from fractions import Fraction as F
from math import comb, factorial
# Scalar Gaussian polynomial cancellation is a finite consistency check, not a proof for all entire scores.
def gm(n):
    return 0 if n%2 else factorial(n)//(2**(n//2)*factorial(n//2))
for n in range(1,15):
    value=sum(comb(n,k)*(-1)**(k//2)*gm(n-k)*gm(k) for k in range(0,n+1,2))
    assert value==0,(n,value)
print('PASS: E[(U+iV)^n]=0, independent standard Gaussians, n=1,...,14 (exact moments).')
vobs,vsim,n,m=F(3),F(5),4,7
assert vobs/n+vsim/(n*m)==F(13,14)
print('PASS: observation/simulation decomposition at v_obs=3, v_sim=5, n=4, m=7 gives 13/14.')
H,c,S=F(2),F(1,2),F(3)
assert c*H==1
endpoint=2*c*c*S
interior=S/(H*H)
assert endpoint==2*interior
print('PASS: scalar cH=1 endpoint covariance = 3/2; interior covariance = 3/4.')
print('Scope: these checks do not establish the general CLTs or rerun experiments.')