(i) Existence of a lowest eigenvector: lambda_min(A) > 0 against the criteria of the derivation kappa_h=1.0, kappa_a=0.7, levels M_k={0: 1.0, 1: 1.5}, grips (F0,F1,F2)={'square': (1.0, 0.6, 0.09), 'nonsquare': (1.0, 0.6, 0.03), 'eight': (1.0, 0.6, 0.045), 'linear': (1.0, 0.6, 0.0), 'negF2': (1.0, 0.6, -0.05)} --- ring L=32 no anchor, one body at a place: lambda_min(A0) and prediction [mu F2 > 0] grip=square k=0 sigma=+1: lambda_min=+3.717e-03 exists=True predicted=True b.1=+0.60 grip=square k=0 sigma=-1: lambda_min=-9.615e-03 exists=False predicted=False b.1=-0.60 grip=square k=1 sigma=+1: lambda_min=+4.725e-03 exists=True predicted=True b.1=+0.90 grip=square k=1 sigma=-1: lambda_min=-1.904e-02 exists=False predicted=False b.1=-0.90 grip=negF2 k=0 sigma=+1: lambda_min=-4.161e-03 exists=False predicted=False b.1=+0.60 grip=negF2 k=0 sigma=-1: lambda_min=+2.445e-03 exists=True predicted=True b.1=-0.60 grip=negF2 k=1 sigma=+1: lambda_min=-7.289e-03 exists=False predicted=False b.1=+0.90 grip=negF2 k=1 sigma=-1: lambda_min=+3.291e-03 exists=True predicted=True b.1=-0.90 grip=linear k=0 sigma=+1: lambda_min=+7.846e-17 exists=False predicted=False b.1=+0.60 grip=linear k=0 sigma=-1: lambda_min=+7.846e-17 exists=False predicted=False b.1=-0.60 grip=linear k=1 sigma=+1: lambda_min=+7.846e-17 exists=False predicted=False b.1=+0.90 grip=linear k=1 sigma=-1: lambda_min=+7.846e-17 exists=False predicted=False b.1=-0.90 no anchor, two bodies (separation 4): alpha1=0.2700, r12=3.500000: threshold alpha2 numerical=-0.1388174807 predicted=-0.1388174807 rel.dev=1.8e-14 anchor at h, one body at distance d: threshold alpha* (lambda_min=0) vs -1/Gamma(x), and which (k, sigma) of grip negF2 / square have a lowest eigenvector d=2: Gamma=3.303571 alpha* numerical=-0.3027027027 predicted=-0.3027027027 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y d=5: Gamma=5.647321 alpha* numerical=-0.1770750988 predicted=-0.1770750988 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=N square:k1+=Y square:k1-=N d=10: Gamma=8.303571 alpha* numerical=-0.1204301075 predicted=-0.1204301075 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=N negF2:k1-=Y square:k0+=Y square:k0-=N square:k1+=Y square:k1-=N anchor, two bodies, random (alpha1, alpha2): criterion (5.10) agrees with lambda_min in 40/40, explicit form (5.26) in 40/40 anchor, two bodies at [1], [3] (Gamma = 2.397321, 4.147321, G12 = 2.334821) with alpha1 = alpha2 = -a, a = 0.207279 between 1/lambda_max(Gamma_2) = 0.173439 and 1/max Gamma = 0.241119: one body alone: 1 - a Gamma(x_i) = +0.5031, +0.1403; lambda_min = +5.295e-03, +1.888e-03 (exists) pair: lambda_min(A) = -3.507e-03; (5.26) predicts existence = False; sufficient condition a (max Gamma + G12) < 1: False --- ring L=64 no anchor, one body at a place: lambda_min(A0) and prediction [mu F2 > 0] grip=square k=0 sigma=+1: lambda_min=+1.363e-03 exists=True predicted=True b.1=+0.60 grip=square k=0 sigma=-1: lambda_min=-8.183e-03 exists=False predicted=False b.1=-0.60 grip=square k=1 sigma=+1: lambda_min=+1.606e-03 exists=True predicted=True b.1=+0.90 grip=square k=1 sigma=-1: lambda_min=-1.816e-02 exists=False predicted=False b.1=-0.90 grip=negF2 k=0 sigma=+1: lambda_min=-2.849e-03 exists=False predicted=False b.1=+0.60 grip=negF2 k=0 sigma=-1: lambda_min=+9.931e-04 