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4457 bytes, as of the version from 2026-11-09 12:01 (current). Recorded here, not run.
# Argument selection for T689
Policy read: the coordinated local and live canonical skills archived by
`agents/runs/20261109T113247Z-build-skills/manifest.json`, copied to the
attachment `skill-manifest.json`. Both skill SHA-256 values are
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
This note is the builder's justification, not the required independent critique.
The table has exactly two free parameters, in order `multidegree`, `d`.
Neither chooses between named quantities. Genus is fixed at zero; only the
Gopakumar–Vafa invariant is stored. Its rational Gromov–Witten companion is T688.
## Multidegree: discrete_index
A smooth complete intersection of r hypersurfaces in complex P^(r+3) is a
threefold. Adjunction makes its first Chern class zero exactly when
sum(l_i-1)=4. Remove linear equations and sort the defining degrees. The five
partitions of 4 give precisely (5), (2,4), (3,3), (2,2,3), (2,2,2,2).
Thus the target class is exhaustive. These are constrained tuples describing
geometric types, not independently varied continuous evaluation arguments.
The invariant is deformation invariant within a smooth type. There is no
complex-structure fibre parameter. The ordinary-projective scope is deliberate;
weighted-projective, product-projective and noncompact targets are other families.
The classification is stated in Macaulay2 EnumerationCurves `rationalCurve`,
Description, and agrees with the geometry of Givental's projective-complete-
intersection mirror theorem (1996, Section 11). T332 and T333 use the same
nondecreasing multidegree and hyperplane polarization. All five types are
included before extending any one's degree range. Their order follows issue
306 and T688, starting with the quintic, then increasing codimension.
## Curve degree: discrete_index
The degree is the positive integer d=integral_beta c1(O(1)|X). It is the
actual degree of a curve class, not a denominator-bound reference sample or
an index disguising a real input. This is the degree in the multiple-cover
formula of Chou–Lee, Section 1.1, equation (1.1), specialized to the primitive
hyperplane polarization. Every degree 1 through 20 is selected for each target.
Degree zero is the classical intersection term, not a GV row.
Twenty is a noncanonical computational and readability cutoff, independent of
numeral base. It provides successive prime and composite degrees with different
multiple-cover contributions, beyond the published low-degree controls, across
the entire five-type class. The degree axis is a useful reference sequence of
curve invariants; no claim of an application for each member is needed. The
bound is not chosen to meet an entry quota. The finite target-by-degree product
is ordinary coverage of two discrete indices, not a mesh of evaluation inputs.
The 100 resulting values have a maximum length of 62 digits at ((5),20).
The dry-run block estimate is 7.8 KB. All entries remain easy to display, and
increasing the cutoff would add depth without adding another geometric type.
## Enumeration, conventions and checks
The generator's defaults enumerate exactly these five types, each with d in
ascending order from 1 to 20. `checks.py` independently constructs the five
partitions, checks the ordered enumeration and both bounds, and compares every
value. Parameter constraints describe the full domain d>=1; only the
completeness note specifies the current cutoff. There are no mixed argument
roles or exploratory reference samples and no quantile-related quantities.
The geometric normalization is kappa=product(l_i), and q/x tends to 1. With
z=Cx, the corresponding period/mirror-coordinate templates are respectively
5, 2.2.4, 3.3, 2.2.3 and 2.2.2.2, and C is respectively 3125, 1024, 729,
432 and 256. This agrees with the live T679/T680 definitions, including the
integral coordinate q_A=Q_A/C. The template (2,4) includes two upper parameters
1/2; its key is not 2.4. Exact coefficient checks verify the correspondence.
The later independent critic must assess both declared parameters and obtain
fresh fingerprints from the live draft. The generator is self-contained:
`generate.py` is the only source dependency defining or controlling selection.
`checks.py`, `check.py` and the JSON evidence are validation artifacts, not
inputs to the generator. The independent reviewer must use its own run ID and
must not treat this builder note or its checks as an independent acceptance.