PART A. Exact rational arithmetic, 1 <= J <= 2000.
  For every J: asserted rho_J < 1 exactly, and n*Q_J(E_{J+1}) < 0 exactly whenever E_{J+1} > E_1.
  interval (0,pi), E = m^2 (n=1): max rho_J = 0.562500 (= 9/16) at J = 1; Yang strict for 2000 values of J, E_(J+1) = E_1 for 0
      rho_J at J = 1: 0.562500, 2: 0.454545, 3: 0.375000, 10: 0.163636, 100: 0.019575, 1000: 0.001996, 2000: 0.000999
  square (0,pi)^2, E = a^2+b^2 (n=2): max rho_J = 0.562500 (= 9/16) at J = 1; Yang strict for 2000 values of J, E_(J+1) = E_1 for 0
      rho_J at J = 1: 0.562500, 2: 0.000000, 3: 0.225000, 10: 0.008389, 100: 0.000000, 1000: 0.000000, 2000: 0.000005
  rectangle (0,pi)x(0,pi/sqrt2), E = a^2+2b^2 (n=2): max rho_J = 0.250000 (= 1/4) at J = 1; Yang strict for 2000 values of J, E_(J+1) = E_1 for 0
      rho_J at J = 1: 0.250000, 2: 0.166667, 3: 0.055556, 10: 0.033215, 100: 0.000000, 1000: 0.000007, 2000: 0.000003
  cube (0,pi)^3, E = a^2+b^2+c^2 (n=3): max rho_J = 0.562500 (= 9/16) at J = 1; Yang strict for 2000 values of J, E_(J+1) = E_1 for 0
      rho_J at J = 1: 0.562500, 2: 0.000000, 3: 0.000000, 10: 0.018450, 100: 0.000000, 1000: 0.000000, 2000: 0.000000
  interval: rho_J = 3(2J+1)/((J+1)(3J+5)) for all J <= 2000: True

PART B. Double precision (Bessel zeros), 1 <= J <= 2000.
  unit disk (n=2): E_1 = 5.7831859629 (j_01^2), max rho_J = 0.591926 at J = 1; Yang strict for 2000 values of J; min relative Yang deficit 5.933e-03
      rho_J at J = 1: 0.591926, 2: 0.000000, 3: 0.404656, 10: 0.063750, 100: 0.005596, 1000: 0.000003, 2000: 0.000000
  unit ball in R^3 (n=3): E_1 = 9.8696044011 (pi^2), max rho_J = 0.615144 at J = 1; Yang strict for 2000 values of J; min relative Yang deficit 1.249e-02
      rho_J at J = 1: 0.615144, 2: 0.000000, 3: 0.000000, 10: 0.139237, 100: 0.000000, 1000: 0.000000, 2000: 0.000000
  two disjoint unit disks (n=2): E_1 = E_2; J = 1: rho_1 = 0.000000, n*Q_1(E_2) = 0.000e+00 (Yang's first inequality is an equality 0 = 0); max rho_J = 0.591926 at J = 2; J with E_(J+1) = E_1: 1

PART C. Harmonic oscillator -Delta+|x|^2 (exact), normalisation (n+1)/n and (n+2)/n.
  n = 1: J <= 2000: 2000 values of J with E_J < E_(J+1); the ratio equals 1 for 2000 of them (other values: []). For E_J = E_(J+1): M~1^2 - M~2^2 > 0 for 0 J, = 0 for 0 J.
  n = 2: J <= 2000: 62 values of J with E_J < E_(J+1); the ratio equals 1 for 62 of them (other values: []). For E_J = E_(J+1): M~1^2 - M~2^2 > 0 for 1907 J, = 0 for 31 J.

PART D. J = 1 on the unit ball in R^n: rho_1 = (n^2/16)(E_2/E_1 - 1)^2 (double precision).
  n=1: E2/E1=4.000000 (1+4/n=5.000000), rho_1=0.562500
  n=2: E2/E1=2.538734 (1+4/n=3.000000), rho_1=0.591926
  n=3: E2/E1=2.045749 (1+4/n=2.333333), rho_1=0.615144
  n=4: E2/E1=1.796395 (1+4/n=2.000000), rho_1=0.634245
  n=5: E2/E1=1.645184 (1+4/n=1.800000), rho_1=0.650410
  n=6: E2/E1=1.543396 (1+4/n=1.666667), rho_1=0.664377
  n=8: E2/E1=1.414590 (1+4/n=1.500000), rho_1=0.687538
  n=10: E2/E1=1.336141 (1+4/n=1.400000), rho_1=0.706192
  n=20: E2/E1=1.174965 (1+4/n=1.200000), rho_1=0.765318
  n=50: E2/E1=1.073223 (1+4/n=1.080000), rho_1=0.837749
  n=100: E2/E1=1.037584 (1+4/n=1.040000), rho_1=0.882836
  n=200: E2/E1=1.019163 (1+4/n=1.020000), rho_1=0.918053
  n=500: E2/E1=1.007801 (1+4/n=1.008000), rho_1=0.950799
  n=1000: E2/E1=1.003934 (1+4/n=1.004000), rho_1=0.967251

PART E. Identities on the interval (0,pi) (double precision, l <= L = 10^6, with tail estimates).
  j=1: sum (E_l-E_j) g^2 = 1.0000000000 (expected 1); sum (E_l-E_j)^2 g^2 + tail = 4.000000 (expected 4E_j = 4)
  j=2: sum (E_l-E_j) g^2 = 1.0000000000 (expected 1); sum (E_l-E_j)^2 g^2 + tail = 16.000000 (expected 4E_j = 16)
  j=3: sum (E_l-E_j) g^2 = 1.0000000000 (expected 1); sum (E_l-E_j)^2 g^2 + tail = 36.000000 (expected 4E_j = 36)
  j=4: sum (E_l-E_j) g^2 = 1.0000000000 (expected 1); sum (E_l-E_j)^2 g^2 + tail = 64.000000 (expected 4E_j = 64)
  j=5: sum (E_l-E_j) g^2 = 1.0000000000 (expected 1); sum (E_l-E_j)^2 g^2 + tail = 100.000000 (expected 4E_j = 100)
  J=1, z=E_(J+1)=4: -n Q_J(z) = 3.000000;  sum_(j<=J<l) terms + tail = 3.000000
  J=1, z=midpoint=2.5: -n Q_J(z) = 3.750000;  sum_(j<=J<l) terms + tail = 3.750000
  J=2, z=E_(J+1)=9: -n Q_J(z) = 23.000000;  sum_(j<=J<l) terms + tail = 23.000000
  J=2, z=midpoint=6.5: -n Q_J(z) = 25.500000;  sum_(j<=J<l) terms + tail = 25.500000
  J=3, z=E_(J+1)=16: -n Q_J(z) = 86.000000;  sum_(j<=J<l) terms + tail = 86.000000
  J=3, z=midpoint=12.5: -n Q_J(z) = 91.250000;  sum_(j<=J<l) terms + tail = 91.250000
  J=4, z=E_(J+1)=25: -n Q_J(z) = 230.000000;  sum_(j<=J<l) terms + tail = 230.000000
  J=4, z=midpoint=20.5: -n Q_J(z) = 239.000000;  sum_(j<=J<l) terms + tail = 239.000000

ALL ASSERTIONS PASSED
