k=2: identities OK=True; sigma=0.563299 lambda=0.408248 mu_k=0.310102; mu_k < 16/(32k-13) < 1/(2k-1): True; mu_k > 1/(2k+1): True; k*mu_k = 0.62020
k=3: identities OK=True; sigma=0.542705 lambda=0.447214 mu_k=0.190983; mu_k < 16/(32k-13) < 1/(2k-1): True; mu_k > 1/(2k+1): True; k*mu_k = 0.57295
k=4: identities OK=True; sigma=0.531967 lambda=0.462910 mu_k=0.138114; mu_k < 16/(32k-13) < 1/(2k-1): True; mu_k > 1/(2k+1): True; k*mu_k = 0.55246
k=5: identities OK=True; sigma=0.525502 lambda=0.471405 mu_k=0.108194; mu_k < 16/(32k-13) < 1/(2k-1): True; mu_k > 1/(2k+1): True; k*mu_k = 0.54097
k=6: identities OK=True; sigma=0.521199 lambda=0.476731 mu_k=0.088936; mu_k < 16/(32k-13) < 1/(2k-1): True; mu_k > 1/(2k+1): True; k*mu_k = 0.53362
k=10: identities OK=True; sigma=0.512643 lambda=0.486664 mu_k=0.051957; mu_k < 16/(32k-13) < 1/(2k-1): True; mu_k > 1/(2k+1): True; k*mu_k = 0.51957
k=40: identities OK=True; sigma=0.503135 lambda=0.496825 mu_k=0.012618; mu_k < 16/(32k-13) < 1/(2k-1): True; mu_k > 1/(2k+1): True; k*mu_k = 0.50474
k=100: identities OK=True; sigma=0.501252 lambda=0.498742 mu_k=0.005019; mu_k < 16/(32k-13) < 1/(2k-1): True; mu_k > 1/(2k+1): True; k*mu_k = 0.50188
k=200: identities OK=True; sigma=0.500625 lambda=0.499373 mu_k=0.002505; mu_k < 16/(32k-13) < 1/(2k-1): True; mu_k > 1/(2k+1): True; k*mu_k = 0.50094
k=3 closed forms (sigma=(15+3sqrt5)/40, lambda=1/sqrt5, (2s-1)^2+4/3 s^2 = 2/5, mu_3=(3-sqrt5)/4): True
integral formula (exact, 2000 random integer cases) and |sum - integral| <= n: True
grid max of phi, k=3: 0.190983 at sigma,lambda=0.5427,0.4472; mu_k=0.190983; grid<=mu+1e-9: True
grid max of phi, k=4: 0.138114 at sigma,lambda=0.5320,0.4630; mu_k=0.138114; grid<=mu+1e-9: True
grid max of phi, k=10: 0.051957 at sigma,lambda=0.5127,0.4867; mu_k=0.051957; grid<=mu+1e-9: True
ALL ASYMPTOTIC CHECKS PASSED: True
