range of r: -L..L with L=120 (open ends); convergence: comparison with L=240 and weight at |r|>L/2
finite differences: 5-point, step 0.01 in P, relative step 0.001 in M_i, tau_i, theta; convention eta=1/2

=== set A_equal_R: {'M1': 1.0, 'M2': 1.0, 'tau1': 0.1, 'tau2': 0.1, 'w': [-0.004, 0.003, -0.002, 0.006]}  max tau_i/M_i = 0.100
P=-0.30  E_b=1.998771748883  isolation=1.452e-03  in a band: False  weight(r=0,11)=0.6408  Gamma=+1.000000  E_b'=-1.133090e-03  <V1>=-1.133090e-03  <V2>=-1.133090e-03  E_b''=+3.517649e-03  (11.9)=+3.517649e-03
P=-0.15  E_b=1.998642641792  isolation=1.414e-03  in a band: False  weight(r=0,11)=0.6278  Gamma=+1.000000  E_b'=-5.810445e-04  <V1>=-5.810445e-04  <V2>=-5.810445e-04  E_b''=+3.809572e-03  (11.9)=+3.809572e-03
P=+0.00  E_b=1.998598883527  isolation=1.401e-03  in a band: False  weight(r=0,11)=0.6235  Gamma=+1.000000  E_b'=+1.850372e-15  <V1>=+7.893878e-21  <V2>=-2.053590e-20  E_b''=+3.905596e-03  (11.9)=+3.905596e-03
P=+0.15  E_b=1.998642641792  isolation=1.414e-03  in a band: False  weight(r=0,11)=0.6278  Gamma=+1.000000  E_b'=+5.810445e-04  <V1>=+5.810445e-04  <V2>=+5.810445e-04  E_b''=+3.809572e-03  (11.9)=+3.809572e-03
P=+0.30  E_b=1.998771748883  isolation=1.452e-03  in a band: False  weight(r=0,11)=0.6408  Gamma=+1.000000  E_b'=+1.133090e-03  <V1>=+1.133090e-03  <V2>=+1.133090e-03  E_b''=+3.517649e-03  (11.9)=+3.517649e-03
max |deviation| over the P values:
  2(a): dE/dM_i - <sigma3_i> 1.25e-12; tau_i dE/dtau_i - <A_i> 1.37e-12; dE/dtheta - <U> 8.96e-13; Euler residual 4.22e-15
  2(b): E_b' - <V1> 1.59e-12; E_b' - <V2> 1.59e-12; <V1> - <V2> 6.51e-19
  2(c): E_b'' (finite difference) - formula (11.9) 1.59e-11
  3(a): coefficients of (11.10) against direct inversion (relative) 1.19e-14; root of (11.10) - E_b 1.55e-15
  numerics: |E_b(L) - E_b(2L)| 2.00e-15; weight at |r|>L/2 3.17e-20; dependence on eta 2.44e-15; dense - banded 2.44e-15
closed forms (items 3, 4), regime R: |w11|/a = 0.800, (max tau/M)^2 = 1.000e-02
  max_P |E_b - (11.12)| = 2.053e-06  = 0.021 * (tau/M)^4;  max_P |E_b' - d(11.12)/dP| = 2.354e-07
  E_b''(0): numeric +3.90559613e-03  closed +3.90434405e-03  rel.dev 3.207e-04  = 0.032 * (tau/M)^2
  s3_1   numeric +9.9945154586e-01  closed +9.9945217202e-01  diff -6.26e-07
  s3_2   numeric +9.9945154586e-01  closed +9.9945217202e-01  diff -6.26e-07
  A1     numeric +1.0948993397e-03  closed +1.0956559528e-03  diff -7.57e-07
  A2     numeric +1.0948993397e-03  closed +1.0956559528e-03  diff -7.57e-07
  U      numeric -2.4940068754e-03  closed -2.4987801902e-03  diff +4.77e-06
  Ebind  numeric -1.4011164735e-03  closed -1.4031242374e-03  diff +2.01e-06
  Epp0   numeric +3.9055961250e-03  closed +3.9043440472e-03  diff +1.25e-06
  m_b/(m_1+m_2): numeric +1.280214  (11.16) +1.280625;  m_b-(m_1+m_2): numeric +5.604286e+01  -4 m_1 m_2 E_bind +5.604466e+01

=== set B_unequal_R: {'M1': 1.0, 'M2': 1.7, 'tau1': 0.2, 'tau2': 0.12, 'w': [-0.01, 0.005, -0.004, 0.008]}  max tau_i/M_i = 0.200
P=-0.30  E_b=2.696652071452  isolation=3.657e-03  in a band: False  weight(r=0,11)=0.6441  Gamma=+1.000000  E_b'=-1.546134e-03  <V1>=-1.546134e-03  <V2>=-1.546134e-03  E_b''=+4.673251e-03  (11.9)=+4.673251e-03
P=-0.15  E_b=2.696475088077  isolation=3.603e-03  in a band: False  weight(r=0,11)=0.6368  Gamma=+1.000000  E_b'=-8.001290e-04  <V1>=-8.001290e-04  <V2>=-8.001290e-04  E_b''=+5.213785e-03  (11.9)=+5.213785e-03
P=+0.00  E_b=2.696414739677  isolation=3.585e-03  in a band: False  weight(r=0,11)=0.6344  Gamma=+1.000000  E_b'=+1.110223e-14  <V1>=-1.752348e-19  <V2>=+3.687192e-20  E_b''=+5.394417e-03  (11.9)=+5.394417e-03
