Check (ii): formation of the place-dependent part (item 2)
kappa_h = 1.0, kappa_a = 1.0, c_0 = 1.0

CUBIC TORUS (window below the wrap time; reference = L->infinity limit)
  g_inf(0) = 0.2527310 (gamma_3/kappa_h = 0.2527310)
  L=64 anchor far: h-x_s=(32, 32, 32), |h-x_s|=55.43; finite-L place average U(lambda_max)=-3.052e-03 (-lambda_max^2/2L^3 = -3.052e-03; -> 0 in order (O1))
    rho= 3 b= 17.92: plateau(1.6<w<=min(3,w_max))=+0.119  pred(finite rho) +0.057  pred 1/(b-1) +0.059  wake max_(w>1.3)|D-pred| 0.318 at w=3.27  free-field wake max|u_0|/g_inf(rho e1): w>1.3 0.301 at w=3.27, 1.3<w<=2.5 0.251  min D(w<0.6)=-1.000  lambda-mean (O1) +0.057  w_max=13.3  c0*l_form/rho within window (eps=.2,.3,.5) = 3.82, 3.73, 1.00
    rho= 4 b= 13.30: plateau(1.6<w<=min(3,w_max))=+0.086  pred(finite rho) +0.080  pred 1/(b-1) +0.081  wake max_(w>1.3)|D-pred| 0.291 at w=1.54  free-field wake max|u_0|/g_inf(rho e1): w>1.3 0.270 at w=1.54, 1.3<w<=2.5 0.270  min D(w<0.6)=-1.000  lambda-mean (O1) +0.079  w_max=10.0  c0*l_form/rho within window (eps=.2,.3,.5) = 3.55, 2.34, 1.04
    rho= 6 b=  8.70: plateau(1.6<w<=min(3,w_max))=+0.113  pred(finite rho) +0.129  pred 1/(b-1) +0.130  wake max_(w>1.3)|D-pred| 0.249 at w=2.34  free-field wake max|u_0|/g_inf(rho e1): w>1.3 0.220 at w=2.34, 1.3<w<=2.5 0.220  min D(w<0.6)=-1.000  lambda-mean (O1) +0.129  w_max=6.7  c0*l_form/rho within window (eps=.2,.3,.5) = 3.49, 3.09, 1.03
    rho= 8 b=  6.40: plateau(1.6<w<=min(3,w_max))=+0.176  pred(finite rho) +0.184  pred 1/(b-1) +0.185  wake max_(w>1.3)|D-pred| 0.279 at w=2.73  free-field wake max|u_0|/g_inf(rho e1): w>1.3 0.235 at w=2.73, 1.3<w<=2.5 0.227  min D(w<0.6)=-1.000  lambda-mean (O1) +0.184  w_max=5.0  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, 4.22, 1.02
    rho=12 b=  4.12: plateau(1.6<w<=min(3,w_max))=+0.321  pred(finite rho) +0.319  pred 1/(b-1) +0.320  wake max_(w>1.3)|D-pred| 0.233 at w=2.35  free-field wake max|u_0|/g_inf(rho e1): w>1.3 0.177 at w=2.35, 1.3<w<=2.5 0.177  min D(w<0.6)=-1.000  lambda-mean (O1) +0.319  w_max=3.3  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, 3.12
    rho=16 b=  3.00: plateau(1.6<w<=min(3,w_max))=+0.501  pred(finite rho) +0.499  pred 1/(b-1) +0.500  wake max_(w>1.3)|D-pred| 0.236 at w=1.35  free-field wake max|u_0|/g_inf(rho e1): w>1.3 0.157 at w=1.35, 1.3<w<=2.5 0.157  min D(w<0.6)=-1.000  lambda-mean (O1) +0.499  w_max=2.5  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, not formed
  L=64 anchor near: h-x_s=(-8, 0, 0), |h-x_s|=8.00; finite-L place average U(lambda_max)=-3.036e-03 (-lambda_max^2/2L^3 = -3.052e-03; -> 0 in order (O1))
