== Part A: Lemma 2.2 (bound 3 log X + 2 log(1/a_1) + 2 + 2 log 2), all thresholds
   families: a_n = n: 1; a_n = 1: 1; a_n ~ sqrt n: 1; a_n ~ 1/n: 1; a_n = 2^-n: 1; lacunary spikes: 15; late spike: 18; late spikes: 36; blocks a_n = n: 4; greedy to the cap, then stop: 28; greedy to the cap, then a_n = n: 28; greedy, gap, greedy: 28; late burst: 96; exact greedy (ties): 24; exact greedy (ties), gap, a_n = n: 24; random multi-phase greedy: 300; random sparse: 300; random short: 3000; exhaustive N=8: 65536
   sequences 69442, (sequence, threshold) pairs 588035, checks of (i) 521493, of (ii) 273493, of the split 475863
   max W/bound = 0.4978 at ('greedy to the cap, then a_n = n', 1000.0, 80.7699, 162.2647)
   max b_n/(2X) in (i) = 1 - 5.0e-12 (always < 1, exact test) ; max n a_1/(2X^2) in (ii) = 0.1893 ; max |G_X|/(sqrt(X) W(X)) in (v) = 1.0000 (equality at X = 1)
   split at N/2: max of the sum over 2n <= N divided by 1 + (1/2) log(4X^3/a_1^2) = 0.8502 ; over 2n > N: 0.8156
   max of (W - 2 log(1/a_1) - 2 - 2 log 2)/log X over thresholds X >= 4: 1.0813
== Part B: Theorem 3.1 (A = 3, B = 2 log(1/a_1) + 2 + 2 log 2) and Theorem 1.1(a)
   sequences 254
   Theorem 3.1: 1016 tests (random x_0 >= 1, random non-increasing step functions g), max LHS/RHS = 0.3890
   Theorem 1.1(a), x_0 = 1: 14253 tests (powers x^-alpha, alpha in (0.51, 0.55, 0.75, 1.0, 2.0); cut-off functions), max LHS/RHS = 0.4530
   constant 2 + 5 log 2 = 5.465736
== Part C: Lemma 4.1 and Proposition 4.2 (exact rational arithmetic)
   s_l = 2^(l+1), l <= 8: m_l = [2, 4, 10, 21, 43, 88, 176, 354], N = 699
   Proposition 6.1(a), s_l = 2^(l^2+1), l <= 3: m_l = [2, 64, 3545], N = 3612, max a_n (n >= 2) = 0.217778
   Proposition 6.1(b), Y_l = 3*4^(l+1), l <= 8: m_l = [4, 8, 18, 37, 76, 152, 306, 613], N = 1215, max a_n (n >= 2) = 0.118050
   target lists 403, indices checked exactly 98036: c_n = Y_l a_n, 0 < a_n <= 5/16, b_n <= n, c_n <= s_l b_n
   max a_n (n >= 2) = 0.238772 (bound 5/16 = 0.3125) ; min m_l^max / ((3/16) sqrt(Y_l) log(Y_l/Y_{l-1})) = 1.6220
   max of sum_l m_l/s_l divided by (1/2) log(Y_L/Y_0) = 0.9710 (must be <= 1)
== Part D: Proposition 6.1
   (a) m_1..m_6 = [2, 64, 3545, 635960, 418646872, 1047919654072]
   (a) partial sums of sum f_0(c_n/a_n) over complete phases: L=1: 0.7500, L=2: 1.2500, L=3: 1.6347, L=4: 1.9379, L=5: 2.1874, L=10: 3.0228, L=20: 3.9177, L=40: 4.8448
   (a) lower bound (3 log 2/8) H_40 = 1.1121
   (a) sum_n f_0(n^2) = 1/2 + sum_l (1 - 2^(1-2l))/l^2 = 1/2 + pi^2/6 - 2 sum_l 4^-l/l^2 = 1.609629 ; bound 2 + pi^2/6 = 3.644934
   (a) theta = 0.00: sum_{n <= 2^17} f_theta(n^2) = 1.3884 <= 2 + (pi^2/6)/(1 - 2 theta) = 3.6449
   (a) theta = 0.10: sum_{n <= 2^17} f_theta(n^2) = 1.5898 <= 2 + (pi^2/6)/(1 - 2 theta) = 4.0562
