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# Argument selection for T552
This table holds one named quantity, the unscaled chiral unitary Weingarten
coefficient Wg^{AIII}_{a,b}(mu). The convention is Matsumoto's Hermitian
involution W=U* diag(I_a,-I_b) U, with Haar probability measure on U(a+b).
It is not the matrix V=diag(I_a,-I_b)W used for another realization of the
same symmetric space. See arXiv:1301.5401v3, section 4.1, Theorem 4.1,
Example 4.1 and the character expansion on page 18.
All three parameters have role `discrete_index`:
* `a` is the integer multiplicity of +1 in W. It indexes the dimension of
the positive eigenspace, not a continuous input sampled on an integer grid.
* `b` is the integer multiplicity of -1. The pair indexes a signature and
the complex Grassmannian U(a+b)/(U(a) x U(b)). The convention a>=b>=1 is
the source's convention. Interchanging a and b changes a degree-k
coefficient by (-1)^k, so it supplies no new signature up to overall sign.
* `mu` is a nonempty partition of k, specifying ordinary permutation cycle
type in S_k and hence the invariant contraction pattern in the moment
theorem. It is not a coset type. All partitions are included, including
mixed types such as (2,1,1) and (2,2).
The finite selection is precisely a>=b>=1, a+b<=8, and every partition mu
with 1<=|mu|<=min(4,a+b). There are 16 signatures and 163 entries. The
parameters describe their full mathematical domains; these bounds describe
coverage only. Enumeration is lexicographic in (a,b), then increasing k,
then reverse lexicographic on partitions; parameter order is (a,b,mu).
Breadth was reviewed before depth: every signature at each selected total
matrix dimension is kept, covering balanced eigenspaces, rank-one negative
complements and intermediate ranks. Every contraction pattern at each selected
degree is kept, rather than keeping only identity and full-cycle coefficients.
The restriction k<=a+b makes the permutation Gram matrix invertible and all
character denominators nonzero. The generalized-inverse definition extends to
larger degrees, which are an explicit coverage gap.
Total dimension eight and degree four are practical, noncanonical budgets.
They are independent of numeral notation, and degree four permits an
independent exact permutation check involving at most 24 permutations per
signature and degree. They are not claims that every coefficient has a
separate documented application. This is a bounded discrete matrix-integral
family, not exploratory continuous reference coverage, so `reference_sample`
and `mixed` are inapplicable. It contains no quantiles or critical values.
The default enumeration, completeness note and stored identities must agree
with this set. `generate.py` alone defines the enumeration and computations;
it imports no local numerical helpers or datasets. `checks.py` is independent
verification code, and `argument-selection.md` records the selection's meaning.
A later independent critique must bind all three attachments with
agents/argument_review.py fingerprint and supply its own argument-review
sidecar. This build does not write or accept that sidecar.
Policy evidence: agents/runs/20261027T144347Z-build-skills/manifest.json.
The archived local and live skill files both have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.