    rho= 3 b=  3.67: plateau(1.6<w<=min(3,w_max))=+0.437  pred(finite rho) +0.357  pred 1/(b-1) +0.375  wake max_(w>1.3)|D-pred| 0.408 at w=3.27  image part max_(w>1.3)|I|/|ref| 0.004 (image front at w=6.33); |u_0|/|ref| at the wake maximum 0.408  min D(w<0.6)=-1.000  lambda-mean (O1) +0.354  w_max=13.3  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, 3.83
    rho= 4 b=  3.00: plateau(1.6<w<=min(3,w_max))=+0.496  pred(finite rho) +0.487  pred 1/(b-1) +0.500  wake max_(w>1.3)|D-pred| 0.401 at w=1.54  image part max_(w>1.3)|I|/|ref| 0.005 (image front at w=5.00); |u_0|/|ref| at the wake maximum 0.401  min D(w<0.6)=-1.000  lambda-mean (O1) +0.483  w_max=10.0  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, 9.91
    rho= 6 b=  2.33: plateau(1.6<w<=min(3,w_max))=+0.717  pred(finite rho) +0.742  pred 1/(b-1) +0.750  wake max_(w>1.3)|D-pred| 0.384 at w=2.34  image part max_(w>1.3)|I|/|ref| 0.007 (image front at w=3.67); |u_0|/|ref| at the wake maximum 0.384  min D(w<0.6)=-1.000  lambda-mean (O1) +0.736  w_max=6.7  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, not formed
    rho= 8 b=  2.00: plateau(1.6<w<=min(3,w_max))=+0.980  pred(finite rho) +0.994  pred 1/(b-1) +1.000  wake max_(w>1.3)|D-pred| 0.469 at w=2.73  image part max_(w>1.3)|I|/|ref| 0.010 (image front at w=3.00); |u_0|/|ref| at the wake maximum 0.469  min D(w<0.6)=-1.000  lambda-mean (O1) +0.986  w_max=5.0  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, not formed
    rho=12 b=  1.67: plateau(1.6<w<=min(3,w_max))=+1.494  pred(finite rho) +1.496  pred 1/(b-1) +1.500  wake max_(w>1.3)|D-pred| 0.447 at w=2.35  image part max_(w>1.3)|I|/|ref| 0.015 (image front at w=2.33); |u_0|/|ref| at the wake maximum 0.442  min D(w<0.6)=-1.000  lambda-mean (O1) +1.484  w_max=3.3  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, not formed
    rho=16 b=  1.50: plateau(1.6<w<=min(3,w_max))=+1.991  pred(finite rho) +1.997  pred 1/(b-1) +2.000  wake max_(w>1.3)|D-pred| 0.472 at w=1.35  image part max_(w>1.3)|I|/|ref| 0.020 (image front at w=2.00); |u_0|/|ref| at the wake maximum 0.472  min D(w<0.6)=-1.000  lambda-mean (O1) +1.981  w_max=2.5  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, not formed
  L=96 anchor far: h-x_s=(48, 48, 48), |h-x_s|=83.14; finite-L place average U(lambda_max)=-9.042e-04 (-lambda_max^2/2L^3 = -9.042e-04; -> 0 in order (O1))
    rho= 3 b= 27.15: plateau(1.6<w<=min(3,w_max))=+0.097  pred(finite rho) +0.037  pred 1/(b-1) +0.038  wake max_(w>1.3)|D-pred| 0.312 at w=3.27  free-field wake max|u_0|/g_inf(rho e1): w>1.3 0.301 at w=3.27, 1.3<w<=2.5 0.251  min D(w<0.6)=-1.000  lambda-mean (O1) +0.037  w_max=13.3  c0*l_form/rho within window (eps=.2,.3,.5) = 3.80, 2.92, 1.00
