(1) unit disk: E = j_{m,k}^2, multiplicity 2 for m >= 1
  2456 eigenvalues below 10000; sign-change count agrees for every order m
  unit disk                            J<=2000: rho_J < 1 OK; max rho_J = 0.591926 at J = 1; min(1-rho_J) = 0.4081; min relative Yang deficit -Q_J(E_J+1)/(J E_J+1^2) = 5.933e-03 at J = 2000

(2) unit ball in R^3: E = j_{l+1/2,k}^2, multiplicity 2l+1
  4854 eigenvalues below 1764
  unit ball in R^3                     J<=2000: rho_J < 1 OK; max rho_J = 0.615144 at J = 1; min(1-rho_J) = 0.3849; min relative Yang deficit -Q_J(E_J+1)/(J E_J+1^2) = 1.249e-02 at J = 1973

(3) two disjoint unit disks
  E_1 = E_2 = 5.783185962947; Q_1(E_2) = (E2-E1)(E2-(1+4/n)E1) = -0.0 (Yang's first inequality is an equality)
    [two disjoint unit disks] J=1: E_(J+1) = E_1 (to 1e-9), n*Q_J(E_(J+1)) = 0.000e+00
  two disjoint unit disks              J<=2000: rho_J < 1 OK; max rho_J = 0.591926 at J = 2; min(1-rho_J) = 0.4081; min relative Yang deficit -Q_J(E_J+1)/(J E_J+1^2) = 8.390e-03 at J = 2000

(4) rho_1 for the unit ball in R^n
  n =     1: rho_1 = 0.562500
  n =     2: rho_1 = 0.591926
  n =     3: rho_1 = 0.615144
  n =    10: rho_1 = 0.706192
  n =   100: rho_1 = 0.882836
  n =  1000: rho_1 = 0.967251
  => agree with Remark 5.3 to the 4 printed digits; rho_1 increases with n and stays < 1
  nu =    50: j_nu,1 (brentq) = 57.1168991601, DLMF 10.21.40 six terms = 57.1168991311
  nu =   200: j_nu,1 (brentq) = 211.0291665106, DLMF 10.21.40 six terms = 211.0291665801
  nu =   499: j_nu,1 (brentq) = 513.8495725707, DLMF 10.21.40 six terms = 513.8495726860
  nu =   500: j_nu,1 (brentq) = 514.8593116905, DLMF 10.21.40 six terms = 514.8593118059
  n =     10000 (asymptotic zeros): rho_1 ~ 0.992170
  n =    100000 (asymptotic zeros): rho_1 ~ 0.998239
  n =   1000000 (asymptotic zeros): rho_1 ~ 0.999613
  n = 100000000 (asymptotic zeros): rho_1 ~ 0.999982
  => rho_1 -> 1 as n -> infinity (roughly 1 - 2.47 (n/2)^(-2/3)), as stated in Remark 5.3

ALL BESSEL CHECKS PASSED (71.2 s)
