1. Bands (Prop. 5.1) vs Floquet-Bloch spectra of w* and w^1
   lam=+3.0: bands=[(np.float64(-1.3028), np.float64(-1.1926)), (np.float64(-1.0), np.float64(0.0)), (np.float64(0.382), np.float64(0.6972)), (np.float64(2.3028), np.float64(2.618)), (np.float64(3.0), np.float64(4.0)), (np.float64(4.1926), np.float64(4.3028))]  all Bloch values in bands: True  max endpoint miss=8.88e-16
   lam=+5.0: bands=[(np.float64(-1.1926), np.float64(-1.1401)), (np.float64(-0.7016), np.float64(0.0)), (np.float64(0.6972), np.float64(0.8074)), (np.float64(4.1926), np.float64(4.3028)), (np.float64(5.0), np.float64(5.7016)), (np.float64(6.1401), np.float64(6.1926))]  all Bloch values in bands: True  max endpoint miss=1.78e-15
   lam=-3.0: bands=[(np.float64(-4.3028), np.float64(-4.1926)), (np.float64(-4.0), np.float64(-3.0)), (np.float64(-2.618), np.float64(-2.3028)), (np.float64(-0.6972), np.float64(-0.382)), (np.float64(0.0), np.float64(1.0)), (np.float64(1.1926), np.float64(1.3028))]  all Bloch values in bands: True  max endpoint miss=8.88e-16
   lam=+2.5: bands=[(np.float64(-1.3508), np.float64(-1.2122)), (np.float64(-1.1085), np.float64(0.0)), (np.float64(0.2192), np.float64(0.6492)), (np.float64(1.8508), np.float64(2.2808)), (np.float64(2.5), np.float64(3.6085)), (np.float64(3.7122), np.float64(3.8508))]  all Bloch values in bands: True  max endpoint miss=1.33e-15
   lam=+1.5: bands=[(np.float64(-1.5), np.float64(0.5)), (np.float64(1.0), np.float64(3.0))]  all Bloch values in bands: True  max endpoint miss=4.44e-16
2. Random configurations on rings (eigvalsh)
   lam=+2.2: min distance of |x(E)| to forbidden set = -2.398e-14 (none inside by more than 1e-9: True; negative values ~1e-13 are round-off at band edges); per-band counts [813, 2574, 813, 813, 2574, 813]
   lam=+3.0: min distance of |x(E)| to forbidden set = -9.015e-14 (none inside by more than 1e-9: True; negative values ~1e-13 are round-off at band edges); per-band counts [780, 2640, 780, 780, 2640, 780]
   lam=+5.0: min distance of |x(E)| to forbidden set = -8.793e-14 (none inside by more than 1e-9: True; negative values ~1e-13 are round-off at band edges); per-band counts [801, 2598, 801, 801, 2598, 801]
   lam=-4.0: min distance of |x(E)| to forbidden set = -9.948e-14 (none inside by more than 1e-9: True; negative values ~1e-13 are round-off at band edges); per-band counts [811, 2578, 811, 811, 2578, 811]
   lam=+12.0: min distance of |x(E)| to forbidden set = -4.352e-13 (none inside by more than 1e-9: True; negative values ~1e-13 are round-off at band edges); per-band counts [822, 2556, 822, 822, 2556, 822]
   lam=+1.5: min distance of |x(E)| to forbidden set = 5.684e-14 (none inside by more than 1e-9: True; negative values ~1e-13 are round-off at band edges); per-band counts [4200, 4200]
3. Random periodic configurations, Floquet-Bloch, original model
   lam=+2.2: 300 random periods, min distance = -1.510e-14 (none inside by more than 1e-9: True)
   lam=+3.0: 300 random periods, min distance = -1.554e-14 (none inside by more than 1e-9: True)
   lam=+4.0: 300 random periods, min distance = -3.464e-14 (none inside by more than 1e-9: True)
   lam=+7.0: 300 random periods, min distance = -1.270e-13 (none inside by more than 1e-9: True)
   lam=-5.0: 300 random periods, min distance = -2.842e-14 (none inside by more than 1e-9: True)
4. Domain wall (exact matching of decaying solutions, float)
   lam=+3.0: mismatch (0 = eigenvalue): E+=8.3e-01, E-=1.1e-16, G+=8.3e-01, G-=0.0e+00
   lam=+5.0: mismatch (0 = eigenvalue): E+=9.3e-01, E-=2.8e-16, G+=9.3e-01, G-=0.0e+00
   lam=-3.0: mismatch (0 = eigenvalue): E+=1.1e-16, E-=8.3e-01, G+=0.0e+00, G-=8.3e-01
   lam=+2.5: mismatch (0 = eigenvalue): E+=7.8e-01, E-=1.1e-16, G+=7.8e-01, G-=1.1e-16
   lam=+1.0: mismatch (0 = eigenvalue): E+=4.5e-01, E-=0.0e+00, G+=4.5e-01, G-=8.3e-16
5. Sharpness example Delta + L(delta_0+delta_1)
   L=2.5: eigenvalues in (2,L): [2.1666666667]  predicted 2.1666666667
   L=3.0: eigenvalues in (2,L): [2.5]  predicted 2.5000000000
   L=5.0: eigenvalues in (2,L): [4.25]  predicted 4.2500000000
   L=10.0: eigenvalues in (2,L): [9.1111111111]  predicted 9.1111111111