    rho= 4 b= 20.22: plateau(1.6<w<=min(3,w_max))=+0.058  pred(finite rho) +0.051  pred 1/(b-1) +0.052  wake max_(w>1.3)|D-pred| 0.284 at w=1.54  free-field wake max|u_0|/g_inf(rho e1): w>1.3 0.270 at w=1.54, 1.3<w<=2.5 0.270  min D(w<0.6)=-1.000  lambda-mean (O1) +0.051  w_max=10.0  c0*l_form/rho within window (eps=.2,.3,.5) = 2.98, 1.62, 1.05
    rho= 6 b= 13.30: plateau(1.6<w<=min(3,w_max))=+0.065  pred(finite rho) +0.081  pred 1/(b-1) +0.081  wake max_(w>1.3)|D-pred| 0.238 at w=2.34  free-field wake max|u_0|/g_inf(rho e1): w>1.3 0.220 at w=2.34, 1.3<w<=2.5 0.220  min D(w<0.6)=-1.000  lambda-mean (O1) +0.081  w_max=6.7  c0*l_form/rho within window (eps=.2,.3,.5) = 3.47, 3.06, 1.04
    rho= 8 b=  9.85: plateau(1.6<w<=min(3,w_max))=+0.105  pred(finite rho) +0.112  pred 1/(b-1) +0.113  wake max_(w>1.3)|D-pred| 0.262 at w=2.73  free-field wake max|u_0|/g_inf(rho e1): w>1.3 0.235 at w=2.73, 1.3<w<=2.5 0.227  min D(w<0.6)=-1.000  lambda-mean (O1) +0.112  w_max=5.0  c0*l_form/rho within window (eps=.2,.3,.5) = 4.24, 2.77, 1.03
    rho=12 b=  6.40: plateau(1.6<w<=min(3,w_max))=+0.187  pred(finite rho) +0.185  pred 1/(b-1) +0.185  wake max_(w>1.3)|D-pred| 0.210 at w=2.35  free-field wake max|u_0|/g_inf(rho e1): w>1.3 0.177 at w=2.35, 1.3<w<=2.5 0.177  min D(w<0.6)=-1.000  lambda-mean (O1) +0.185  w_max=3.3  c0*l_form/rho within window (eps=.2,.3,.5) = 3.33, 3.14, 1.02
    rho=16 b=  4.69: plateau(1.6<w<=min(3,w_max))=+0.272  pred(finite rho) +0.271  pred 1/(b-1) +0.271  wake max_(w>1.3)|D-pred| 0.200 at w=1.35  free-field wake max|u_0|/g_inf(rho e1): w>1.3 0.157 at w=1.35, 1.3<w<=2.5 0.157  min D(w<0.6)=-1.000  lambda-mean (O1) +0.270  w_max=2.5  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, 1.23
  L=96 anchor near: h-x_s=(-8, 0, 0), |h-x_s|=8.00; finite-L place average U(lambda_max)=-8.996e-04 (-lambda_max^2/2L^3 = -9.042e-04; -> 0 in order (O1))
    rho= 3 b=  3.67: plateau(1.6<w<=min(3,w_max))=+0.437  pred(finite rho) +0.357  pred 1/(b-1) +0.375  wake max_(w>1.3)|D-pred| 0.408 at w=3.27  image part max_(w>1.3)|I|/|ref| 0.004 (image front at w=6.33); |u_0|/|ref| at the wake maximum 0.408  min D(w<0.6)=-1.000  lambda-mean (O1) +0.354  w_max=13.3  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, 3.83
    rho= 4 b=  3.00: plateau(1.6<w<=min(3,w_max))=+0.496  pred(finite rho) +0.487  pred 1/(b-1) +0.500  wake max_(w>1.3)|D-pred| 0.401 at w=1.54  image part max_(w>1.3)|I|/|ref| 0.005 (image front at w=5.00); |u_0|/|ref| at the wake maximum 0.401  min D(w<0.6)=-1.000  lambda-mean (O1) +0.483  w_max=10.0  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, 9.91
