== Ring: numerical (4.3) vs closed form (4.5); M2 * t_f * L^2 / (N_f g1 g2) vs d(L-d) - (L^2-1)/6 == L= 3 t_f=1.0: max_d |M2 - (4.5)| / max|(4.5)| = 5.62e-16 L= 3 t_f=2.0: max_d |M2 - (4.5)| / max|(4.5)| = 5.62e-16 L= 4 t_f=1.0: max_d |M2 - (4.5)| / max|(4.5)| = 1.78e-16 L= 4 t_f=2.0: max_d |M2 - (4.5)| / max|(4.5)| = 1.78e-16 L= 16 t_f=1.0: max_d |M2 - (4.5)| / max|(4.5)| = 1.50e-15 L= 16 t_f=2.0: max_d |M2 - (4.5)| / max|(4.5)| = 1.50e-15 L= 101 t_f=1.0: max_d |M2 - (4.5)| / max|(4.5)| = 6.56e-14 L= 101 t_f=2.0: max_d |M2 - (4.5)| / max|(4.5)| = 6.56e-14 L= 400 t_f=1.0: max_d |M2 - (4.5)| / max|(4.5)| = 1.47e-12 L= 400 t_f=2.0: max_d |M2 - (4.5)| / max|(4.5)| = 1.47e-12 Ring L=3, d=1, t_f=1: M2 = 0.074074074074, 2/27 = 0.074074074074 == Even rings: lowest + highest state (sum over the two ends, (4.13)); odd ring: top degeneracy == L=4: max_d |M2_lowest + M2_highest| = 1.39e-17 L=16: max_d |M2_lowest + M2_highest| = 1.30e-15 L=64: max_d |M2_lowest + M2_highest| = 2.78e-15 L=15 (odd): two highest eigenvalues of C1 = 1.956295201468, 1.956295201468 (degenerate) kappa_2 = 1.310533, xi_3 = -0.225785 == Square torus: m = M2 t_f/(n g1 g2) vs leading form (4.6): m_lead = -(1/pi)[ln(L/|d|) - kappa_2] == -- D=2, L=64: |X|=4096, eps0=-4.000000, ||C1 phi - eps0 phi|| = 0.0e+00, phi>0 (Perron) d vector |d| m_num m_lead m_num-m_lead (2, 0) 2.00 -0.695059 -0.686022 -9.04e-03 (4, 0) 4.00 -0.469300 -0.465387 -3.91e-03 (8, 0) 8.00 -0.253040 -0.244751 -8.29e-03 (16, 0) 16.00 -0.056436 -0.024116 -3.23e-02 (2, 2) 2.83 -0.573480 -0.575705 2.22e-03 (4, 4) 5.66 -0.358125 -0.355069 -3.06e-03 (8, 8) 11.31 -0.149590 -0.134433 -1.52e-02 (16, 16) 22.63 0.027659 0.086202 -5.85e-02 (4, 2) 4.47 -0.431846 -0.429872 -1.97e-03 (8, 4) 8.94 -0.218862 -0.209237 -9.63e-03 (16, 8) 17.89 -0.027187 0.011399 -3.86e-02 max |m_num - m_lead| over 4 <= |d| <= L/8: 8.29e-03 -- D=2, L=128: |X|=16384, eps0=-4.000000, ||C1 phi - eps0 phi|| = 0.0e+00, phi>0 (Perron) d vector |d| m_num m_lead m_num-m_lead (2, 0) 2.00 -0.915339 -0.906658 -8.68e-03 (4, 0) 4.00 -0.688478 -0.686022 -2.46e-03 (8, 0) 8.00 -0.467770 -0.465387 -2.38e-03 (16, 0) 16.00 -0.252726 -0.244751 -7.97e-03 (32, 0) 32.00 -0.056370 -0.024116 -3.23e-02 (2, 2) 2.83 -0.793395 -0.796340 2.95e-03 (4, 4) 5.66 -0.575856 -0.575705 -1.52e-04 (8, 8) 11.31 -0.358750 -0.355069 -3.68e-03 (16, 16) 22.63 -0.149758 -0.134433 -1.53e-02 (32, 32) 45.25 0.027599 0.086202 -5.86e-02 (4, 2) 4.47 -0.650663 -0.650508 -1.55e-04 (8, 4) 8.94 -0.432210 -0.429872 -2.34e-03 (16, 8) 17.89 -0.218948 -0.209237 -9.71e-03 (32, 16) 35.78 -0.027219 0.011399 -3.86e-02 max |m_num - m_lead| over 4 <= |d| <= L/8: 7.97e-03 -- D=2, L=256: |X|=65536, eps0=-4.000000, ||C1 phi - eps0 phi|| = 0.0e+00, phi>0 (Perron) d vector |d| m_num m_lead m_num-m_lead (2, 0) 2.00 -1.135885 -1.127294 -8.59e-03 (4, 0) 4.00 -0.908750 -0.906658 -2.09e-03 (8, 0) 8.00 -0.686940 -0.686022 -9.18e-04 (16, 0) 16.00 -0.467447 -0.465387 -2.06e-03 (32, 0) 32.00 -0.252650 -0.244751 -7.90e-03 (2, 2) 2.83 -1.013850 -1.016976 3.13e-03 (4, 4) 5.66 -0.795763 -0.796340 