P=+0.15  E_b=2.696475088077  isolation=3.603e-03  in a band: False  weight(r=0,11)=0.6368  Gamma=+1.000000  E_b'=+8.001290e-04  <V1>=+8.001290e-04  <V2>=+8.001290e-04  E_b''=+5.213785e-03  (11.9)=+5.213785e-03
P=+0.30  E_b=2.696652071452  isolation=3.657e-03  in a band: False  weight(r=0,11)=0.6441  Gamma=+1.000000  E_b'=+1.546134e-03  <V1>=+1.546134e-03  <V2>=+1.546134e-03  E_b''=+4.673251e-03  (11.9)=+4.673251e-03
max |deviation| over the P values:
  2(a): dE/dM_i - <sigma3_i> 1.49e-12; tau_i dE/dtau_i - <A_i> 1.56e-12; dE/dtheta - <U> 6.98e-13; Euler residual 4.44e-15
  2(b): E_b' - <V1> 8.50e-13; E_b' - <V2> 8.50e-13; <V1> - <V2> 7.35e-17
  2(c): E_b'' (finite difference) - formula (11.9) 1.83e-11
  3(a): coefficients of (11.10) against direct inversion (relative) 4.28e-14; root of (11.10) - E_b 4.88e-15
  numerics: |E_b(L) - E_b(2L)| 3.11e-15; weight at |r|>L/2 9.86e-21; dependence on eta 6.22e-15; dense - banded 3.55e-15
closed forms (items 3, 4), regime R: |w11|/a = 0.825, (max tau/M)^2 = 4.000e-02
  max_P |E_b - (11.12)| = 8.267e-06  = 0.005 * (tau/M)^4;  max_P |E_b' - d(11.12)/dP| = 6.697e-07
  E_b''(0): numeric +5.39441742e-03  closed +5.39148145e-03  rel.dev 5.446e-04  = 0.014 * (tau/M)^2
  s3_1   numeric +9.9771238948e-01  closed +9.9771281374e-01  diff -4.24e-07
  s3_2   numeric +9.9971332360e-01  closed +9.9971509098e-01  diff -1.77e-06
  A1     numeric +4.5625732894e-03  closed +4.5743725191e-03  diff -1.18e-05
  A2     numeric +9.7104322032e-04  closed +9.6869065111e-04  diff +2.35e-06
  U      numeric -6.3439164227e-03  closed -6.3649433780e-03  diff +2.10e-05
  Ebind  numeric -3.5852603230e-03  closed -3.5934117929e-03  diff +8.15e-06
  Epp0   numeric +5.3944174247e-03  closed +5.3914814496e-03  diff +2.94e-06
  m_b/(m_1+m_2): numeric +1.295838  (11.16) +1.296544;  m_b-(m_1+m_2): numeric +4.232127e+01  -4 m_1 m_2 E_bind +4.232599e+01

=== set C_above_band_R: {'M1': 1.0, 'M2': 0.8, 'tau1': 0.1, 'tau2': 0.15, 'w': [0.02, -0.005, 0.003, -0.01]}  max tau_i/M_i = 0.187
P=-0.30  E_b=1.831346531561  isolation=1.274e-02  in a band: False  weight(r=0,11)=0.9037  Gamma=+1.000000  E_b'=+8.867732e-04  <V1>=+8.867732e-04  <V2>=+8.867732e-04  E_b''=-2.629234e-03  (11.9)=-2.629234e-03
P=-0.15  E_b=1.831448371991  isolation=1.260e-02  in a band: False  weight(r=0,11)=0.8994  Gamma=+1.000000  E_b'=+4.619112e-04  <V1>=+4.619112e-04  <V2>=+4.619112e-04  E_b''=-2.996418e-03  (11.9)=-2.996418e-03
P=+0.00  E_b=1.831483249143  isolation=1.255e-02  in a band: False  weight(r=0,11)=0.8980  Gamma=+1.000000  E_b'=+1.850372e-15  <V1>=-1.604183e-19  <V2>=+5.735736e-19  E_b''=-3.121011e-03  (11.9)=-3.121011e-03
P=+0.15  E_b=1.831448371991  isolation=1.260e-02  in a band: False  weight(r=0,11)=0.8994  Gamma=+1.000000  E_b'=-4.619112e-04  <V1>=-4.619112e-04  <V2>=-4.619112e-04  E_b''=-2.996418e-03  (11.9)=-2.996418e-03
P=+0.30  E_b=1.831346531561  isolation=1.274e-02  in a band: False  weight(r=0,11)=0.9037  Gamma=+1.000000  E_b'=-8.867732e-04  <V1>=-8.867732e-04  <V2>=-8.867732e-04  E_b''=-2.629234e-03  (11.9)=-2.629234e-03
max |deviation| over the P values:
  2(a): dE/dM_i - <sigma3_i> 1.04e-12; tau_i dE/dtau_i - <A_i> 1.30e-12; dE/dtheta - <U> 1.11e-12; Euler residual 1.78e-15
  2(b): E_b' - <V1> 2.74e-12; E_b' - <V2> 2.74e-12; <V1> - <V2> 8.55e-17
  2(c): E_b'' (finite difference) - formula (11.9) 1.26e-11
  3(a): coefficients of (11.10) against direct inversion (relative) 4.97e-15; root of (11.10) - E_b 8.88e-16
  numerics: |E_b(L) - E_b(2L)| 2.22e-16; weight at |r|>L/2 5.56e-28; dependence on eta 1.33e-15; dense - banded 4.44e-16