    rho= 6 b=  2.33: plateau(1.6<w<=min(3,w_max))=+0.717  pred(finite rho) +0.742  pred 1/(b-1) +0.750  wake max_(w>1.3)|D-pred| 0.384 at w=2.34  image part max_(w>1.3)|I|/|ref| 0.007 (image front at w=3.67); |u_0|/|ref| at the wake maximum 0.384  min D(w<0.6)=-1.000  lambda-mean (O1) +0.736  w_max=6.7  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, not formed
    rho= 8 b=  2.00: plateau(1.6<w<=min(3,w_max))=+0.980  pred(finite rho) +0.994  pred 1/(b-1) +1.000  wake max_(w>1.3)|D-pred| 0.469 at w=2.73  image part max_(w>1.3)|I|/|ref| 0.010 (image front at w=3.00); |u_0|/|ref| at the wake maximum 0.469  min D(w<0.6)=-1.000  lambda-mean (O1) +0.986  w_max=5.0  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, not formed
    rho=12 b=  1.67: plateau(1.6<w<=min(3,w_max))=+1.494  pred(finite rho) +1.496  pred 1/(b-1) +1.500  wake max_(w>1.3)|D-pred| 0.447 at w=2.35  image part max_(w>1.3)|I|/|ref| 0.015 (image front at w=2.33); |u_0|/|ref| at the wake maximum 0.442  min D(w<0.6)=-1.000  lambda-mean (O1) +1.484  w_max=3.3  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, not formed
    rho=16 b=  1.50: plateau(1.6<w<=min(3,w_max))=+1.991  pred(finite rho) +1.997  pred 1/(b-1) +2.000  wake max_(w>1.3)|D-pred| 0.472 at w=1.35  image part max_(w>1.3)|I|/|ref| 0.020 (image front at w=2.00); |u_0|/|ref| at the wake maximum 0.472  min D(w<0.6)=-1.000  lambda-mean (O1) +1.981  w_max=2.5  c0*l_form/rho within window (eps=.2,.3,.5) = not formed, not formed, not formed
  size check (far): max_l,rho |da_64 - da_96| = 1.95e-07 (max |da| = 3.55e-02)
  size check (near): max_l,rho |da_64 - da_96| = 1.93e-07 (max |da| = 3.55e-02)
  time-step check (L=64 near): max |da(dt=0.1) - da(dt=0.05)| = 8.74e-06
  ABOVE THRESHOLD: kappa_a = 8.0 > kappa_h/gamma_3 = 3.9568; L=64, near anchor h-x_s=(-8, 0, 0)
    localized mode: nu_b from 1 + kappa_a gamma_b(0) = 0 on Z^3: 14.78961903; sparse eigen-solver (L=64): 14.78961903
    rho= 3 b= 3.67: plateau=+0.437  pred(finite rho) +0.357  1/(b-1) +0.375  lambda-mean (O1) +0.348 (beta=0.0264)  min D(w<0.6)=-1.000  c0*l_form/rho (eps=.5, window) = 3.83;  max_window|I|*d_x*d_s^(1/6): kappa_a=1 0.0011, kappa_a=8 0.0037;  localized share psi_b(x)psi_b(x_s)/nu_b = -9.45e-17
    rho= 4 b= 3.00: plateau=+0.496  pred(finite rho) +0.487  1/(b-1) +0.500  lambda-mean (O1) +0.474 (beta=0.0264)  min D(w<0.6)=-1.000  c0*l_form/rho (eps=.5, window) = 9.90;  max_window|I|*d_x*d_s^(1/6): kappa_a=1 0.0011, kappa_a=8 0.0037;  localized share psi_b(x)psi_b(x_s)/nu_b = +1.88e-17
    rho= 6 b= 2.33: plateau=+0.717  pred(finite rho) +0.742  1/(b-1) +0.750  lambda-mean (O1) +0.722 (beta=0.0264)  min D(w<0.6)=-1.000  c0*l_form/rho (eps=.5, window) = not formed;  max_window|I|*d_x*d_s^(1/6): kappa_a=1 0.0011, kappa_a=8 0.0037;  localized share psi_b(x)psi_b(x_s)/nu_b = +7.64e-19
