# Argument selection for T688

This table stores one quantity: the connected, unmarked, genus-zero
Gromov–Witten invariant N_(ell,d), in the primitive hyperplane degree. Its
rational multiple-cover weights are retained. Gopakumar–Vafa invariants are a
separate quantity and no rows for them are submitted here.

## Multidegree: discrete_index

A smooth complete intersection of r nonlinear hypersurfaces in P^(r+3) is a
threefold. Adjunction gives c1 = (r+4-sum ell_i)H. Setting c1=0 and taking
ell_i>=2 gives sum(ell_i-1)=4. The nondecreasing partitions of 4 therefore
produce exactly (5), (2,4), (3,3), (2,2,3), and (2,2,2,2). Removing linear
hypersurfaces removes duplicate presentations in a larger projective space.
The checks independently enumerate the partitions and compare the full set.

This is exhaustive coverage of a finite geometric class, rather than five
separately chosen simple arguments. All five targets receive equal degree
coverage. Ordinary projective complete intersections are a deliberate scope
boundary: product-projective, weighted-projective, and noncompact targets are
not the same class. No complex-structure fibre is a free parameter; smooth
members of each deformation type have the same invariants.

Checked support: Givental, arXiv:alg-geom/9701016, introduction and Theorem 0.1;
Macaulay2 EnumerationCurves rationalCurve documentation, description and target
list; live T332/T333, definitions and adjunction/hyperplane conventions.

## Curve degree d: discrete_index

Degree is intersection with H=c1(O(1)|X), the primitive ample generator. It is
a positive integer and the domain in the parameter declaration is d>=1.
Enumeration includes every d=1,...,20, in increasing order, within each target
in the family order displayed above. Degree zero is the classical Yukawa
coefficient kappa, not an entry of this quantity.

Twenty is a noncanonical practical bound. It retains the initial localization
controls, successive multiple-cover contributions, and prime and composite
degrees across the whole target class before increasing depth for one target.
These exact curve invariants are the coefficients encountered in mirror-symmetry
and stable-map calculations; no separate application for every degree is
claimed or needed. The maximum serialized rational is 68 characters, so this
initial extent remains readable; neither the 100-entry total nor the size
limit supplied the selection rule. There is no arbitrary real evaluation
mesh and no decimal-base dependence. The target-by-degree product is coverage
of actual discrete indices, not independent sampling of continuous inputs.

Checked support: Chou–Lee, arXiv:2305.08480v4, Section 1.1, for GW and GV
conventions; Givental, arXiv:alg-geom/9603021, introductory theorem and Section
11, for curve-degree indexing in the mirror expansion; EnumerationCurves,
rationalCurve outputs o3, o5, o9, o12 and o13, for low-degree geometric checks.

## Scope and review

Both parameters are discrete_index, without mixed or exploratory groups.
The completeness note and generate.py enumerate exactly the selection stated
here. Approximate 100-digit output would weaken these exact rational values.
The generator's only source dependency controlling enumeration is generate.py
itself; checks.py independently validates the chosen finite class and degree
extent. No random choices, external numerical ledgers, or rounded input
arguments affect generation.

This is the builder's justification, not the independent argument-selection
review. The controller must arrange a separate critique actor, including
numerical checks and a fingerprint bound to current attachments, before any
offer. The build actor has not written an independent-review sidecar.

The instructions read were the local and live copies in
agents/runs/20261109T110159Z-build-skills/manifest.json. Both have SHA-256
886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b.
The manifest is attached as skill-manifest.json.