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# Argument selection for Orthogonal Weingarten functions (T550)
This build follows family issue 221 and the coordinated skill archived in
`agents/runs/20261027T133445Z-build-skills/manifest.json`. Both local and live
copies have SHA-256 `e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
This is the builder's justification, not the independent critique's acceptance.
The single quantity is the unscaled, signed orthogonal Weingarten coefficient
for the full real group O(N), with Haar probability measure. Parameters have
order (N, mu), where mu is a partition coset type and k is the sum of its parts.
All stored values are exact rationals, so a decimal precision target is inapplicable.
## N: discrete_index
N is the size of the orthogonal matrix, not a continuous argument sampled at
integers. Orthogonal tensor integration contracts 2k matrix entries in pairs;
the Gram entry for two pairings is N to the number of loops of their union.
This is the family-specific reason that every positive integer dimension has
a mathematical role. We retain every N from 1 through 12, at all degrees
1 <= k <= min(6,N). This includes both even and odd dimensions and intermediate
sizes, without preferring powers of two or a particular decimal notation.
Twelve is a practical, noncanonical small-matrix coverage limit, consistent
with the companion unitary proposal. It is not a claim of a special threshold
or a separately documented application of every matrix size.
## mu: discrete_index
A nonempty partition mu of k specifies a double coset of the hyperoctahedral
subgroup in S_(2k), equivalently a contraction pattern: overlay the standard
pairing and its image, and halve the connected-component sizes. All partitions
at every selected k are retained, in reverse lexicographic order. Retaining
only the identity and single-part cases would omit mixed patterns needed by
the Haar integration formula. Thus breadth across all contraction types comes
before depth into high moments. The finite bound k <= 6 covers integrands of
total degree at most twelve and keeps the independent exact pairing check
small: the largest pairing set has 11!! = 10395 members. This is an explicit
computation budget, not an intrinsic maximum degree. No root parameter,
continuous reference grid, or quantity selector is concealed here.
## Interaction and gaps
The bound k <= N selects the range in which all representation denominators
are nonzero and the pairing Gram matrix is invertible. It couples degree to
dimension; these are not independently meshed inputs. The conventional
pseudoinverse for N < k remains defined in the table's character-sum formula,
but those singular cases are a declared coverage gap. The empty partition is
omitted because it only gives the constant integral 1. Larger dimensions and
higher degrees are also omitted. No campaign-excluded probability quantiles,
percentiles, or statistical critical values occur.
Enumeration is increasing N, increasing k, and reverse lexicographic mu.
The counts per degree are p(1),...,p(6) = 1,2,3,5,7,11; these bounds give
242 entries at 57 dimension-degree pairs. These counts describe the selection,
not its justification. The completeness note must state these bounds, their
contraction-coverage purpose, and their noncanonical computational nature.
## Checked primary sources
- Sho Matsumoto, *Weingarten calculus for matrix ensembles associated with
compact symmetric spaces*, https://arxiv.org/abs/1301.5401, section 2.2.1
(coset type and representative (2.3)), section 2.2.2 (normalization and
spherical expansion), Theorem 2.2 (Haar O(N) integration), Example 2.3.
- Benoit Collins and Sho Matsumoto, *On some properties of orthogonal
Weingarten functions*, https://arxiv.org/abs/0903.5143, section 2.1
(pairing Gram matrix), equations (3.5)-(3.7) and Theorem 3.1 (zonal/Jack
normalization and coefficients), section 4's concluding invertibility
statement for dimension at least degree.
These sources establish the function, discrete parameter meanings and
conventions. They do not declare our finite bounds canonical. The independent
critique must review this justification and fingerprint all relevant live
attachments; this build creates no argument-acceptance sidecar.