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# Argument selection for T566
This build follows the coordinated canonical NumberDB skill archived in
`agents/runs/20261028T001715Z-build-skills/manifest.json`. Both the local and
live copies have SHA-256
`e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c`.
The attachment `skill-evidence.json` records each verified digest and the manifest
path and digest. This is a build rationale, not an independent acceptance.
## Named quantity and convention
The sole quantity is the full Mehta normalization integral
Z(beta,n) = integral over R^n of exp(-sum(x_i^2)/2)
times product over i<j of |x_i-x_j|^beta, with Lebesgue measure.
Coordinates are labelled. There is no factor 1/n!, and no Gaussian-probability
measure factor (2*pi)^(-n/2). We store Z, not its reciprocal or logarithm.
Forrester and Warnaar (2008), equations (1.4)-(1.6), call the version divided
by (2*pi)^(n/2) F_n(beta/2). DLMF 5.14.6 fixes the same normalization.
The convention agrees with proposal family #225 and its Jacobi, Laguerre and
Morris companions; the issue is workflow provenance and is not table prose.
## Every independently chosen parameter
* **beta** is the Dyson exponent in {1,2,4}. This is a documented application
family: the orthogonal, unitary and symplectic Gaussian ensembles represent
real, complex and quaternion matrix symmetry. All three are included.
This is not a mesh of a continuous coupling. The eigenvalues have been
scaled so their common one-body weight is exp(-x^2/2). Forrester and Warnaar,
section 1, pages 3-4, equations (1.4)-(1.6), explicitly give this association.
* **n** is a positive integer counting eigenvalues, with a quaternion eigenvalue
counted once, not twice in its complex matrix representation. It is a genuine
discrete size index. The selection includes every n from 1 through 50 in
each of the three symmetry classes. The cutoff 50 is noncanonical: it gives
a contiguous reference for small and moderate finite matrices while avoiding
indefinite extension into huge partition functions that are usually handled
by logarithms or asymptotics. It does not claim a distinct application for
every dimension. The full infinite domain n>=1 remains in Parameters.
## Breadth, bounds and ordering
There are exactly 150 rows, ordered by beta=1,2,4, then increasing n. The
selection first covers all three classical classes at the same sizes, rather
than giving extra depth only to the unitary class. Both odd and even sizes
matter, particularly for the orthogonal ensemble and its bordered skew-moment
formula. All positive sizes up to the stated bound are retained, including
n=1 even though the three classes coincide there. These are three occurrences
of the same scalar Gaussian integral in distinct ensemble contexts.
There is no extra weight-scale parameter, no second quantity selector, and no
exploratory continuous reference group. The product of the three symmetry
classes with a contiguous dimension family is intrinsic to the classical
ensemble question, not a Cartesian mesh of arbitrary inputs. The finite
extent is a practical choice, independent of decimal notation; the 150-row
count is a consequence rather than a target. Quantiles and critical values
are not involved.
A 100-significant-digit scientific-notation serialization keeps every value
readable even when its integer part is very large. `numerical-evidence.json`
records the measured entry lengths, block size and precision. Those costs
leave extension room; fitting more entries is not itself a reason to extend
beyond n=50. The generator's default enumeration, the program example and the
completeness note use precisely this selection. `checks.py` independently
integrates every selected member, and confirms the stored enumeration.
## Sources checked
* NIST DLMF 5.14.3 and 5.14.6, https://dlmf.nist.gov/5.14.E6:
Vandermonde convention, Gaussian measure and gamma product.
* P. J. Forrester and S. O. Warnaar, *The importance of the Selberg integral*,
Bull. Amer. Math. Soc. 45 (2008), 489-534, arXiv:0710.3981,
section 1, pages 3-4, equations (1.4)-(1.6): classical symmetry classes,
eigenvalue count and normalization.
* P. J. Forrester, *Meet Andreief, Bordeaux 1886, and Andreev, Kharkov 1882-83*,
arXiv:1806.10411, equations (1.7) and (3.1): integration identities supporting
the independent moment checks, not evidence for an arbitrarily chosen mesh.
The independent critique must assess beta and n separately and fingerprint
`generate.py`, the sole source controlling the enumeration. The accompanying
checks and rationale are also attachments so the critic can inspect them.