exp(1/8+e^(1/8)-1) = 1.2945310  (< 1.2946 claimed): True
2 e^(3/2) * 1.2946 = 11.60399  (< 11.61 < 12 claimed): True
(e/8)^(1/4) e^(1/64) = 0.77551  (< 0.78 claimed): True
2+e = 4.71828  (< 4.72 claimed): True
eps_n >= 1 for some n up to 590086; eps_n < 1 for all n in (590086, 5e6]; at n=590087: log terms = (-149994.42, -0.0, -13216.278)
   n=10000: eps_n = 8.267e+05
   n=100000: eps_n = 5.303e+05
   n=500000: eps_n = 13.24
   n=600000: eps_n = 0.7514
   n=1000000: eps_n = 6.255e-06
   n=2000000: eps_n = 6.977e-19
Range I: beta_1<=1 and beta_2<=2/n for all 10<=n<=3000 with i-2<=n^2/64 (exact): True
Range I: q=1/(n-2-floor(n/4)) <= 2/n and 1/(n-1-L) <= 2/n for 8<=n<=1e5 (exact): True
Range II: (2n-5)/(2(n-2)^2) >= 1/n for 3<=n<=1e5 (exact): True
Range II: n>=23 => n^2/64 > 8, so i>=9: 8.266
(eq:f-lower) exact check, 3<=n<=30, 2<=i<=n(n-1), 0<=c'<=i-2: 2421006 cases, failures: 0
f increasing in m (integer steps), 3<=n<60: True
(i-2)/(n-2) <= i/n + 3i/n^2 for 6<=n<200, 0<=i<n^2 (exact): True
(n+1) e^(-sqrt n) <= 1 for 3<=n<1e6: True
1/(e^(1/n)-1) >= n-1/2 for 3<=n<1e6: False
1-e^(-1/n) >= 1/(n+1) for 3<=n<1e6: True
Lemma 3.4 randomized test: 515896 (path, i) pairs on 3000 random trees/paths (4<=n<=25), violations of S_i >= f(m): 0
recheck with expm1: 1/(e^(1/n)-1) >= n-1/2 fails for 0 values of n in [3,1e6); first few: []
(the earlier False came from cancellation in exp(1/n)-1; first 'failures' with exp()-1: [91254, 91255, 91572] )
analytic: e^x - 1 <= x/(1-x/2) for 0<x<2 (since sum x^k/k! <= x sum (x/2)^(k-1)), hence 1/(e^x-1) >= 1/x - 1/2: holds