   (a) theta = 0.25: sum_{n <= 2^17} f_theta(n^2) = 2.0052 <= 2 + (pi^2/6)/(1 - 2 theta) = 5.2899
   (a) theta = 0.40: sum_{n <= 2^17} f_theta(n^2) = 2.6381 <= 2 + (pi^2/6)/(1 - 2 theta) = 10.2247
   (a) theta = 0.49: sum_{n <= 2^17} f_theta(n^2) = 3.1910 <= 2 + (pi^2/6)/(1 - 2 theta) = 84.2467
   (b) Y_l = 3*4^(l+1) is not a square for l < 200 ; m_1..m_10 = [4, 8, 18, 37, 76, 152, 306, 613, 1228, 2458]
   (b) partial sums sum_{l <= L} m_l/s_l: L=1: 0.5774, L=2: 1.1547, L=3: 1.8042, L=10: 6.6096, L=40: 27.4037 ; (3 log 2/8) * 40 = 10.3972
   (c) sum_n f_0(n^2/2) = 2.0689 (blocks l <= 30; the rest is < 0.0471) <= 1/2 + sqrt(2) pi^2/6 = 2.8263
== Part E: Proposition 6.2 (a_n <= n psi(n), spikes at M_k, zeros elsewhere), exact
   psi(n) = n                   c/a at the spikes M_2, M_3, ...: 1.5000, 1.7222, 1.6134, 1.6203, 1.6172, 1.6184, 1.6179
   psi(n) = 1 + floor(log2 n)   c/a at the spikes M_2, M_3, ...: 1.5000, 1.8125, 1.5661
   psi(n) = ceil(sqrt n)        c/a at the spikes M_2, M_3, ...: 1.5000, 1.8077, 1.5707, 1.6366, 1.6110
== Part F: Proposition 7.1
   (a) b_n/(2X) and n a_1/(2X^2): X=4: 0.500000, 0.281250 ; X=5: 0.600000, 0.400000 ; X=10: 0.800000, 0.675000 ; X=100: 0.980000, 0.965250 ; X=1000: 0.998000, 0.996502 ; X=1000000: 0.999998, 0.999997
   (b) a_n = n: W(X) >= (sqrt(6)/2) log X - 3/2: 1e0: 1.0000 >= -1.5000 ; 1e1: 3.4073 >= 1.3201 ; 1e2: 6.1991 >= 4.1402 ; 1e3: 8.9547 >= 6.9602 ; 1e4: 11.7898 >= 9.7803 ; 1e5: 14.6107 >= 12.6004 ; 1e6: 17.4290 >= 15.4205 ; 1e7: 20.2496 >= 18.2406 ; 1e8: 23.0697 >= 21.0606
   (c) X=4, a_1=1e-10: K=33, W >= 17.500 >= theta log(1/a_1) = 16.610 (theta = 0.7213, max a_n = 0.322)
   (c) X=4, a_1=1e-60: K=199, W >= 100.500 >= theta log(1/a_1) = 99.658 (theta = 0.7213, max a_n = 0.301)
   (c) X=100, a_1=1e-10: K=218, W >= 22.800 >= theta log(1/a_1) = 21.854 (theta = 0.9491, max a_n = 0.052)
   (c) X=100, a_1=1e-60: K=1311, W >= 132.100 >= theta log(1/a_1) = 131.126 (theta = 0.9491, max a_n = 0.054)
   (c) X=10000, a_1=1e-30: K=6873, W >= 69.730 >= theta log(1/a_1) = 68.732 (theta = 0.9950, max a_n = 0.005)
== Part G: Lemma 8.1 (bound 5 log X + 2 log(1/a_1) + 2 + 4 log 2) and Theorem 8.2; zero tail included
   sequences 12752 (of which exhaustive, length 7: 12288), (sequence, threshold) pairs 347759
   max of the sum divided by the bound = 0.6002 at ('greedy for c_n <= X b_n', 23.8659, 12.3845, 20.6349)
   c_n >= b_n^2/(2n): 168161 exact checks, max b_n^2/(2 n c_n) = 0.9592
   step (3): 168161 exact checks, max b_n/(4X^2) = 0.2838, max n a_1/(4X^3) = 0.1055
   Theorem 8.2, x_0 = 1, f = x^-alpha (alpha = 1.1, 1.5, 2): 38256 tests, max LHS/RHS = 0.3389
== Part H: elementary examples (exact)
   closed forms for a_n = n, a_n = 1, a_n = 2^-n; c_M = 2M for the late spike; spikes at 2^k; r_n >= 1 + a_1: ok
TOTAL failures: 0