closed forms (items 3, 4), regime R: |w11|/a = 2.098, (max tau/M)^2 = 3.516e-02
  max_P |E_b - (11.12)| = 2.030e-04  = 0.164 * (tau/M)^4;  max_P |E_b' - d(11.12)/dP| = 1.488e-05
  E_b''(0): numeric -3.12101105e-03  closed -3.17366056e-03  rel.dev 1.659e-02  = 0.472 * (tau/M)^2
  s3_1   numeric +9.9650760383e-01  closed +9.9642448170e-01  diff +8.31e-05
  s3_2   numeric +9.8795706438e-01  closed +9.8742981847e-01  diff +5.27e-04
  A1     numeric +7.0534612009e-03  closed +7.1510366031e-03  diff -9.76e-05
  A2     numeric +1.9596946814e-02  closed +2.0112290446e-02  diff -5.15e-04
  U      numeric +1.7959585791e-02  closed +1.8054602308e-02  diff -9.50e-05
  Ebind  numeric +3.1483249143e-02  closed +3.1686265833e-02  diff -2.03e-04
  Epp0   numeric -3.1210110472e-03  closed -3.1736605620e-03  diff +5.26e-05
  m_b/(m_1+m_2): numeric -2.363673  (11.16) -2.324461;  m_b-(m_1+m_2): numeric -4.559645e+02  -4 m_1 m_2 E_bind -4.477618e+02

=== set D_wide_R: {'M1': 1.3, 'M2': 1.0, 'tau1': 0.15, 'tau2': 0.2, 'w': [-0.004, 0.002, 0.001, 0.003]}  max tau_i/M_i = 0.200
P=-0.30  E_b=2.299969840671  isolation=5.677e-04  in a band: False  weight(r=0,11)=0.2780  Gamma=+1.000000  E_b'=-3.402251e-03  <V1>=-3.402251e-03  <V2>=-3.402251e-03  E_b''=+1.074662e-02  (11.9)=+1.074662e-02
P=-0.15  E_b=2.299583345287  isolation=5.522e-04  in a band: False  weight(r=0,11)=0.2707  Gamma=+1.000000  E_b'=-1.734310e-03  <V1>=-1.734310e-03  <V2>=-1.734310e-03  E_b''=+1.141561e-02  (11.9)=+1.141561e-02
P=+0.00  E_b=2.299452860723  isolation=5.471e-04  in a band: False  weight(r=0,11)=0.2684  Gamma=+1.000000  E_b'=+1.110223e-14  <V1>=-2.235218e-19  <V2>=+5.295572e-19  E_b''=+1.163513e-02  (11.9)=+1.163513e-02
P=+0.15  E_b=2.299583345287  isolation=5.522e-04  in a band: False  weight(r=0,11)=0.2707  Gamma=+1.000000  E_b'=+1.734310e-03  <V1>=+1.734310e-03  <V2>=+1.734310e-03  E_b''=+1.141561e-02  (11.9)=+1.141561e-02
P=+0.30  E_b=2.299969840671  isolation=5.677e-04  in a band: False  weight(r=0,11)=0.2780  Gamma=+1.000000  E_b'=+3.402251e-03  <V1>=+3.402251e-03  <V2>=+3.402251e-03  E_b''=+1.074662e-02  (11.9)=+1.074662e-02
max |deviation| over the P values:
  2(a): dE/dM_i - <sigma3_i> 2.05e-12; tau_i dE/dtau_i - <A_i> 1.96e-12; dE/dtheta - <U> 2.07e-12; Euler residual 5.33e-15
  2(b): E_b' - <V1> 5.04e-12; E_b' - <V2> 5.04e-12; <V1> - <V2> 8.46e-17
  2(c): E_b'' (finite difference) - formula (11.9) 9.92e-11
  3(a): coefficients of (11.10) against direct inversion (relative) 2.34e-13; root of (11.10) - E_b 1.78e-15
  numerics: |E_b(L) - E_b(2L)| 4.44e-15; weight at |r|>L/2 4.85e-08; dependence on eta 4.44e-15; dense - banded 2.22e-15
closed forms (items 3, 4), regime R: |w11|/a = 0.279, (max tau/M)^2 = 4.000e-02
  max_P |E_b - (11.12)| = 7.729e-07  = 0.000 * (tau/M)^4;  max_P |E_b' - d(11.12)/dP| = 1.014e-06
  E_b''(0): numeric +1.16351317e-02  closed +1.16355522e-02  rel.dev 3.615e-05  = 0.001 * (tau/M)^2
  s3_1   numeric +9.9987712967e-01  closed +9.9987739882e-01  diff -2.69e-07
  s3_2   numeric +9.9963191441e-01  closed +9.9963165158e-01  diff +2.63e-07
  A1     numeric +3.1883906936e-04  closed +3.1876305792e-04  diff +7.60e-08
  A2     numeric +7.3524924751e-04  closed +7.3669684496e-04  diff -1.45e-06
  U      numeric -1.0734105787e-03  closed -1.0756421628e-03  diff +2.23e-06
  Ebind  numeric -5.4713927724e-04  closed -5.4791221131e-04  diff +7.73e-07
  Epp0   numeric +1.1635131656e-02  closed +1.1635552241e-02  diff -4.21e-07
  m_b/(m_1+m_2): numeric +1.038281  (11.16) +1.038244;  m_b-(m_1+m_2): numeric +3.168822e+00  -4 m_1 m_2 E_bind +3.161249e+00

