(U) exact f_K(0) = f_{2m}(m) versus analytic bound  B(m) = sqrt(3/(pi m)) + (2/pi) pi^(1-2m)/(2m-1)
  m=  1  f_K(0)=1.00000000  sqrt(3/(pi m))=0.97720502  B(m)=1.17984739  ratio=0.847567
  m=  2  f_K(0)=0.66666667  sqrt(3/(pi m))=0.69098830  B(m)=0.69783229  ratio=0.955339
  m=  3  f_K(0)=0.55000000  sqrt(3/(pi m))=0.56418958  B(m)=0.56460565  ratio=0.974131
  m=  5  f_K(0)=0.43041777  sqrt(3/(pi m))=0.43701937  B(m)=0.43702175  ratio=0.984889
  m= 10  f_K(0)=0.30669310  sqrt(3/(pi m))=0.30901936  B(m)=0.30901936  ratio=0.992472
  m= 20  f_K(0)=0.21768872  sqrt(3/(pi m))=0.21850969  B(m)=0.21850969  ratio=0.996243
  m= 40  f_K(0)=0.15421970  sqrt(3/(pi m))=0.15450968  B(m)=0.15450968  ratio=0.998123
  m= 80  f_K(0)=0.10915237  sqrt(3/(pi m))=0.10925484  B(m)=0.10925484  ratio=0.999062
  max ratio f_K(0)/B(m) over m=1..80: 0.999062  (must be <= 1)  -> OK

(L) exact P(|K|<eps) versus (3/4) eps/(eps+2 sqrt(m/6))
  failures: 0;  min ratio P/LB = 1.8777
  m=  1: P(|K|<1/100) = 1.990000e-02;  sqrt(m)*P = 1.990000e-02  (LLT prediction 2*0.01*sqrt(6/(2 pi))/1 = 1.954410e-02)
  m= 10: P(|K|<1/100) = 6.133803e-03;  sqrt(m)*P = 1.939679e-02  (LLT prediction 2*0.01*sqrt(6/(2 pi))/1 = 1.954410e-02)
  m= 60: P(|K|<1/100) = 2.519972e-03;  sqrt(m)*P = 1.951962e-02  (LLT prediction 2*0.01*sqrt(6/(2 pi))/1 = 1.954410e-02)

(S) sin(u)/u <= exp(-u^2/6) on [0.01,pi], grid of 200000 points (near 0 the difference is -u^4/180+O(u^6),
    below float resolution; the proof is analytic: log(sin u/u) = sum_k log(1-u^2/(k pi)^2) <= -u^2/6):
  max of sin(u)/u - exp(-u^2/6) on grid = -5.556e-11  (must be <= 0)

(M) Monte Carlo (evidence only): Y = sgn(JJ')min(|J|,|J'|), J,J' ~ U(-1,1), 10^6 samples, seed 1
  empirical bin masses vs exact (density 1-|y|):
   [-1.0,-0.8): emp 0.0199  exact 0.0200
   [-0.8,-0.6): emp 0.0601  exact 0.0600
   [-0.6,-0.4): emp 0.0999  exact 0.1000
   [-0.4,-0.2): emp 0.1395  exact 0.1400
   [-0.2,+0.0): emp 0.1804  exact 0.1800
   [+0.0,+0.2): emp 0.1801  exact 0.1800
   [+0.2,+0.4): emp 0.1399  exact 0.1400
   [+0.4,+0.6): emp 0.1001  exact 0.1000
   [+0.6,+0.8): emp 0.0600  exact 0.0600
   [+0.8,+1.0): emp 0.0199  exact 0.0200
  empirical Var(Y) = 0.16662  (exact 1/6 = 0.16667)
