# Sources and normalization evidence for T564

Skill manifest: agents/runs/20261027T230956Z-build-skills/manifest.json.
Both archived copies have SHA-256
e37efb7e2ededf6c1d7134c673f152f648f26e0267c556aebdf3ec65765d8b4c.

Checked the following primary sources during this build:

- NIST DLMF 5.14.5, https://dlmf.nist.gov/5.14#E5. Setting its insertion
  index to zero gives the full positive-orthant Laguerre integral. The
  factors are Gamma(a+(n-k)c)*Gamma(1+kc)/Gamma(1+c), k=1,...,n.
  Its convergence hypotheses hold since a=c(r+1)>0 and c>0.
- Dumitriu and Edelman, https://arxiv.org/pdf/math-ph/0206043, Section 1.2,
  Figure 1 and equations (3)–(4): beta=1,2,4 for the classical scalar fields,
  the rectangular Wishart exponent, and the reciprocal normalization with
  weight exp(-lambda/2). Rename the paper's m rows and n columns to this
  table's n and n+r. Substitute lambda=2*x. The homogeneous factor is
  2^[a*n+c*n*(n-1)]=2^[c*n*(n+r)], cancelling the source's power of 2.
- Forrester and Warnaar, https://people.smp.uq.edu.au/OleWarnaar/pubs/Selberg_review.pdf,
  equations (3.18)–(3.19), checked against the search engine's extracted
  primary-source page. The extra 1/n! in (3.19) disagrees with its stated
  ordered-chamber density. This table uses the agreeing DLMF and
  Dumitriu–Edelman normalizations. Direct control: expanding (x-y)^2 gives
  integral 2, whereas the printed (3.19) yields 1/2 for a=c=1,n=2.
- Forrester, https://arxiv.org/pdf/1806.10411, equation (1.7) and Section 3.1,
  equation (3.1), for the determinant and skew-kernel Pfaffian integral
  identities. The odd-dimensional extension uses a bordering vector of
  ordinary moments.
- Ishikawa and Zeng, https://www.math.okayama-u.ac.jp/~mi/fpsac/fpsac14_poster1.pdf,
  the stated de Bruijn formula for paired columns, for the beta=4 confluent
  determinant. Its chamber integral equals Pf(Q); symmetry multiplies it by
  n! for this table's domain.

External access limitations: a direct urllib download of DLMF's TeX returned
HTTP 403. The DLMF HTML mathematical equation was accessible through the web
tool and was read. The MIT author PDF of Dumitriu–Edelman had corrupted text
extraction; the arXiv PDF had legible mathematical text and was read instead.
No inaccessible source was silently treated as evidence. An initially guessed
arXiv path math/0206043 was an unrelated biography; it was discarded, and the
correct primary source is math-ph/0206043.

Corpus evidence is in screen.json and lookup-success.json. Public searches
for Wishart and Mehta were empty; Selberg matched a singular elliptic-integral
table. Laguerre hits concerned polynomials, quadrature or other functions.
T563 was read through the authenticated API and is the Jacobi integral,
not the requested positive-orthant integral. The title claim created T564.
API numeric lookups used numberdb.search, which serializes the API record
correctly, after two raw record-shape attempts were refused. The successful
lookups of 2, sqrt(pi), pi and 135 did not reveal a Wishart integral family.
The pi lookup is a general normalization-neighbor probe; pi itself is not a
value asserted for this selected Wishart family.

Existing tags integral, probability theory and special values were verified
on the live list and used. The tag random matrix theory is absent from both
pages. Proposed subject grouping: this table, T563, T379, T441 and T451.
The proposal does not modify any published table or create a singleton tag.