=== set E_order_one: {'M1': 1.0, 'M2': 1.3, 'tau1': 1.0, 'tau2': 0.8, 'w': [-0.6, 0.2, -0.15, 0.3]}  max tau_i/M_i = 1.000
P=-0.30  E_b=2.020243065365  isolation=2.944e-01  in a band: False  weight(r=0,11)=0.7017  Gamma=+1.000000  E_b'=-5.330716e-02  <V1>=-5.330716e-02  <V2>=-5.330716e-02  E_b''=+1.580211e-01  (11.9)=+1.580211e-01
P=-0.15  E_b=2.014120790386  isolation=2.896e-01  in a band: False  weight(r=0,11)=0.6956  Gamma=+1.000000  E_b'=-2.776997e-02  <V1>=-2.776997e-02  <V2>=-2.776997e-02  E_b''=+1.801270e-01  (11.9)=+1.801270e-01
P=+0.00  E_b=2.012023935088  isolation=2.880e-01  in a band: False  weight(r=0,11)=0.6936  Gamma=+1.000000  E_b'=-7.401487e-15  <V1>=-3.140157e-17  <V2>=+1.914190e-17  E_b''=+1.876436e-01  (11.9)=+1.876436e-01
P=+0.15  E_b=2.014120790386  isolation=2.896e-01  in a band: False  weight(r=0,11)=0.6956  Gamma=+1.000000  E_b'=+2.776997e-02  <V1>=+2.776997e-02  <V2>=+2.776997e-02  E_b''=+1.801270e-01  (11.9)=+1.801270e-01
P=+0.30  E_b=2.020243065365  isolation=2.944e-01  in a band: False  weight(r=0,11)=0.7017  Gamma=+1.000000  E_b'=+5.330716e-02  <V1>=+5.330716e-02  <V2>=+5.330716e-02  E_b''=+1.580211e-01  (11.9)=+1.580211e-01
max |deviation| over the P values:
  2(a): dE/dM_i - <sigma3_i> 7.62e-13; tau_i dE/dtau_i - <A_i> 1.52e-12; dE/dtheta - <U> 1.15e-12; Euler residual 1.33e-15
  2(b): E_b' - <V1> 1.74e-10; E_b' - <V2> 1.74e-10; <V1> - <V2> 3.54e-16
  2(c): E_b'' (finite difference) - formula (11.9) 2.40e-10
  3(a): coefficients of (11.10) against direct inversion (relative) 3.59e-15; root of (11.10) - E_b 1.33e-15
  numerics: |E_b(L) - E_b(2L)| 8.88e-16; weight at |r|>L/2 2.26e-30; dependence on eta 8.88e-16; dense - banded 2.66e-15
closed forms (items 3, 4), control, outside R: |w11|/a = 1.608, (max tau/M)^2 = 1.000e+00
  max_P |E_b - (11.12)| = 4.605e-02  = 0.046 * (tau/M)^4;  max_P |E_b' - d(11.12)/dP| = 3.589e-03
  E_b''(0): numeric +1.87643618e-01  closed +1.74198841e-01  rel.dev 7.718e-02  = 0.077 * (tau/M)^2
  s3_1   numeric +8.5646967127e-01  closed +8.8201005947e-01  diff -2.55e-02
  s3_2   numeric +9.3859089119e-01  closed +9.5531741897e-01  diff -1.67e-02
  A1     numeric +2.2543811170e-01  closed +2.3597988106e-01  diff -1.05e-02
  A2     numeric +1.2495039252e-01  closed +1.1617471068e-01  diff +8.78e-03
  U      numeric -4.1500239894e-01  closed -5.0953161098e-01  diff +9.45e-02
  Ebind  numeric -2.8797606491e-01  closed -3.3345431511e-01  diff +4.55e-02
  Epp0   numeric +1.8764361764e-01  closed +1.7419884136e-01  diff +1.34e-02
  m_b/(m_1+m_2): numeric +1.758104  (11.16) +1.893795;  m_b-(m_1+m_2): numeric +2.298001e+00  -4 m_1 m_2 E_bind +2.339806e+00

=== epsilon scan on set B_unequal_R (tau -> eps*tau, w -> eps^2*w)
eps= 1.000  E_b-(11.12): P=0 +8.1515e-06, P=0.3 +8.2675e-06;  rel.dev E_b''(0) +5.4456e-04;  <sigma3_i>-(11.14): -4.2426e-07, -1.7674e-06
eps= 0.500  E_b-(11.12): P=0 +5.0943e-07, P=0.3 +5.1674e-07;  rel.dev E_b''(0) +1.3708e-04;  <sigma3_i>-(11.14): -2.7203e-08, -1.1020e-07
          ratios to the previous eps: dE_P0 16.00, dE_P03 16.00, rel_dEpp0 3.97, d_s3_1 15.60, d_s3_2 16.04
eps= 0.250  E_b-(11.12): P=0 +3.1839e-08, P=0.3 +3.2297e-08;  rel.dev E_b''(0) +3.4328e-05;  <sigma3_i>-(11.14): -1.7109e-09, -6.8832e-09
          ratios to the previous eps: dE_P0 16.00, dE_P03 16.00, rel_dEpp0 3.99, d_s3_1 15.90, d_s3_2 16.01
eps= 0.125  E_b-(11.12): P=0 +1.9899e-09, P=0.3 +2.0186e-09;  rel.dev E_b''(0) +8.5858e-06;  <sigma3_i>-(11.14): -1.0710e-10, -4.3014e-10
          ratios to the previous eps: dE_P0 16.00, dE_P03 16.00, rel_dEpp0 4.00, d_s3_1 15.97, d_s3_2 16.00
