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# Argument selection for T563
This is the author's justification for the independent critique, not an
independent acceptance. The authoritative instruction archive is
`agents/runs/20261027T223837Z-build-skills/manifest.json`. Both skill-local.md
and skill-live.md have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The run read both and verified byte equality with the snapshot's canonical
`.claude/skills/numberdb-table/SKILL.md`.
## Quantity and normalization
The sole quantity is the full-cube Selberg integral Z for the classical
Jacobi ensemble. Put c=beta/2, a=c(r+1), b=c(s+1). Integrate
prod x_i^(a-1)(1-x_i)^(b-1) times prod_(i<j)|x_i-x_j|^beta over [0,1]^n
with ordinary Lebesgue measure and labelled coordinates. No n! factor is
removed. The integral is stored, not its reciprocal density coefficient.
The order of the integer parameters is (beta,n,r,s). Quaternion eigenvalues
are counted once. Swapping r and s reflects every eigenvalue about 1/2, so
only r<=s is enumerated. No quantiles or statistical critical values occur.
## Evidence checked
* Forrester and Warnaar, *The importance of the Selberg integral*,
arXiv:0710.3981, equation (1.1), gives exactly the cube integral and gamma
product. Equation (1.2) gives convergence; all selected arguments satisfy
it strictly. Equation (1.3) gives the n=1 beta integral.
https://arxiv.org/html/0710.3981v1
* NIST DLMF 5.14.4 with insertion index m=0 gives the same normalization.
The m in that source is an insertion index, not a matrix column count.
https://dlmf.nist.gov/5.14.E4
* Edelman and Sutton, *The beta-Jacobi matrix model, the CS decomposition,
and generalized singular value problems*, Introduction/Table 1 and
Proposition 1.2, identifies the density exponents and real/complex matrix
dimensions. The Introduction identifies beta=1,2,4 with the real, complex,
and quaternion symmetry classes. The source's excess parameters a,b are
this table's r,s, not the a,b used in the gamma product.
DOI 10.1007/s10208-006-0215-9.
https://math.mit.edu/~edelman/publications/beta-jacobi.pdf
The family issue #225 was read first. Its linked batch file returned HTTP
404, so its suggested finite bounds could not be read. The selection here
was assessed afresh from the available family conventions and primary
sources. The sources establish the family, conventions and relationships;
they do not prescribe our finite cutoffs or every aspect-ratio example.
## Every free parameter
* beta: documented application. All three classical symmetry classes
{1,2,4} are included, rather than a mesh of a continuous coupling. This is
exhaustive for the named classical symmetry classes, but is not a claim
about arbitrary positive beta ensembles.
* n: discrete index, the number of eigenvalues of n-dimensional matrices.
The scalar case connects to the beta integral; higher n introduces
eigenvalue interaction. The upper bound depends on the selected slice,
as listed next. Each initial segment is complete within that slice.
* r: discrete index, the nonnegative excess column count of the first
n-row rectangular Gaussian matrix. It sets a=beta(r+1)/2, rather than
being an independently sampled continuous exponent. Square matrices,
near-square matrices and aspect-ratio slices have distinct dimension
meanings. r<=s removes the reflection duplicate, not a second object.
* s: discrete index, the excess column count of the second matrix. Both
balanced and asymmetric matrix pairs are retained. Its independent
small-excess bounds and its couplings to n on aspect-ratio slices are
specified explicitly next. It is not silently set equal to r everywhere.
These parameter roles apply throughout each parameter's selected values.
There is no continuous reference_sample group hidden under discrete_index.
The **bounds and choice of discrete slices are noncanonical reference
choices**. Discrete dimensions justify the family, not the assertion that
any particular finite cutoff or aspect ratio is uniquely natural.
## Exact enumeration and coverage tradeoff
For every beta in {1,2,4}, take the union of:
1. Small matrix dimensions: 1<=n<=6 and every 0<=r<=s<=4. This includes
scalar normalization, both square matrices, one square matrix, balanced
rectangular pairs and all small asymmetric excesses. Breadth in matrix
shapes is covered before extending any one dimension.
2. Near-square matrices: 1<=n<=20 and every 0<=r<=s<=2. Fixed small excess
is a useful regime distinct from a fixed positive aspect ratio. The
smaller excess bound permits a longer initial dimension segment.
3. Fixed aspect ratios: 1<=n<=12 and (r,s)=(0,n),(n,n),(n,2n). The ratios
of column count to row count are respectively (1,2), (2,2), (2,3).
These are simple dimension relationships, independent of numeral base,
covering a square/rectangular pair, a balanced rectangular pair and an
asymmetric pair with neither matrix square. The choice of ratios and
cutoff 12 is ours, not a claim of a particular published application for
each tuple. Additional ratios would increase breadth but are deferred
to keep a bounded reference rather than a four-dimensional dump.
Deduplicate by the full (beta,n,r,s) tuple. Enumerate beta=1,2,4, then
increasing (n,r,s). The result is 200 tuples per symmetry class, 600 total.
The three families have 90, 120 and 36 triples respectively before union;
their overlaps reduce the union to 200. `generate.py:selection` is the sole
executable selection dependency. `selection.json` records its exact output.
`checks.py` does not add or remove rows.
The measured longest written value is 518 characters at beta=4,n=20,r=s=2;
the checker measures an entries block of about 80 KiB, before final API
serialization. Thus the chosen dimension depths remain readable with room
for extension. The cutoff is a coverage and readability choice, not a
storage target. Nothing was removed merely because it was exploratory or
not individually famous. Extending only n would miss the matrix-shape
breadth supplied by the first and third families.
## Corpus and independent-review boundary
Fresh already_here/already_asked searches and api/lookup values found no
same-family table. The n=1 beta integrals and the gamma factors have related
corpus tables; these are linked. Their overlap follows the mathematical
specialization and is not copied numerical data requiring a repeats flag.
Existing tags integral, probability theory and special values apply.
The absent random matrix theory tag is proposed separately in the handoff,
with existing tables T379, T441 and T451 as well as this family as candidates.
The separate critique must check this rationale, the 600 stored tuples,
complete-note and every selection-relevant attachment. Use
`python3 agents/argument_review.py fingerprint T563 --generator generate.py`
and include checks.py and argument-selection.md if binding the entire
verification/justification bundle. No argument-review verdict or independent
sidecar has been authored by this build.
The mixed real table triggers the API's 500-digit soft threshold because
the longest exact fraction writes 517 digits. A documented Size exception
retains this 518-character exact fraction; it does not increase approximate
precision or change the selected arguments.