# Argument selection for T619

This build follows family issue 246 and the coordinated canonical skill archived
in `../20261030T054723Z-build-skills/manifest.json`. Both archived versions have
SHA-256 `28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.
This is the builder's justification, not an independent critique or acceptance.

The object is the static Newtonian tidal Love number in the apsidal-motion
convention (one half the surface induced/applied potential ratio). Each row is
one value of this quantity at polytropic index n and multipole degree ell.
Potential Love numbers would be twice these values; they are not added as a
second quantity. No radius, density, rotation, compactness or EOS scale is a
free input. Scale invariance is explicit in Yip and Leung, equations (19)–(20).

## Polytropic index n (Q): mixed selection

The domain is rational 0 <= n < 5, following the positive Lane–Emden profile to
its first zero. Default enumeration is exactly
0, 1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2, in that order.
The following groups are disjoint and exhaustive:

* Intrinsic: `0`, `1`. These have the incompressible and linear analytic
  profiles and exact tidal solutions. Yip and Leung, sections 3 and 5.3,
  equations (5), (6), (28), (33). The n=0 surface density jump is included.
* Documented application: `3/2`, `3`. These are idealized nonrelativistic and
  ultrarelativistic stellar models. Princeton's Polytropes, pp. 2–3; Lu,
  Polytropic Star, pp. 1–2, gives n=3/2 convective envelopes and discusses
  radiative-envelope n=3. These are not precision models of actual stars.
* Reference sample: `1/2`, `2`, `5/2`, `7/2`, `4`, `9/2`. These are exploratory
  references for response constants of low-complexity algebraic density laws,
  not intrinsic indices and not a claim of documented demand for each member.

The reference_policy is:

```json
{
  "purpose": "Identify tidal response constants of simple algebraic density–enthalpy laws beyond the analytic and distinguished stellar cases.",
  "bounds": "Only integer and half-integer n between 0 and 9/2, excluding 0, 1, 3/2 and 3 assigned to the other groups; ell is restricted to 2 and 3.",
  "coverage_rationale": "Integral and square-root powers span stiff through centrally concentrated finite models before increasing denominators or moving closer to the divergent-radius endpoint n=5. The laws become monomials in sqrt(theta); selection depends on algebraic exponent complexity, not decimal notation. The two leading tidal degrees provide multipolar breadth before higher orders. This is sparse reference coverage, not interpolation or a count target.",
  "noncanonical_acknowledged": true,
  "notation_independent": true,
  "sources_checked": true
}
```

Denominator two and the cutoff 9/2 are practical, noncanonical reference
choices. They do not assert that other rational indices are uninteresting.
The n>3 rows describe mathematical equilibria without claiming radial dynamical
stability. The noncompact n=5 endpoint has no finite stellar surface and is
excluded rather than assigned an artificial zero. Yip and Leung, sections 3,
5.1–5.3, supplies the function, surface conventions and endpoint context.

## Multipole degree ell (Z): discrete index

The domain is integer ell >= 2. The finite prefix ell=2,3 comprises quadrupolar
and octupolar static deformation, the two leading tidal orders studied in
Yip and Leung, section 7. The cutoff is a practical coverage choice supported
by analytic n=0 and n=1 controls for both orders. It is not an additional
continuous coupling or a renamed quantity selector.

Every chosen background has both orders. This small product compares the two
leading multipoles of the same models; it does not combine arbitrary simple
values of several continuous inputs. Indices ascend in n and then ell. The
six companion tables cover background scales and shape before this table adds
tidal response; we do not extend their ranges or absorb their quantities.

## Checked sources and live companions

* Yip and Leung, 2017, arXiv:1709.02469, DOI:10.1093/mnras/stx2363,
  sections 3, 5.1–5.3 and 7. Downloaded PDF and extracted text retained.
* Princeton Polytropes, https://www.astro.princeton.edu/~gk/A403/polytrop.pdf,
  pp. 1–3, model, analytic cases and stellar applications.
* Wenbin Lu, https://wenbinlu.github.io/files/LaneEmden.pdf, pp. 1–2,
  normalized mass and radius, nonrelativistic and convective applications.
* Live T613–T618 documents were read. Their shared n, xi, theta, q, mu and
  normalization conventions are retained. T187 was read for the Bessel family
  connection; it is not asserted to store the spherical Bessel ratio at pi.

An independent critique must assess both parameters and their interaction,
check all 20 rows and this enumeration, and bind its own argument-review
sidecar to `generate.py`, `tidal_solver.py`, and any additional reviewed source
dependencies. The builder does not produce or accept that sidecar.