exists=True predicted=True b.1=-0.60 grip=negF2 k=1 sigma=+1: lambda_min=-5.792e-03 exists=False predicted=False b.1=+0.90 grip=negF2 k=1 sigma=-1: lambda_min=+1.248e-03 exists=True predicted=True b.1=-0.90 grip=linear k=0 sigma=+1: lambda_min=+2.264e-16 exists=False predicted=False b.1=+0.60 grip=linear k=0 sigma=-1: lambda_min=+2.264e-16 exists=False predicted=False b.1=-0.60 grip=linear k=1 sigma=+1: lambda_min=+2.264e-16 exists=False predicted=False b.1=+0.90 grip=linear k=1 sigma=-1: lambda_min=+2.264e-16 exists=False predicted=False b.1=-0.90 no anchor, two bodies (separation 4): alpha1=0.2700, r12=3.750000: threshold alpha2 numerical=-0.1341614907 predicted=-0.1341614907 rel.dev=7.3e-14 anchor at h, one body at distance d: threshold alpha* (lambda_min=0) vs -1/Gamma(x), and which (k, sigma) of grip negF2 / square have a lowest eigenvector d=2: Gamma=3.366071 alpha* numerical=-0.2970822281 predicted=-0.2970822281 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y d=5: Gamma=6.037946 alpha* numerical=-0.1656192237 predicted=-0.1656192237 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=N square:k1+=Y square:k1-=N d=10: Gamma=9.866071 alpha* numerical=-0.1013574661 predicted=-0.1013574661 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=N negF2:k1-=Y square:k0+=Y square:k0-=N square:k1+=Y square:k1-=N anchor, two bodies, random (alpha1, alpha2): criterion (5.10) agrees with lambda_min in 40/40, explicit form (5.26) in 40/40 anchor, two bodies at [1], [3] (Gamma = 2.412946, 4.287946, G12 = 2.381696) with alpha1 = alpha2 = -a, a = 0.201208 between 1/lambda_max(Gamma_2) = 0.169204 and 1/max Gamma = 0.233212: one body alone: 1 - a Gamma(x_i) = +0.5145, +0.1372; lambda_min = +1.739e-03, +7.853e-04 (exists) pair: lambda_min(A) = -2.105e-03; (5.26) predicts existence = False; sufficient condition a (max Gamma + G12) < 1: False --- cubic torus L=8 no anchor, one body at a place: lambda_min(A0) and prediction [mu F2 > 0] grip=square k=0 sigma=+1: lambda_min=+3.379e-04 exists=True predicted=True b.1=+0.60 grip=square k=0 sigma=-1: lambda_min=-3.664e-04 exists=False predicted=False b.1=-0.60 grip=square k=1 sigma=+1: lambda_min=+4.972e-04 exists=True predicted=True b.1=+0.90 grip=square k=1 sigma=-1: lambda_min=-5.614e-04 exists=False predicted=False b.1=-0.90 grip=negF2 k=0 sigma=+1: lambda_min=-1.998e-04 exists=False predicted=False b.1=+0.60 grip=negF2 k=0 sigma=-1: lambda_min=+1.910e-04 exists=True predicted=True b.1=-0.60 grip=negF2 k=1 sigma=+1: lambda_min=-3.032e-04 exists=False predicted=False b.1=+0.90 grip=negF2 k=1 sigma=-1: lambda_min=+2.834e-04 exists=True predicted=True b.1=-0.90 grip=linear k=0 sigma=+1: lambda_min=-8.867e-16 exists=False predicted=False b.1=+0.60 grip=linear k=0 sigma=-1: lambda_min=-8.867e-16 exists=False predicted=False b.1=-0.60 grip=linear k=1 sigma=+1: lambda_min=-8.867e-16 exists=False predicted=False b.1=+0.90 grip=linear k=1 sigma=-1: lambda_min=-8.867e-16 exists=False predicted=False b.1=-0.90 no anchor, two bodies (separation 4): alpha1=0.2700, r12=0.449684: threshold alpha2 numerical=-0.2407673308 predicted=-0.2407673308 rel.dev=2.1e-13 anchor at h, one body at distance d: threshold alpha* (lambda_min=0) vs -1/Gamma(x), and which (k, sigma) of grip negF2 / square have a lowest