Independent verification run 2: exact checks (fractions only)
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(c) interval (0,pi): E_j = j^2, closed form of rho_J
  rho_J == 3(2J+1)/((J+1)(3J+5)) exactly for all J <= 2000   (rho_1 = 9/16)

(a,b,d) domains of Remark 5.3 (exact), J <= 2000
  interval (0,pi)                                  n=1  J<= 2000:  rho_J<1 OK, Yang strict OK;  max rho = 9/16 = 0.562500 at J = 1
  square (0,pi)^2                                  n=2  J<= 2000:  rho_J<1 OK, Yang strict OK;  max rho = 9/16 = 0.562500 at J = 1
  rectangle (0,pi)x(0,pi/sqrt2)                    n=2  J<= 2000:  rho_J<1 OK, Yang strict OK;  max rho = 1/4 = 0.250000 at J = 1
  cube (0,pi)^3                                    n=3  J<= 2000:  rho_J<1 OK, Yang strict OK;  max rho = 9/16 = 0.562500 at J = 1
  => maxima 9/16, 9/16, 1/4, 9/16 at J = 1: matches Remark 5.3

(a,b) further boxes and hypercubes (exact)
  hypercube (0,pi)^4                               n=4  J<= 800:  rho_J<1 OK, Yang strict OK;  max rho = 9/16 = 0.562500 at J = 1
  hypercube (0,pi)^5                               n=5  J<= 800:  rho_J<1 OK, Yang strict OK;  max rho = 9/16 = 0.562500 at J = 1
  hypercube (0,pi)^6                               n=6  J<= 800:  rho_J<1 OK, Yang strict OK;  max rho = 9/16 = 0.562500 at J = 1
  hypercube (0,pi)^7                               n=7  J<= 800:  rho_J<1 OK, Yang strict OK;  max rho = 9/16 = 0.562500 at J = 1
  box with 1/L_i^2 = 1,4                           n=2  J<= 1200:  rho_J<1 OK, Yang strict OK;  max rho = 25/142 = 0.176056 at J = 2
  box with 1/L_i^2 = 1,9/4                         n=2  J<= 1200:  rho_J<1 OK, Yang strict OK;  max rho = 225/1012 = 0.222332 at J = 2
  box with 1/L_i^2 = 1,100                         n=2  J<= 1200:  rho_J<1 OK, Yang strict OK;  max rho = 363/23930 = 0.015169 at J = 16
  box with 1/L_i^2 = 1,2,3                         n=3  J<= 1200:  rho_J<1 OK, Yang strict OK;  max rho = 9/64 = 0.140625 at J = 1
  box with 1/L_i^2 = 1,1,1/25                      n=3  J<= 1200:  rho_J<1 OK, Yang strict OK;  max rho = 2025/55888 = 0.036233 at J = 7

(a,b) disconnected sets (exact); equality case of Yang's first inequality
  two disjoint congruent intervals                 n=1  J<= 1200:  rho_J<1 OK, Yang strict OK;  max rho = 9/16 = 0.562500 at J = 2
  two congruent intervals, J=1: E1=E2=1, n*Q_1(E_2) = 0  (Yang 1 is an equality)
  two disjoint congruent squares                   n=2  J<= 1200:  rho_J<1 OK, Yang strict OK;  max rho = 9/16 = 0.562500 at J = 2
  three disjoint congruent cubes                   n=3  J<= 700:  rho_J<1 OK, Yang strict OK;  max rho = 9/16 = 0.562500 at J = 3
  60 random disjoint unions of 2-5 boxes, each duplicated with prob. 0.3 (n = 1,2,3; rational squared sides), J <= 300: rho_J < 1 and strict Yang whenever E_(J+1) > E_1 -- all OK

(e) harmonic oscillator -Delta+|x|^2 on R^n, normalisation (n+p)/n (Remark 5.2)
  R^1: analogue of rho_J equals 1 exactly at all 2000 shell closures with J <= 2000
        inside degenerate shells M1~^2 - M2~^2 > 0 for all J <= 2000
  R^2: analogue of rho_J equals 1 exactly at all 62 shell closures with J <= 2000
        and, INSIDE degenerate shells (E_J = E_(J+1)), M1~^2 - M2~^2 = 0 = gap^2/4 at 31 values of J <= 2000; first ones (J, E_J): [(4, 6), (12, 10), (24, 14), (40, 18), (60, 22), (84, 26)]
  R^3: analogue of rho_J equals 1 exactly at all 21 shell closures with J <= 2000
        inside degenerate shells M1~^2 - M2~^2 > 0 for all J <= 2000

ALL EXACT CHECKS PASSED  (1.8 s)