expected for halving eps: 16 for E_b and <sigma3_i> (next order eps^4), 4 for the relative deviation of E_b''(0)

=== new in v2, I3: reading statistics of the clock of a part, (11.19), (11.25), (11.26)
start: bound eigenvector of the level 0 on every level, clock label in |0>, total label P=0 sharp, no records;
readings at lambda = t*lambda_0, t = 0.25, 0.5, ..., 2N; deviation = max over a and t of |q - P_N(a; Gamma*t)|
set E_order_one, clock of part 1, N=4, spread D=0.0600 (D/g0=0.208, D/m=0.0600, lambda_0=78.54)
  Gamma: level 0 (v1) 0.8564696713; mean level (11.19) 0.8621903823; first order (11.25) 0.8623337320; least-squares fit 0.8621456724
  d2E_b/dM^2 at m: spectral sum +1.95468691e-01; closed form of (11.26) +1.72184621e-01;  closed (11.20) at the mean level 0.8870173806; closed (11.26) 0.8871755981
  deviation of the statistics: Gamma at the mean level 9.468e-04 (t<=N: 3.598e-04); first order 2.645e-03; Gamma of the level 0 7.098e-02 (t<=N: 3.790e-02)
set E_order_one, clock of part 1, N=4, spread D=0.0300 (D/g0=0.104, D/m=0.0300, lambda_0=157.08)
  Gamma: level 0 (v1) 0.8564696713; mean level (11.19) 0.8593655981; first order (11.25) 0.8594017016; least-squares fit 0.8593541025
  d2E_b/dM^2 at m: spectral sum +1.95468691e-01; closed form of (11.26) +1.72184621e-01;  closed (11.20) at the mean level 0.8845527554; closed (11.26) 0.8845928288
  deviation of the statistics: Gamma at the mean level 2.476e-04 (t<=N: 9.235e-05); first order 6.867e-04; Gamma of the level 0 3.606e-02 (t<=N: 1.923e-02)
  ratios to the previous spread: dev_mean_level 3.82, dev_level0 1.97, dev_first_order 3.85
set E_order_one, clock of part 1, N=4, spread D=0.0150 (D/g0=0.052, D/m=0.0150, lambda_0=314.16)
  Gamma: level 0 (v1) 0.8564696713; mean level (11.19) 0.8579266274; first order (11.25) 0.8579356864; least-squares fit 0.8579237667
  d2E_b/dM^2 at m: spectral sum +1.95468691e-01; closed form of (11.26) +1.72184621e-01;  closed (11.20) at the mean level 0.8832913598; closed (11.26) 0.8833014441
  deviation of the statistics: Gamma at the mean level 6.077e-05 (t<=N: 2.322e-05); first order 1.707e-04; Gamma of the level 0 1.816e-02 (t<=N: 9.686e-03)
  ratios to the previous spread: dev_mean_level 4.07, dev_level0 1.99, dev_first_order 4.02
set B_unequal_R, clock of part 1, N=4, spread D=0.1000 (D/g0=27.860, D/m=0.1000, lambda_0=47.12)
  Gamma: level 0 (v1) 0.9977123895; mean level (11.19) 0.9978093968; first order (11.25) 0.9978119033; least-squares fit 0.9978328312
  d2E_b/dM^2 at m: spectral sum +1.99027585e-03; closed form of (11.26) +1.99577457e-03;  closed (11.20) at the mean level 0.9978100720; closed (11.26) 0.9978126025
  deviation of the statistics: Gamma at the mean level 4.686e-04 (t<=N: 4.128e-04); first order 4.707e-04; Gamma of the level 0 1.426e-03 (t<=N: 9.115e-04)
set B_unequal_R, clock of part 1, N=4, spread D=0.0500 (D/g0=13.930, D/m=0.0500, lambda_0=94.25)
  Gamma: level 0 (v1) 0.9977123895; mean level (11.19) 0.9977615160; first order (11.25) 0.9977621464; least-squares fit 0.9977620037
  d2E_b/dM^2 at m: spectral sum +1.99027585e-03; closed form of (11.26) +1.99577457e-03;  closed (11.20) at the mean level 0.9977620716; closed (11.26) 0.9977627081
  deviation of the statistics: Gamma at the mean level 1.307e-04 (t<=N: 1.251e-04); first order 1.309e-04; Gamma of the level 0 5.874e-04 (t<=N: 3.517e-04)
  ratios to the previous spread: dev_mean_level 3.59, dev_level0 2.43, dev_first_order 3.60
set B_unequal_R, clock of part 1, N=4, spread D=0.0250 (D/g0=6.965, D/m=0.0250, lambda_0=188.50)