5.77e-04 (8, 8) 11.31 -0.576473 -0.575705 -7.68e-04 (16, 16) 22.63 -0.358907 -0.355069 -3.84e-03 (32, 32) 45.25 -0.149800 -0.134433 -1.54e-02 (4, 2) 4.47 -0.870843 -0.871144 3.00e-04 (8, 4) 8.94 -0.651019 -0.650508 -5.11e-04 (16, 8) 17.89 -0.432288 -0.429872 -2.42e-03 (32, 16) 35.78 -0.218969 -0.209237 -9.73e-03 (64, 32) 71.55 -0.027227 0.011399 -3.86e-02 max |m_num - m_lead| over 4 <= |d| <= L/8: 7.90e-03 == Cubic torus: m vs leading form (4.7): m_lead = -2[1/(4 pi |d|) + xi_3/L] == -- D=3, L=32: |X|=32768, eps0=-6.000000, ||C1 phi - eps0 phi|| = 0.0e+00, phi>0 (Perron) d vector |d| m_num m_lead m_num-m_lead (2, 0, 0) 2.00 -0.071711 -0.065466 -6.24e-03 (3, 0, 0) 3.00 -0.041074 -0.038940 -2.13e-03 (4, 0, 0) 4.00 -0.026635 -0.025677 -9.58e-04 (6, 0, 0) 6.00 -0.013006 -0.012414 -5.91e-04 (8, 0, 0) 8.00 -0.006579 -0.005783 -7.96e-04 (2, 2, 0) 2.83 -0.042082 -0.042158 7.58e-05 (3, 3, 0) 4.24 -0.023502 -0.023402 -1.01e-04 (4, 4, 0) 5.66 -0.014303 -0.014023 -2.79e-04 (6, 6, 0) 8.49 -0.005343 -0.004645 -6.98e-04 (8, 8, 0) 11.31 -0.001184 0.000044 -1.23e-03 (2, 2, 2) 3.46 -0.031314 -0.031833 5.18e-04 (3, 3, 3) 5.20 -0.016597 -0.016518 -7.90e-05 (4, 4, 4) 6.93 -0.009248 -0.008860 -3.88e-04 (6, 6, 6) 10.39 -0.002170 -0.001203 -9.67e-04 (8, 8, 8) 13.86 0.001009 0.002626 -1.62e-03 max |m_num - m_lead| over 4 <= |d| <= L/8: 9.58e-04 -- D=3, L=48: |X|=110592, eps0=-6.000000, ||C1 phi - eps0 phi|| = 4.3e-14, phi>0 (Perron) d vector |d| m_num m_lead m_num-m_lead (2, 0, 0) 2.00 -0.076384 -0.070170 -6.21e-03 (3, 0, 0) 3.00 -0.045711 -0.043644 -2.07e-03 (4, 0, 0) 4.00 -0.031220 -0.030381 -8.38e-04 (6, 0, 0) 6.00 -0.017433 -0.017118 -3.15e-04 (8, 0, 0) 8.00 -0.010770 -0.010487 -2.83e-04 (2, 2, 0) 2.83 -0.046727 -0.046862 1.35e-04 (3, 3, 0) 4.24 -0.028077 -0.028105 2.88e-05 (4, 4, 0) 5.66 -0.018780 -0.018727 -5.23e-05 (6, 6, 0) 8.49 -0.009549 -0.009349 -2.00e-04 (8, 8, 0) 11.31 -0.005031 -0.004660 -3.71e-04 (2, 2, 2) 3.46 -0.035932 -0.036536 6.05e-04 (3, 3, 3) 5.20 -0.021112 -0.021222 1.10e-04 (4, 4, 4) 6.93 -0.013626 -0.013564 -6.21e-05 (6, 6, 6) 10.39 -0.006195 -0.005907 -2.88e-04 (8, 8, 8) 13.86 -0.002602 -0.002078 -5.23e-04 max |m_num - m_lead| over 4 <= |d| <= L/8: 8.38e-04 -- D=3, L=64: |X|=262144, eps0=-6.000000, ||C1 phi - eps0 phi|| = 0.0e+00, phi>0 (Perron) d vector |d| m_num m_lead m_num-m_lead (2, 0, 0) 2.00 -0.078728 -0.072522 -6.21e-03 (3, 0, 0) 3.00 -0.048046 -0.045996 -2.05e-03 (4, 0, 0) 4.00 -0.033543 -0.032733 -8.10e-04 (6, 0, 0) 6.00 -0.019720 -0.019470 -2.50e-04 (8, 0, 0) 8.00 -0.013004 -0.012839 -1.65e-04 (2, 2, 0) 2.83 -0.049065 -0.049214 1.49e-04 (3, 3, 0) 4.24 -0.030397 -0.030457 6.04e-05 (4, 4, 0) 5.66 -0.021076 -0.021079 3.44e-06 (6, 6, 0) 8.49 -0.011777 -0.011701 -7.63e-05 (8, 8, 0) 11.31 -0.007166 -0.007012 -1.55e-04 (2, 2, 2) 3.46 -0.038262 -0.038888 6.26e-04 (3, 3, 3) 5.20 -0.023417 -0.023574 1.56e-04 (4, 4, 4) 6.93 -0.015897 -0.015916 1.97e-05 (6, 6, 6) 10.39 -0.008369 -0.008259 -1.10e-04 (8, 8, 8) 13.86 -0.004654 -0.004430 -2.23e-04 max |m_num - m_lead| over 4 <= |d| <= L/8: 8.10e-04