# Unitary Weingarten functions: argument selection for T549

This build follows numberdb-data family #221, read before building. It uses the
coordinated canonical skill archived at
`agents/runs/20261027T125634Z-build-skills/manifest.json`. Both its local and live
versions have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The manifest is attached as `skill-manifest.json`. This is a builder's rationale,
not an independent argument-review acceptance.

## Quantity and normalization

We store one named quantity, the unscaled unitary Weingarten coefficient, with
its sign, for the full group U(N) with Haar mass one. The parameters are ordered
(N, mu); N is the complex matrix dimension, and mu is an ordinary permutation
cycle type, not an orthogonal coset type. The nonempty partition mu is written
as decreasing comma-separated positive parts, without parentheses or spaces.
Its size k is determined by mu and is not an additional free parameter.

Checked primary sources: Sho Matsumoto, arXiv:1301.5401, Sections 2.1.1–2.1.3,
equation (2.1), Theorem 2.1 and Example 2.1; Benoît Collins and Piotr Śniady,
arXiv:math-ph/0402073, Section 2, equation (9), Corollary 2.4 and Proposition 2.5.
The archived Matsumoto PDF page 4 was also inspected visually, including the
complex conjugation in the integration formula. These sources agree.

## N: discrete_index

Positive integer N is a genuine matrix dimension in the integration theorem.
We select every dimension 1 through 12. Including all small sizes covers the
transition from the first admissible size N=k through several larger dimensions
for each contraction pattern, rather than only selected powers of two. Twelve
is a practical, noncanonical finite bound, independent of numeral notation.
There is no claim that every selected coefficient has an individually documented
application. Matrix-entry moments and invariant contractions supply the family;
the theorem, rather than a grid of simple real inputs, justifies its indices.

## mu: discrete_index

For each N, include every integer partition mu with
1 <= |mu| <= min(6,N). Every partition is one conjugacy class in S_k and hence
one invariant contraction type of the integration formula. Complete breadth in
cycle type includes identity, single-cycle and mixed-cycle contractions. Keeping
only exceptional types would discard valid distinct coefficients. The degree
cutoff k=6 covers integrands of total degree 2k through twelve and permits an
independent check enumerating at most 6!=720 permutations. Extending dimension
or k beyond these bounds is possible but not required for this finite reference.

The selection has 242 entries. Dimensions increase first, then k increases,
then partitions run in reverse lexicographic order. The two bounds are coupled
by k<=N, the ordinary-inverse range. There is no Cartesian product of continuous
reference inputs and no exploratory reference_sample component. Both parameters
are discrete indices of the same mathematical quantity. No quantiles,
percentiles, thresholds, or arbitrary decimal grids enter the selection.

## Domain versus coverage

The table's domain allows all positive N and all nonempty partitions. The
character-sum definition restricts its internal partition lambda to length at
most N, thereby defining the conventional generalized inverse also for k>N.
The stored finite selection deliberately omits that singular range. No poles
are evaluated, and no small-dimensional pseudoinverse values are silently
identified with ordinary inverse values. Dimensions beyond twelve and degrees
beyond six are also gaps. The complete-note states this finite set and rationale.

## Measured extent and coverage review

The unchanged proposal covers every selected contraction type and every selected
admissible dimension. It does not deepen only identity or single-cycle slices.
The longest exact fraction is 20 characters at (N,mu)=(12,(3,1,1,1)), and the
entries occupy about 13 KB, well below the recommended block margin. These
measurements confirm readability, not a reason to inflate the range.

The independent critique must inspect both parameters, the coupled bounds,
this note, generate.py and its complete default enumeration, and the actual
stored parameter identities. The builder does not write its acceptance sidecar.