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# Argument selection for T564
The quantity is the full labelled Laguerre–Selberg integral with weight
exp(-x), not the reciprocal density coefficient or an ordered-chamber integral.
The parameter order is (beta, n, r), where r=m-n. The domain is beta in
{1,2,4}, positive integer n, and nonnegative integer r. Quaternion eigenvalues
are counted once.
The current policy is the two matching copies recorded in
agents/runs/20261027T230956Z-build-skills/manifest.json. Both have SHA-256
e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.
The build checked these bytes against the canonical source snapshot. The
selection requires a separate independent critique; this document is the
builder's justification, not an acceptance sidecar.
## Parameters and mathematical roles
* beta is a documented application selection: the complete real, complex and
quaternion classical Wishart symmetry classes. It is not a sample of the
continuous beta-ensemble coupling. Dumitriu and Edelman, *Matrix Models for
Beta Ensembles*, Section 1.2, Figure 1 and equations (3)–(4), establish the
three classes and the rectangular matrix exponent.
* n is a discrete index: the number of eigenvalues of an n by (n+r) matrix.
Every positive integer is meaningful; the finite extent is a practical
reference choice. Dimensions are covered across matrix shapes before
increasing the length of any one dimension sequence.
* r is a discrete index: an actual excess column count, not a continuous weight
exponent sampled at convenient arguments. The exponent is forced to be
beta*(r+1)/2-1 by the Wishart matrix model. The near-square and fixed-ratio
selections are subfamilies of these integer matrix shapes. No claim is made
that each finite matrix shape has a separate documented application.
Sources checked: https://arxiv.org/pdf/math-ph/0206043, Section 1.2 and Figure 1;
https://dlmf.nist.gov/5.14.E5, with insertion index zero. The paper's n columns
and m rows are renamed to n+r columns and n rows here. Its eigenvalues are
scaled by lambda=2*x to obtain exp(-x). These choices agree with issue #225.
## Finite selection and breadth
For each of beta=1,2,4, take the union of:
1. Small rectangular shapes: 1 <= n <= 6 and 0 <= r <= 6 (42 shapes).
This covers both parities of the real exponent and several excess counts
for each small dimension, including the scalar gamma integral and square
matrices. It gives a compact reference across actual matrix sizes.
2. Near-square sequences: 1 <= n <= 16 and r in {0,1,2} (30 additional shapes).
Fixed small excess probes the square boundary and its closest rectangular
neighbors while increasing dimension. It is a dimension family with
documented Wishart meaning, not an arbitrary continuous input grid.
3. Aspect-ratio sequences: 1 <= n <= 10 and r in {n,2*n} (11 additional shapes).
These keep m/n exactly 2 or 3, covering rectangular growth outside the
near-square regime. The ratios are simple, noncanonical reference choices
among discrete matrix shapes. No asymptotic approximation is stored.
The union has 83 shapes, hence 249 entries. Intersections appear once.
All tuples are ordered first by beta=1,2,4, then by increasing n and r.
The cutoffs and the ratios 2 and 3 are not canonical endpoints or uniquely
privileged applications. They are notation-independent choices that give
breadth over square, almost square and rectangular matrix families without
forming a large box of dimensions and excesses. An arbitrary mesh in a real
coupling or in a weight exponent is not included.
The dry run measures a longest exact value of 727 characters at (4,10,20),
well within the readability scale discussed by the skill. The entries block is
about 34 KiB. Extending n further rapidly lengthens exact factorial products;
using 100-digit rounded integers to conceal that growth would be wrong. The
finite bounds retain the complete chosen slices and allow later extensions.
The completeness note and generator enumerate exactly this union.
## Quantity and exclusions
Every row holds one quantity, Z_beta,n(r). Beta chooses the scalar field of a
matrix, not a separate quantity. Exact rational values stay exact. Irrational
values are rational multiples of sqrt(pi) and have 100 significant certified
digits. No quantiles, percentiles, critical values, or inverse CDFs occur.
## Source normalization discrepancy
Forrester and Warnaar, equation (3.19) in the author PDF, prints 1/n! multiplying
product Gamma(a+j*c)*Gamma((j+1)*c)/Gamma(c), while its preceding density is on
an ordered chamber. That product already equals the full-orthant integral
divided by n!, so the displayed additional factor cannot be used here. DLMF
5.14.5 and Dumitriu–Edelman equation (4), after rescaling, agree. Direct
expansion at beta=2,n=2,r=0 gives 2 on the quadrant, 1 on one chamber, while
the printed expression gives 1/2. The discrepancy has a named cause and is
not resolved by choosing whichever numeric value looks convenient.