(a) pi*0.45^2 = 0.636173 >= 0.63: True ;  1 - 0.63/2 = 0.685 <= 0.7: True
    1/c_H = 13.428991701438
(b) covering scheme with exact geometry (levels <= 7):
    step 1:     1 squares (total area 1.0000 >= |U|/2 = 0.5000),      1 discs in all, |U_1| = 0.36383, ratio 0.3638 (<= 0.7: True)
    step 2:    12 squares (total area 0.1875 >= |U|/2 = 0.1819),     13 discs in all, |U_2| = 0.24455, ratio 0.6721 (<= 0.7: True)
    step 3:   105 squares (total area 0.1223 >= |U|/2 = 0.1223),    118 discs in all, |U_3| = 0.16673, ratio 0.6818 (<= 0.7: True)
    step 4:   601 squares (total area 0.0834 >= |U|/2 = 0.0834),    719 discs in all, |U_4| = 0.11369, ratio 0.6819 (<= 0.7: True)
    step 5: free dyadic squares of level <= 7 have total area 0.0148 < |U|/2 = 0.0568 : resolution exhausted, stop
    719 discs, all inside the square: True ; minimal gap between two discs: 7.813e-04 (pairwise disjoint: True) ; uncovered measure 0.11369
(d) F_alpha upper bounds of explicit partitions  vs the claimed limit min{alpha c_H, 1}:
    alpha =   1.0000 : limit 0.07447 ; honeycomb bound at N=10^6: 0.08930 ; single square: 0.18109 ; discs+dust: 1 + 2 alpha eps -> 1
    alpha =   5.0000 : limit 0.37233 ; honeycomb bound at N=10^6: 0.42672 ; single square: 0.90545 ; discs+dust: 1 + 2 alpha eps -> 1
    alpha =  13.0000 : limit 0.96805 ; honeycomb bound at N=10^6: 1.10156 ; single square: 2.35417 ; discs+dust: 1 + 2 alpha eps -> 1
    alpha =  13.4290 : limit 1.00000 ; honeycomb bound at N=10^6: 1.13775 ; single square: 2.43186 ; discs+dust: 1 + 2 alpha eps -> 1
    alpha =  14.0000 : limit 1.00000 ; honeycomb bound at N=10^6: 1.18592 ; single square: 2.53526 ; discs+dust: 1 + 2 alpha eps -> 1
    alpha =  30.0000 : limit 1.00000 ; honeycomb bound at N=10^6: 2.53561 ; single square: 5.43270 ; discs+dust: 1 + 2 alpha eps -> 1
(c) Phi_beta >= c_H + 0.06 m, beta = 1/alpha, alpha in {1,3,8,13,1/c_H}: 12500 tests, violations 0; min slack by alpha: {1.0: 0.0734, 3.0: 0.0734, 8.0: 0.0734, 13.0: 0.0734, 13.429: 0.0734}
RESULT PASS
