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# Argument selection for T551: Symplectic Weingarten functions
The authoritative instructions read for this build are recorded in
`agents/runs/20261027T140846Z-build-skills/manifest.json`. Both archived local
and live versions have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
This document is a build justification, not an independent acceptance review.
## Quantity and normalization
Each row is Matsumoto's unscaled symplectic Weingarten coefficient at the even
representative sigma_mu of equation (2.3), for Haar probability measure on
USp(2N), with J = [[0,I_N],[-I_N,0]]. The integer N is rank, and 2N is complex
matrix dimension. The coset partition mu describes the connected components
of two pairings, not the ordinary cycle type of a permutation. General
permutations of that coset type have coefficient sign(sigma) times this row.
Source: Matsumoto, arXiv:1301.5401v3, sections 2.2.1 and 2.3, especially
equations (2.3), (2.8), and Theorem 2.4. These locators were read in this run.
## Every free parameter
- N is a `discrete_index`: the rank of a compact symplectic group. Every
positive integer is a legitimate family member. We select every rank
1 through 6, covering complex matrix dimensions 2 through 12, including
intermediate sizes rather than only powers of two. Six is a practical,
noncanonical small-matrix budget, independent of numeral notation.
- mu is a `discrete_index`: a nonempty partition of k, indexing every
inequivalent pairing contraction in a degree-2k Haar moment. For each
selected N we include every partition with 1 <= k <= N, in reverse
lexicographic order within each k. This includes mixed patterns such as
(3,2,1), not just identity and full-cycle contractions. The shared source
explains why all these types appear in the integration theorem; it does
not claim an application for each particular stored fraction.
The bounds interact through k <= N. This is the stable invertible range of
the symplectic pairing Gram matrix, and it keeps every character denominator
nonzero. The total degree limit 12 is a practical independent-check budget:
there are 10,395 pairings at k=6 but only eleven coset types. A complete set
of types is more useful coverage than extending identity coefficients to
larger ranks. The selection has 68 entries, a consequence of these bounds,
not a target used to choose them. No continuous mesh, exploratory reference
sample, statistical quantile, or redundant k parameter is present.
## Gaps and extent
Larger ranks, higher degrees, and singular small-rank pseudoinverses k>N
are not stored. The definition states the standard character-sum convention
for these singular ranks so the family itself is not defined by today's
cutoff. The measured fraction length and block size are recorded in the
numerical evidence; the finite extent is retained if those are readable.
The generator defaults and completeness note use exactly this selection.
An independent critique must assess both parameters, the sign convention,
these coupled bounds, and the generator source. It must create its own
fingerprint-bound sidecar with its own reviewer run. This build creates no
acceptance sidecar and does not offer the draft.