  Gamma: level 0 (v1) 0.9977123895; mean level (11.19) 0.9977371099; first order (11.25) 0.9977372679; least-squares fit 0.9977368502
  d2E_b/dM^2 at m: spectral sum +1.99027585e-03; closed form of (11.26) +1.99577457e-03;  closed (11.20) at the mean level 0.9977376013; closed (11.26) 0.9977377609
  deviation of the statistics: Gamma at the mean level 3.259e-05 (t<=N: 3.259e-05); first order 3.261e-05; Gamma of the level 0 3.021e-04 (t<=N: 1.408e-04)
  ratios to the previous spread: dev_mean_level 4.01, dev_level0 1.94, dev_first_order 4.01
set B_unequal_R, clock of part 2, N=5, spread D=0.1000 (D/g0=27.860, D/m=0.0588, lambda_0=50.27)
  Gamma: level 0 (v1) 0.9997133236; mean level (11.19) 0.9997276451; first order (11.25) 0.9997282035; least-squares fit 0.9997277090
  d2E_b/dM^2 at m: spectral sum +2.97598087e-04; closed form of (11.26) +2.95174829e-04;  closed (11.20) at the mean level 0.9997292967; closed (11.26) 0.9997298497
  deviation of the statistics: Gamma at the mean level 7.255e-06 (t<=N: 5.690e-06); first order 1.305e-05; Gamma of the level 0 2.199e-04 (t<=N: 1.083e-04)
set B_unequal_R, clock of part 2, N=5, spread D=0.0500 (D/g0=13.930, D/m=0.0294, lambda_0=100.53)
  Gamma: level 0 (v1) 0.9997133236; mean level (11.19) 0.9997206216; first order (11.25) 0.9997207635; least-squares fit 0.9997205698
  d2E_b/dM^2 at m: spectral sum +2.97598087e-04; closed form of (11.26) +2.95174829e-04;  closed (11.20) at the mean level 0.9997223298; closed (11.26) 0.9997224704
  deviation of the statistics: Gamma at the mean level 1.889e-06 (t<=N: 1.889e-06); first order 3.433e-06; Gamma of the level 0 1.114e-04 (t<=N: 5.357e-05)
  ratios to the previous spread: dev_mean_level 3.84, dev_level0 1.97, dev_first_order 3.80
set B_unequal_R, clock of part 2, N=5, spread D=0.0250 (D/g0=6.965, D/m=0.0147, lambda_0=201.06)
  Gamma: level 0 (v1) 0.9997133236; mean level (11.19) 0.9997170078; first order (11.25) 0.9997170436; least-squares fit 0.9997169943
  d2E_b/dM^2 at m: spectral sum +2.97598087e-04; closed form of (11.26) +2.95174829e-04;  closed (11.20) at the mean level 0.9997187452; closed (11.26) 0.9997187807
  deviation of the statistics: Gamma at the mean level 4.854e-07 (t<=N: 4.854e-07); first order 8.020e-07; Gamma of the level 0 5.664e-05 (t<=N: 2.673e-05)
  ratios to the previous spread: dev_mean_level 3.89, dev_level0 1.97, dev_first_order 4.28
expected for halving the spread at fixed lambda/lambda_0: 4 with Gamma at the mean level, 2 with Gamma of the level 0

=== new in v2, I4: order of the contribution of w22 (set B, tau -> eps*tau, w -> eps^2*w)
eps= 2.000  E_b(w22) - E_b(w22=0) = +2.867296e-07;  w22*|<22|psi(0)>|^2 = +2.867347e-07;  |<22|psi(0)>|^2 = 8.960461e-06
eps= 1.000  E_b(w22) - E_b(w22=0) = +4.444423e-09;  w22*|<22|psi(0)>|^2 = +4.444423e-09;  |<22|psi(0)>|^2 = 5.555528e-07;  ratios to the previous eps: dE 64.51, weight 16.13
eps= 0.500  E_b(w22) - E_b(w22=0) = +6.930323e-11;  w22*|<22|psi(0)>|^2 = +6.930409e-11;  |<22|psi(0)>|^2 = 3.465205e-08;  ratios to the previous eps: dE 64.13, weight 16.03
eps= 0.250  E_b(w22) - E_b(w22=0) = +1.080025e-12;  w22*|<22|psi(0)>|^2 = +1.082329e-12;  |<22|psi(0)>|^2 = 2.164659e-09;  ratios to the previous eps: dE 64.17, weight 16.01
expected for halving eps: 64 for the contribution of w22 to E_b (order eps^6), 16 for the weight (order eps^4)

=== new in v2, I5: equal data (a_1 = a_2) at P = pi/2, where j(P) = 0 (set A, tau -> eps*tau, w -> eps^2*w)
eps= 1.000  E_b = 2.001000623470  M_1+M_2+a+w11 = 2.001000000000  difference +6.235e-07;  1 - weight(r=0,11) = 2.495e-03;  weight of 11 at r != 0: 3.044e-07
