# Independent integral checks for T566

`checks.py` evaluates the defining integrals without a Gamma function or a
Mehta product. Its exact rational coefficient and exponents of sqrt(2) and
sqrt(pi) are compared with the generator for every beta=1,2,4 and n=1,...,50.
The shared `check.py` is the unchanged campaign checker, used for exactness,
measurement, named imports and exact agreement. This is build verification;
it does not replace the campaign's separate independent critique.

Write w(x)=exp(-x^2/2) and let
mu_k = integral x^k w(x) dx / sqrt(2*pi).
Odd moments vanish. Integration by parts gives mu_0=1 and
mu_(2r)=(2r-1)mu_(2r-2). These elementary Gaussian moments are independent
of the Mehta product. Determinants and Pfaffians are eliminated over QQ,
with explicit pivoting, so no cancellation error enters the integral checks.

## beta=2

Expanding the Vandermonde determinants and integrating term by term gives

Z(2,n) = n! (sqrt(2*pi))^n det[mu_(i+j)]_(i,j=0,...,n-1).

This is Andreief's identity, equation (1.7) in Forrester,
arXiv:1806.10411. The factor n! is for the full labelled-coordinate domain.

## beta=4

The confluent Vandermonde determinant with alternating value and derivative
columns is product_(i<j)(x_j-x_i)^4. Expanding it and integrating the paired
columns gives

Z(4,n) = n! (sqrt(2*pi))^n Pf[(j-i)mu_(i+j-1)]_(i,j=0,...,2n-1).

The zero diagonal is defined as zero without needing a negative moment.
This identity also follows directly by grouping the terms of the determinant
into pairs. In particular n=1 gives sqrt(2*pi), fixing the sign and any
possible powers of two. It is the confluent version of the Pfaffian integration
identity discussed in Forrester, section 3.1.

## beta=1

Let A_ij be integral over R^2 of sign(y-x)x^i y^j w(x)w(y) dx dy.
Reflection makes A_ij zero for even i+j; antisymmetry determines A_ji.
For i even and j=2k+1, integration by parts in y gives

integral_x^infinity y^(2k+1) w(y) dy
 = w(x) sum_(l=0)^k 2^(k-l) k!/l! x^(2l).

The odd full moment is zero, so integrating sign(y-x) in y is twice this
expression. Consequently

A_(i,2k+1)/sqrt(pi)
 = 2 sum_(l=0)^k [2^(k-l) k!/l!] mu_(i+2l) / 2^((i+2l)/2).

These coefficients are exact rationals. For even n, de Bruijn's sign-kernel
identity gives Z(1,n)=n! Pf(A), hence a rational times sqrt(pi)^(n/2).
For odd n, border A by b_i=integral x^i w(x)dx=sqrt(2*pi)mu_i and its
negative transpose, with a zero corner. Every Pfaffian term uses the border
once, giving a rational times sqrt(2)*sqrt(pi)^((n+1)/2). Forrester equation
(3.1) and its following sentence describe the even and odd integration rules.

The normalized matrices in the code remove these uniform factors before
elimination. This is why no transcendental arithmetic or half-integer Gamma
values are needed in the independent exact coefficient comparison.

## Controls and numerical claims

Before evaluating the family, the code checks a determinant equal to 1,
a pivot-swap determinant equal to -6, a singular determinant equal to 0,
a 4-by-4 Pfaffian equal to 8, and a swap-requiring Pfaffian equal to -1.
Gaussian moments mu_0=mu_2=1, mu_4=3 and mu_3=0 are controls.
The first skew coefficients are A_01/sqrt(pi)=2 and A_03/sqrt(pi)=5.

The independently known one-dimensional integral is sqrt(2*pi). For n=2,
the orthogonal change u=(x+y)/sqrt(2), v=(y-x)/sqrt(2) separates the integral:
Z(1,2)=4sqrt(pi), Z(2,2)=4pi, and Z(4,2)=24pi. The absolute-value integral
in the beta=1 case uses integral |v|exp(-v^2/2)dv=2. The n=3 orthogonal
control is 6sqrt(2)pi, obtained from the bordered 4-by-4 matrix.

The exact comparison is followed by 640-bit real-ball evaluation of the
independent components. Both these enclosures and the generator's 429-bit
enclosures must be finite, narrow and wholly contained in each serialized
100-significant-digit decimal interval. Decimal endpoints are constructed
as exact rationals; containment is stronger than overlapping two rounded
answers. `numerical-evidence.json` records the minimum measured relative
accuracy, value lengths, block sizes and software versions.

The dimension recurrence Z(beta,n)/Z(beta,n-1)
 = sqrt(2*pi) Gamma(1+n*beta/2)/Gamma(1+beta/2)
is also checked on all 147 adjacent pairs with independently integrated
values. This recurrence check is supplementary, not the independent method.
With --stored, the script reads every row back from T566, checks its exact
parameter order and text, contains each independent enclosure in that stored
interval, and checks the recurrence on the stored intervals themselves.

The coordinated skill manifest and its verified SHA-256 values are cited in
`skill-evidence.json` and embedded in the numerical evidence. No source's
rounding or printed digit string supplies the values.