    rho= 8 b= 2.00: plateau=+0.980  pred(finite rho) +0.994  1/(b-1) +1.000  lambda-mean (O1) +0.968 (beta=0.0264)  min D(w<0.6)=-1.000  c0*l_form/rho (eps=.5, window) = not formed;  max_window|I|*d_x*d_s^(1/6): kappa_a=1 0.0011, kappa_a=8 0.0037;  localized share psi_b(x)psi_b(x_s)/nu_b = +3.18e-20
    rho=12 b= 1.67: plateau=+1.480  pred(finite rho) +1.496  1/(b-1) +1.500  lambda-mean (O1) +1.456 (beta=0.0264)  min D(w<0.6)=-1.000  c0*l_form/rho (eps=.5, window) = not formed;  max_window|I|*d_x*d_s^(1/6): kappa_a=1 0.0011, kappa_a=8 0.0037;  localized share psi_b(x)psi_b(x_s)/nu_b = +5.75e-23
    rho=16 b= 1.50: plateau=+1.970  pred(finite rho) +1.997  1/(b-1) +2.000  lambda-mean (O1) +1.944 (beta=0.0264)  min D(w<0.6)=-1.000  c0*l_form/rho (eps=.5, window) = not formed;  max_window|I|*d_x*d_s^(1/6): kappa_a=1 0.0011, kappa_a=8 0.0037;  localized share psi_b(x)psi_b(x_s)/nu_b = +9.10e-26

RING (exact eigen-decomposition; window below the wrap time; reference grad = 1/kappa_h)
  (a) anchor at d_s = 2 rho (a = 2): prediction -1 (w<1), -1/2 (1<w<3), 0 (w>3)
    L=1024 rho= 20: plateau(1.4<w<2.6)=-0.496  plateau(w>3.4)=-0.007  max|D|(w>1.4)=0.577  D across image front (2.6<w<3.4) in [-0.544, +0.069]  c0*l_form/rho: eps=0.25 -> 3.08 (pred 3), eps=0.6 -> 1.34 (pred 1);  finite-L ref grad*kappa_h = 0.961
    L=1024 rho= 40: plateau(1.4<w<2.6)=-0.501  plateau(w>3.4)=-0.002  max|D|(w>1.4)=0.561  D across image front (2.6<w<3.4) in [-0.531, +0.097]  c0*l_form/rho: eps=0.25 -> 3.06 (pred 3), eps=0.6 -> 1.20 (pred 1);  finite-L ref grad*kappa_h = 0.922
    L=1024 rho= 80: plateau(1.4<w<2.6)=-0.500  plateau(w>3.4)=+0.000  max|D|(w>1.4)=0.543  D across image front (2.6<w<3.4) in [-0.522, +0.098]  c0*l_form/rho: eps=0.25 -> 3.03 (pred 3), eps=0.6 -> 1.12 (pred 1);  finite-L ref grad*kappa_h = 0.844
    L=1024 rho=160: plateau(1.4<w<2.6)=-0.500  plateau(w>3.4)=-0.000  max|D|(w>1.4)=0.530  D across image front (2.6<w<3.4) in [-0.515, +0.117]  c0*l_form/rho: eps=0.25 -> 3.02 (pred 3), eps=0.6 -> 1.02 (pred 1);  finite-L ref grad*kappa_h = 0.688
    L=2048 rho= 20: plateau(1.4<w<2.6)=-0.496  plateau(w>3.4)=-0.007  max|D|(w>1.4)=0.577  D across image front (2.6<w<3.4) in [-0.544, +0.069]  c0*l_form/rho: eps=0.25 -> 3.08 (pred 3), eps=0.6 -> 1.34 (pred 1);  finite-L ref grad*kappa_h = 0.980
    L=2048 rho= 40: plateau(1.4<w<2.6)=-0.501  plateau(w>3.4)=-0.002  max|D|(w>1.4)=0.561  D across image front (2.6<w<3.4) in [-0.531, +0.097]  c0*l_form/rho: eps=0.25 -> 3.06 (pred 3), eps=0.6 -> 1.20 (pred 1);  finite-L ref grad*kappa_h = 0.961