eps= 0.500  E_b = 2.000250039039  M_1+M_2+a+w11 = 2.000250000000  difference +3.904e-08;  1 - weight(r=0,11) = 6.247e-04;  weight of 11 at r != 0: 1.906e-08;  ratios: dE 15.97, 1-weight 3.99
eps= 0.250  E_b = 2.000062502441  M_1+M_2+a+w11 = 2.000062500000  difference +2.441e-09;  1 - weight(r=0,11) = 1.562e-04;  weight of 11 at r != 0: 1.192e-09;  ratios: dE 15.99, 1-weight 4.00
expected: psi_P = delta_{r,0}|11> up to O(eps) (1 - weight of order eps^2), E_b = M_1+M_2+a+w11 up to O(eps^4)

=== new in v2, I2: twofold bound levels (on-site binding W0 = -G(E0,P0)^(-1) on span(|11>,|22>))
set X_P0_0.3: M = 1.0, 0.7; tau = 0.5, 0.35; P0 = 0.3; W0 on (11,22) = [[-0.870998, 0.169999], [0.169999, 2.488876]]; bands {'+1+1': (1.7046, 1.8965), '+1-1': (0.2832, 0.3515), '-1+1': (-0.3515, -0.2832), '-1-1': (-1.8965, -1.7046)}
  level 1.000000000000, splitting of the two eigenvalues 2.11e-15, distance to the next eigenvalue 0.5368, weight at |r|>L/2 1.3e-30
  M1   : eigenvalues of Pi dH Pi -0.628449130, +0.893076371; slopes of the branches (finite differences) -0.628449130, +0.893076371; max dev 1.8e-12; overlap with the branch vectors >= 0.999997
  M2   : eigenvalues of Pi dH Pi -0.919383668, +0.837955656; slopes of the branches (finite differences) -0.919383668, +0.837955656; max dev 8.3e-13; overlap with the branch vectors >= 0.999999
  tau1 : eigenvalues of Pi dH Pi +0.137013648, +0.601738268; slopes of the branches (finite differences) +0.137013648, +0.601738268; max dev 8.2e-12; overlap with the branch vectors >= 0.999999
  tau2 : eigenvalues of Pi dH Pi -0.156239994, +0.533638710; slopes of the branches (finite differences) -0.156239994, +0.533638710; max dev 1.1e-12; overlap with the branch vectors >= 1.000000
  theta: eigenvalues of Pi dH Pi -0.726064561, +2.016971420; slopes of the branches (finite differences) -0.726064561, +2.016971420; max dev 4.6e-12; overlap with the branch vectors >= 0.999996
  P    : eigenvalues of Pi dH Pi -0.031846375, +0.056264981; slopes of the branches (finite differences) -0.031846375, +0.056264981; max dev 2.9e-10; overlap with the branch vectors >= 0.999955
         E'' of the branches: (11.24) -0.10636246, +0.17322408; finite differences -0.10636246, +0.17322408; max dev 1.3e-10
         (11.9) with Q_P = 1 - |psi><psi|, term of the other vector of the level in the basis of the solver: |<psi'|V_eta|psi>|^2 = 6.549e-04 over E' - E = -2.109e-15
  every eigenvector (5 random vectors of the level): residual of (11.6) 1.6e-15; <V1> - <V2> 1.1e-16; ||Pi (V1 - V2) Pi|| 1.6e-16; <dH> - weighted mean of the branch slopes (M1, theta) 4.1e-12
  |<branch 0 of M1 | branch 0 of xi>|^2: M2 0.9912, tau1 0.2769, tau2 0.7272, theta 0.0013, P 0.5106
  <U> in the two branch vectors of M1: +2.013542, -0.722635  against the slopes of the branches of theta: -0.726065, +2.016971
set X_P0_0: M = 1.0, 0.7; tau = 0.5, 0.35; P0 = 0.0; W0 on (11,22) = [[-0.869086, 0.177794], [0.177794, 2.488618]]; bands {'+1+1': (1.7, 1.9007), '+1-1': (0.3, 0.3354), '-1+1': (-0.3354, -0.3), '-1-1': (-1.9007, -1.7)}
  level 1.000000000000, splitting of the two eigenvalues 3.55e-15, distance to the next eigenvalue 0.5397, weight at |r|>L/2 3.2e-30
  M1   : eigenvalues of Pi dH Pi -0.631196110, +0.895473254; slopes of the branches (finite differences) -0.631196110, +0.895473254; max dev 1.8e-12; overlap with the branch vectors >= 0.999997
  M2   : eigenvalues of Pi dH Pi -0.918901736, +0.845071613; slopes of the branches (finite differences) -0.918901736, +0.845071613; max dev 1.2e-12; overlap with the branch vectors >= 0.999999
