# Build evidence for T550

Run: 20261027T133445Z-build. Author account: zeta3. The draft is not published
or offered by this build. A separate critique and argument-selection review
must precede the controller's offer action.

## Coordinated instructions

Manifest: `agents/runs/20261027T133445Z-build-skills/manifest.json` in the source
snapshot. Archived `skill-local.md` and `skill-live.md` were both read; both
hash to SHA-256 `e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The current canonical source file has the same hash. `skill-evidence.json`
records the exact manifest path and both source hashes.

Family issue 221 was read first using `python3 agents/queue.py show 221`.
The live search, `screen.already_here`, `screen.already_asked`, and exact lookup
checks found no orthogonal Weingarten table. Exact lookups for -1/30, -1/8
and 2/15 found unrelated Bernoulli, Wigner, integration-weight and finite
difference contexts, not this family. The unique-title draft creation returned
T550, `published: false`, `ready_for_review: false`.

## What is computed

242 exact reduced rational coefficients at every integer dimension 1 through
12, every degree 1 through min(6,N), and every partition of that degree.
Parameter order is (N,mu); enumeration order is N, degree, reverse lexicographic
partition. This is the full O(N) with Haar mass one. No sign or factorial
rescaling is applied. The singular range N<k is defined but not enumerated.
See `argument-selection.md` for the two discrete parameter justifications,
finite bounds, breadth review and declared omissions.

`generate.py` uses Sage's Jack J basis at alpha=2, the power-sum coefficients
of the zonal polynomials, and the hook-length dimensions. Collins and
Matsumoto, equations (3.5)-(3.7), fix every factor. All divisions have a Sage
rational operand. There are no decimal approximations or error estimates:
exact rational arithmetic provides more information than 100 digits.

## Offline checks actually run

- The repository's `dry_run.py` computed all 242 entries. `exactness` passed,
  `names_its_rings` passed, no trailing-zero disguises or prose findings arose.
- The longest reduced fraction is 21 characters, at N=10, mu=1,1,1,1,1,1.
  `dry_run.py` measured 12.7 KiB for its standardized flat block; the
  independent helper's different tuple-key labels measure 13.126 KiB.
  These sizes are not a selection target.
- `independent_checks.py` enumerates every pairing, up to 10,395 at k=6.
  It forms one full Gram row for every coset representative, compresses by
  coset type, and solves the resulting equations with rational elimination.
  All 242 independent coefficients agree, and all 242 Gram equations have
  exactly zero residual. Symmetry carries these rows to the full inverse.
- For all 57 selected dimension-degree pairs, the sum of the matrix-integral
  coefficients agrees with E[R_11^(2k)] = (2k-1)!!/[N(N+2)...(N+2k-2)].
  This independent moment follows by writing an isotropic standard Gaussian
  as its independent radius and uniform spherical direction and dividing its
  coordinate moment (2k-1)!! by its radial moment N(N+2)...(N+2k-2).
- All 64 entries with k<=3 agree with the explicit rational formulas in
  Collins and Matsumoto, section 7; the k<=2 subset also appears as Matsumoto's
  Example 2.3.
- Controls precede the general checks: a nonzero 2-by-2 rational solve,
  N=3,k=2 coefficients (-1/30,2/15), the sphere moment 1/5, and rejection
  after perturbing one coefficient by 1/1000.
- `auxiliary_checks.py` compares the table's self-contained Sage program with
  every one of the 242 entries. It independently gets representation
  dimensions from Schur/power-sum scalar products, rather than the generator's
  hook computation. The extension example N=13,mu=3,1 gives 1/2633400 and
  agrees with a separately formed pairing inverse.
- All 348 zonal specializations at N=1,...,12 agree with their content products;
  all 29 zonal polynomials have coefficient 1 on p_(1^k). Their doubled-shape
  representation dimensions sum to the pairing count at each degree.
- All 29 N=1 coefficients through k=6 satisfy the singular convention: the
  Gram matrix is the all-ones matrix of size m=(2k-1)!!, whose pseudoinverse
  entries are 1/m^2. These are controls, not additional table entries.

The check files are attached for repeatability. The independent checks do not
use zonal polynomials, representation characters, or generator helpers in
stored mode. `check.stored` requires Django database access; this build instead
reads the authenticated table API and parses its exact rational entries.

## Resolved failures, before any entries were sent

A Sage Jack conversion initially raised an AttributeError because its matrix
ring dispatch had not imported `sage.rings.polynomial.laurent_polynomial_ring`.
An explicit named import resolved it; the generator and reader program both
include that import. The symbolic ring is also initialized explicitly. Sage's
lazy-import warnings were retained in the logs, and did not alter the checks.

The first hand-written N=3,k=2 control incorrectly used -1/24 and 1/6. The
independent inverse rejected it before any general check ran. Matsumoto's
explicit denominator gives 3*5*2=30, hence -1/30 and 4/30=2/15; the sphere
moment independently gives 1/5. The control was corrected, and the complete
suite passed. No generated value was changed to resolve this failure. The
initial failed log and the corrected log are both preserved. Initial lookup
extras -1/24 and -1/120 were supplemented with the actual expected coefficients.

## Software and sources

Sage 10.9; Python 3.12.3; numberdb 0.1.10. Every Sage invocation used
`agents/sage.sh`, including dry runs. No deployment, Git operation, offer,
publication or review action was performed.

Primary source PDFs were downloaded and retained with these SHA-256 values:

- https://arxiv.org/pdf/1301.5401, Matsumoto, section 2.2 and Theorem 2.2:
  `cdfc30306b1a7e7d745c420fcef0d368d565c4e704f4cfcd06fe0125dc92eda2`.
- https://arxiv.org/pdf/0903.5143, Collins and Matsumoto, sections 2, 3, 4 and 7:
  `d66d0e7be659415e62ed996dc7ec258d97e8b1db435c8df16232ca68b7f5ceb9`.

T379 and T119 were read from the database, and their exact slugs from search
results are used in the document. The companion T549 was read but was not
publicly accessible during the build, so the mathematical document does not
link to it. Once both are published, a human can add their reciprocal relation.
Existing tags groups, probability theory and integral apply. The proposed
random matrix theory tag still has no published tables; it would connect
T379, T441, T451 and this family. This build does not edit those tables.