    L=2048 rho= 80: plateau(1.4<w<2.6)=-0.500  plateau(w>3.4)=+0.000  max|D|(w>1.4)=0.543  D across image front (2.6<w<3.4) in [-0.522, +0.098]  c0*l_form/rho: eps=0.25 -> 3.03 (pred 3), eps=0.6 -> 1.12 (pred 1);  finite-L ref grad*kappa_h = 0.922
    L=2048 rho=160: plateau(1.4<w<2.6)=-0.500  plateau(w>3.4)=-0.000  max|D|(w>1.4)=0.530  D across image front (2.6<w<3.4) in [-0.515, +0.117]  c0*l_form/rho: eps=0.25 -> 3.02 (pred 3), eps=0.6 -> 1.02 (pred 1);  finite-L ref grad*kappa_h = 0.844
  (b) anchor far, d_s = 400: prediction -1/2 until c0*l = 2 d_s - rho, then 0
    L=1024 rho= 10: mean D on 1.4 rho < c0 l < 0.9(2d_s-rho) = -0.500, max|D+1/2| there = 0.111;  c0*l_form(eps=0.25) = 793.2 (pred 2d_s-rho = 790), c0*l_form/rho = 79.3
    L=1024 rho= 20: mean D on 1.4 rho < c0 l < 0.9(2d_s-rho) = -0.500, max|D+1/2| there = 0.087;  c0*l_form(eps=0.25) = 783.2 (pred 2d_s-rho = 780), c0*l_form/rho = 39.2
    L=1024 rho= 40: mean D on 1.4 rho < c0 l < 0.9(2d_s-rho) = -0.500, max|D+1/2| there = 0.062;  c0*l_form(eps=0.25) = 763.5 (pred 2d_s-rho = 760), c0*l_form/rho = 19.1
    L=1024 rho= 80: mean D on 1.4 rho < c0 l < 0.9(2d_s-rho) = -0.500, max|D+1/2| there = 0.045;  c0*l_form(eps=0.25) = 723.0 (pred 2d_s-rho = 720), c0*l_form/rho = 9.0
    L=2048 rho= 10: mean D on 1.4 rho < c0 l < 0.9(2d_s-rho) = -0.500, max|D+1/2| there = 0.111;  c0*l_form(eps=0.25) = 793.2 (pred 2d_s-rho = 790), c0*l_form/rho = 79.3
    L=2048 rho= 20: mean D on 1.4 rho < c0 l < 0.9(2d_s-rho) = -0.500, max|D+1/2| there = 0.087;  c0*l_form(eps=0.25) = 783.2 (pred 2d_s-rho = 780), c0*l_form/rho = 39.2
    L=2048 rho= 40: mean D on 1.4 rho < c0 l < 0.9(2d_s-rho) = -0.500, max|D+1/2| there = 0.062;  c0*l_form(eps=0.25) = 763.5 (pred 2d_s-rho = 760), c0*l_form/rho = 19.1
    L=2048 rho= 80: mean D on 1.4 rho < c0 l < 0.9(2d_s-rho) = -0.500, max|D+1/2| there = 0.045;  c0*l_form(eps=0.25) = 723.0 (pred 2d_s-rho = 720), c0*l_form/rho = 9.0
  (c) uniform form of Step 8: sup_window kappa_h|grad da - F(rho) - F(m)|, a = 2, m = 3 rho - 1, c0 l <= 8 rho, L = 2048
    kappa_a=0.25: rho=20: 0.4402;  rho=40: 0.4055;  rho=80: 0.3688;  rho=160: 0.3305;  localized eigenvalue 4.015564 vs 2kh+sqrt(4kh^2+ka^2) = 4.015564
    kappa_a=1.0: rho=20: 0.2152;  rho=40: 0.1817;  rho=80: 0.1515;  rho=160: 0.1248;  localized eigenvalue 4.236068 vs 2kh+sqrt(4kh^2+ka^2) = 4.236068
    kappa_a=4.0: rho=20: 0.0648;  rho=40: 0.0524;  rho=80: 0.0421;  rho=160: 0.0337;  localized eigenvalue 6.472136 vs 2kh+sqrt(4kh^2+ka^2) = 6.472136
    Bessel form of F (n = rho and m) against the free-ring gradient, max kappa_h|diff|: rho=20: 1.2e-07;  rho=40: 8.3e-08;  rho=80: 5.9e-08;  rho=160: 4.2e-08