  tau1 : eigenvalues of Pi dH Pi +0.120869467, +0.605304949; slopes of the branches (finite differences) +0.120869467, +0.605304949; max dev 2.1e-12; overlap with the branch vectors >= 0.999999
  tau2 : eigenvalues of Pi dH Pi -0.179004420, +0.538130054; slopes of the branches (finite differences) -0.179004420, +0.538130054; max dev 7.5e-13; overlap with the branch vectors >= 1.000000
  theta: eigenvalues of Pi dH Pi -0.725682673, +2.024305435; slopes of the branches (finite differences) -0.725682673, +2.024305435; max dev 3.7e-12; overlap with the branch vectors >= 0.999996
  P    : eigenvalues of Pi dH Pi -0.000000000, +0.000000000; slopes of the branches (finite differences) +0.000000000, -0.000000000; max dev 1.8e-14; overlap with the branch vectors >= 0.999952; branch vectors from the second order
         E'' of the branches: (11.24) -0.10630610, +0.19294341; finite differences -0.10630610, +0.19294341; max dev 4.1e-10
         (11.9) with Q_P = 1 - |psi><psi|, term of the other vector of the level in the basis of the solver: |<psi'|V_eta|psi>|^2 = 9.382e-35 over E' - E = -3.553e-15
  every eigenvector (5 random vectors of the level): residual of (11.6) 1.6e-15; <V1> - <V2> 3.5e-17; ||Pi (V1 - V2) Pi|| 6.4e-17; <dH> - weighted mean of the branch slopes (M1, theta) 2.9e-12
  |<branch 0 of M1 | branch 0 of xi>|^2: M2 0.9904, tau1 0.2843, tau2 0.7171, theta 0.0014, P 0.5112
  <U> in the two branch vectors of M1: +2.020503, -0.721880  against the slopes of the branches of theta: -0.725683, +2.024305

=== new in v2, I1: the operator C_bod = <chi|C|chi> of (11.21), small system in the places (dimension 64 x 3)
listing A: the terms of the question; listing B: h*sigma3 of b_1, a constant and a part of the motion term of b_2 listed in the record terms; <chi|X|chi> = -0.6021 for the medium that is not centred; lambda = 3
kappa=0.040 centred    : ||C_B - C_A|| 1.3e-15; ||C_bod(B) - C_bod(A)|| 5.2e-16; ||C_bod - C_pair(A)|| 1.339e-15 (minus the gradient: 1.3e-15); ||C_bod - C_pair(B)|| 2.589; view against evolution by C_bod 4.141e-02, by C_pair(A) 4.141e-02, by C_pair(B) 1.025
kappa=0.040 not_centred: ||C_B - C_A|| 1.5e-15; ||C_bod(B) - C_bod(A)|| 1.3e-15; ||C_bod - C_pair(A)|| 6.534e-01 (minus the gradient: 1.3e-15); ||C_bod - C_pair(B)|| 2.491; view against evolution by C_bod 4.147e-02, by C_pair(A) 1.526e-01, by C_pair(B) 1.045
kappa=0.020 centred    : ||C_B - C_A|| 1.3e-15; ||C_bod(B) - C_bod(A)|| 1.0e-15; ||C_bod - C_pair(A)|| 1.298e-15 (minus the gradient: 1.3e-15); ||C_bod - C_pair(B)|| 2.589; view against evolution by C_bod 1.052e-02, by C_pair(A) 1.052e-02, by C_pair(B) 1.034
kappa=0.020 not_centred: ||C_B - C_A|| 1.4e-15; ||C_bod(B) - C_bod(A)|| 9.8e-16; ||C_bod - C_pair(A)|| 3.267e-01 (minus the gradient: 1.5e-15); ||C_bod - C_pair(B)|| 2.520; view against evolution by C_bod 1.053e-02, by C_pair(A) 7.442e-02, by C_pair(B) 1.044
kappa=0.010 centred    : ||C_B - C_A|| 1.4e-15; ||C_bod(B) - C_bod(A)|| 8.3e-16; ||C_bod - C_pair(A)|| 1.295e-15 (minus the gradient: 1.3e-15); ||C_bod - C_pair(B)|| 2.589; view against evolution by C_bod 2.642e-03, by C_pair(A) 2.642e-03, by C_pair(B) 1.037
kappa=0.010 not_centred: ||C_B - C_A|| 1.3e-15; ||C_bod(B) - C_bod(A)|| 1.2e-15; ||C_bod - C_pair(A)|| 1.633e-01 (minus the gradient: 1.8e-15); ||C_bod - C_pair(B)|| 2.549; view against evolution by C_bod 2.643e-03, by C_pair(A) 3.695e-02, by C_pair(B) 1.041
expected for halving kappa: the view against the evolution by C_bod falls by 4 (second order); by the listed C_pair(A) with a medium that is not centred it falls by 2