# Sine-process (GUE) check: given points at +-s/2, conditional density of a third point at x is
# rho3(-s/2,s/2,x)/rho2(-s/2,s/2). I(s)=int_{|x|>s/2} x^-6 rho dx  (lower bound for E[R_3^2 | gap s]).
import numpy as np
from mpmath import mp, mpf, sin, pi, quad, matrix, det, inf
mp.dps=40
def S(a,b):
    d=a-b
    return mpf(1) if d==0 else sin(pi*d)/(pi*d)
def rho(pts):
    n=len(pts); return det(matrix([[S(pts[i],pts[j]) for j in range(n)] for i in range(n)]))
for s in [mpf('0.4'),mpf('0.2'),mpf('0.1'),mpf('0.05'),mpf('0.025')]:
    r2=rho([-s/2,s/2])
    f=lambda x: 2*x**-6*rho([-s/2,s/2,x])/r2
    I=quad(f,[s/2,s,4*s,1,10,inf])
    print(float(s), float(I), 's*I=',float(s*I))