eigenvector d=1: Gamma=1.761254 alpha* numerical=-0.5677773671 predicted=-0.5677773671 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y d=2: Gamma=1.845431 alpha* numerical=-0.5418788955 predicted=-0.5418788955 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y d=3: Gamma=1.871738 alpha* numerical=-0.5342627406 predicted=-0.5342627406 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y anchor, two bodies, random (alpha1, alpha2): criterion (5.10) agrees with lambda_min in 40/40, explicit form (5.26) in 40/40 anchor, two bodies at [1, 0, 0], [3, 1, 1] (Gamma = 1.761254, 1.878598, G12 = 1.600983) with alpha1 = alpha2 = -a, a = 0.412270 between 1/lambda_max(Gamma_2) = 0.292228 and 1/max Gamma = 0.532312: one body alone: 1 - a Gamma(x_i) = +0.2739, +0.2255; lambda_min = +3.563e-04, +2.936e-04 (exists) pair: lambda_min(A) = -5.359e-04; (5.26) predicts existence = False; sufficient condition a (max Gamma + G12) < 1: False --- cubic torus L=12 no anchor, one body at a place: lambda_min(A0) and prediction [mu F2 > 0] grip=square k=0 sigma=+1: lambda_min=+9.996e-05 exists=True predicted=True b.1=+0.60 grip=square k=0 sigma=-1: lambda_min=-1.087e-04 exists=False predicted=False b.1=-0.60 grip=square k=1 sigma=+1: lambda_min=+1.470e-04 exists=True predicted=True b.1=+0.90 grip=square k=1 sigma=-1: lambda_min=-1.668e-04 exists=False predicted=False b.1=-0.90 grip=negF2 k=0 sigma=+1: lambda_min=-5.926e-05 exists=False predicted=False b.1=+0.60 grip=negF2 k=0 sigma=-1: lambda_min=+5.655e-05 exists=True predicted=True b.1=-0.60 grip=negF2 k=1 sigma=+1: lambda_min=-8.996e-05 exists=False predicted=False b.1=+0.90 grip=negF2 k=1 sigma=-1: lambda_min=+8.386e-05 exists=True predicted=True b.1=-0.90 grip=linear k=0 sigma=+1: lambda_min=-1.988e-16 exists=False predicted=False b.1=+0.60 grip=linear k=0 sigma=-1: lambda_min=-1.988e-16 exists=False predicted=False b.1=-0.60 grip=linear k=1 sigma=+1: lambda_min=-1.988e-16 exists=False predicted=False b.1=+0.90 grip=linear k=1 sigma=-1: lambda_min=-1.988e-16 exists=False predicted=False b.1=-0.90 no anchor, two bodies (separation 4): alpha1=0.2700, r12=0.461270: threshold alpha2 numerical=-0.2400975469 predicted=-0.2400975468 rel.dev=1.3e-11 anchor at h, one body at distance d: threshold alpha* (lambda_min=0) vs -1/Gamma(x), and which (k, sigma) of grip negF2 / square have a lowest eigenvector d=1: Gamma=1.761712 alpha* numerical=-0.5676297142 predicted=-0.5676297142 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y d=2: Gamma=1.847457 alpha* numerical=-0.5412844656 predicted=-0.5412844656 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y d=3: Gamma=1.877051 alpha* numerical=-0.5327505289 predicted=-0.5327505289 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y anchor, two bodies, random (alpha1, alpha2): criterion (5.10) agrees with lambda_min in 40/40, explicit form (5.26) in 40/40 anchor, two bodies at [1, 0, 0], [3, 1, 1] (Gamma = 1.761712, 1.883815, G12 = 1.602561) with alpha1 = alpha2 = -a, a = 0.411341 between 1/lambda_max(Gamma_2) = 0.291844 and 1/max Gamma = 0.530838: one body alone: 1 - a Gamma(x_i) = +0.2753, +0.2251; lambda_min = +1.059e-04, +8.670e-05 (exists) pair: lambda_min(A) = -1.586e-04; (5.26) predicts existence = False; sufficient condition a (max Gamma + G12) < 1: False