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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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  min D(w<0.6)=-1.000  c0*l_form/rho (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

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  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  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  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  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  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  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  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  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