# Argument selection for T627

This table stores only the eccentricity at an axisymmetric bifurcation of the
Maclaurin sequence. Its sole free parameter is the integer branch index k >= 1.
The index fixes the neutral perturbation's Legendre degree 2k+2. Density, mass,
rotation rate and eccentricity are not independent sampled inputs. The
normalization is e = sqrt(1-c^2/a^2), with equatorial semiaxis a and polar
semiaxis c. The real oblate radial functions are fixed by the positive integral
in the definition, not by a complex logarithm convention.

Category: `discrete_index`. The generator enumerates exactly k=1,...,10,
in increasing order. This is a complete finite prefix of an infinite physical
and mathematical hierarchy, not a continuous mesh disguised as an index.

The source is Ansorg, Kleinwächter and Meinel (2003), Section 2, equations
(3)–(4), and Table 1; https://arxiv.org/html/astro-ph/0208267v1 . Section 1
states that their branch investigations reach k=10 and explains the different
continuations for odd and even branches. Both parity classes are present.
The first branch leads to Dyson rings; the prefix also includes the successive
core-ring and two-ring branches. Later branches remain valid equilibria even
where dynamical instability precludes interpreting them as stable figures.
The table makes no stability claim.

Ten is a practical, noncanonical, notation-independent cutoff. It retains all
branches in the paper's comparison table, giving two independently published
coarse controls (eccentricity and axis ratio) for each entry. Extending farther
would deepen the index without broadening the branch geometries covered here.
The first ten cover the initial Dyson branch, both subsequent parity classes,
and later reference equilibria, rather than keeping only the famous first
value. All ten values are short at 100 significant digits; the measured stored
entry is 102 characters and the unrounded ball is at most 146 characters.
No exploratory continuous-input group or arbitrary coupling is present.

Jacobi and other non-axisymmetric bifurcations have different mode equations
and are outside this named hierarchy. No extra physical invariant, flattening,
or axis-ratio convention is included. The family proposal's other quantities
are separate tables, to be linked when they exist. The completeness note,
parameter constraint, and generator enumeration express the same scope.

This is the builder's justification, not independent acceptance. A separate
critique must inspect the live selection and fingerprint generate.py and
checks.py (plus any added selection-relevant dependencies) using
agents/argument_review.py. The controller owns the review gate.

Instructions: agents/runs/20261030T141911Z-build-skills/manifest.json.
Archived local and live SHA-256 (identical):
28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199.