1365-8711.2003.06190.x, arXiv astro-ph/0208267, Section 2,
equations (3)–(4), Table 1, and Section 4's definitions of T and W.
https://arxiv.org/html/astro-ph/0208267v1
The source fixes the real oblate integral convention, physical model and
branch indexing. Its Table 1 supplies an independently published coarse
energy-ratio comparison for every selected branch; the table does not infer
high precision from those rounded values.

Ten is a practical, noncanonical, notation-independent cutoff. It covers
all branches in the source's comparison table and its branch investigation,
including the first Dyson-ring branch, both parity classes of subsequent
continuations, and later reference equilibria beyond the source's stated
axisymmetric stability threshold. Extending k farther would deepen an already
represented mode hierarchy without adding a new continuation type. Keeping
only the first famous value would miss both the even and later odd branches.
All ten stored values require only 102 characters each at 100 significant
digits; the dry run measured a maximum of 145 characters before rounding and
an entries block of about 1.8 KB. Readability and computation permit this
whole justified prefix without expanding it to a size target.

The table has no mixed or exploratory continuous-input group. Its companion
T627 stores the eccentricities of these same configurations. The shared range
is a shared set of equilibria, not a common mesh imposed on unrelated
functions. Other invariants are outside this table. Jacobi and other
non-axisymmetric bifurcations require different mode equations and remain
outside this hierarchy. Values are recomputed from the root equation; the
rounded numbers in T627 and AKM are not arguments of the generator.

The parameter declaration describes k >= 1. The completeness note states the
finished prefix and its reason. Both this note and the Programs snippet agree
with the generator's enumeration.

This is the builder's justification. Independent acceptance belongs to a
separate critique session. Its fingerprint should cover generate.py and
checks.py; no imported local numerical helper controls the selection.
The controller, not the builder, owns offering and review.

Policy manifest: agents/runs/20261030T190951Z-build-skills/manifest.json.
The archived local and live copies were read and SHA-256 verified, both:
dedecbdca15a728b6e594bfa79f0d149896fca336080ac27b53f513e0b16cd2e.
The manifest is reproduced in skill-evidence.json.