====================================================================================================================== (a) ISOLATED TIE SPIN (h = 0, H = gX; D40 eq. 7.1) frozen: Gamma_m = kappa, nu_th = nu ====================================================================================================================== Bloch matrix from jumps vs eq.(7.1) [(z,y) block, x decoupled at -(nu/2+Gamma_m), unital]: max |diff| = 1.42e-14 parameters: g = 0.3, nu_th = 1.0; w_eig := -(lambda_slow + nu_th)/2 from the full 4x4 Lindbladian Gamma_m w_eig |w_eig-closed| 2g2/(G-nu/2) (w_eig-2g2/D)/(8g4/D3) 2g2/Gamma_m G*(w_eig/w0-1) w_Schur0 |w_S0-2g2/(G+nu/2)| G*(w_S0/w0-1) ---------------------------------------------------------------------------------------------------------------------- 5 0.0407375779 1.11e-16 0.0400000000 1.037219 0.03600000 0.657997 0.0327272727 1.04e-16 -0.454545 10 0.0190235570 1.11e-16 0.0189473684 1.008058 0.01800000 0.568643 0.0171428571 9.37e-17 -0.476190 20 0.0092395250 5.55e-16 0.0092307692 1.001898 0.00900000 0.532278 0.0087804878 1.20e-16 -0.487805 50 0.0036368981 0.00e+00 0.0036363636 1.000294 0.00360000 0.512473 0.0035643564 1.14e-16 -0.495050 100 0.0018091110 1.44e-15 0.0018090452 1.000073 0.00180000 0.506167 0.0017910448 6.29e-17 -0.497512 reading: G*(w/w0-1) -> +nu_th/2 = +0.5 for the eigenvalue definition, -> -nu_th/2 = -0.5 for the zero-frequency Schur definition: same leading 2g^2/Gamma_m, opposite first thermal correction. overdamped (real-rate) threshold Gamma_m > nu_th/2 + 4g = 1.700. At Gamma_m = 0.8 (Attack-11 value) the (z,y) eigenvalues are -1.3000+0.0000j, -1.1500+0.5809j: underdamped, no Zeno rate. ====================================================================================================================== (b) 2x2 PLAQUETTE, frozen exterior e = (0, 0, 2, -2) (exterior pairs {0: (1, -1), 1: (-1, 1), 2: (1, 1), 3: (-1, -1)}), beta J = 0.6, nu_th = 1.0, g_i = (0.3, 0.4, 0.25, 0.35) ====================================================================================================================== directed tie edges (s, i) with h_i(s) = 0: 24 (per site: [8, 8, 4, 4]) thermal rate on every tie edge c_i = nu_th/2 = 0.5 (checked): True distinct Gamma_th^R1 values: ['0.9204', '1.1663', '2.0000', '2.1709', '2.4168', '2.6627'] distinct Gamma_loc^R2 values: ['0.8373', '1.0832', '1.2541', '1.5000', '1.7459', '1.9168'] --- convention R1 (rank-one, fully configuration-resolved flips; Attack-11 style) Gamma_m stat.res maxrel[edge vs exact O(g2)] maxrel[edge vs 2g2/(G+G_R1)] maxrel[edge vs 2g2/G] maxrel[edge vs 2g2/(G+G_loc)] O(g2) analytic-vs-num ---------------------------------------------------------------------------------------------------------------------- 5 3.9e-17 6.494e-03 6.494e-03 3.497e-01 1.311e-01 4.0e-16 10 5.3e-17 1.891e-03 1.891e-03 2.112e-01 7.643e-02 4.3e-16 20 4.2e-17 5.121e-04 5.121e-04 1.178e-01 4.168e-02 3.7e-16 50 2.7e-17 8.608e-05 8.608e-05 5.062e-02 1.763e-02 2.8e-16 100 3.3e-17 2.188e-05 2.188e-05 2.595e-02 8.989e-03 4.1e-16 log-log slope (Gamma_m 50->100): |edge - exact O(g^2)| ~ Gamma_m^-2.960 (expect 3: O(g^4/G^3)); rel. dev. from 2g^2/Gamma_m ~ Gamma_m^-0.964 (expect 1) g -> g/2 at Gamma_m = 20: max|edge - exact O(g^2)| 7.420e-06 -> 4.640e-07, ratio 15.993 (expect ~16 = O(g^4)) structure: max|G| on >=2-spin moves: 9.75e-05, 1.53e-05, 2.16e-06, 1.49e-07, 1.91e-08 | min off-diagonal G: +0.00e+00, +0.00e+00, +0.00e+00, +0.00e+00, +0.00e+00 pi-detailed-balance violation of G_Schur (rel.): 3.6e-17, 3.7e-17, 3.8e-17, 3.9e-17, 3.9e-17 | max Im G: 0.0e+00 --- convention R2 (neighbour-resolved flips; D40 section 0) Gamma_m stat.res maxrel[edge vs exact O(g2)] maxrel[edge vs 2g2/(G+G_R1)] maxrel[edge vs 2g2/G] maxrel[marg vs 2g2/(G+G_loc)] O(g2) analytic-vs-num ---------------------------------------------------------------------------------------------------------------------- 5 3.3e-17 6.508e-03 4.792e-03 3.487e-01 9.003e-03 6.8e-16 10 2.6e-17 1.892e-03 1.332e-03 2.108e-01 2.919e-03 6.0e-16 20 1.9e-17 5.122e-04 3.498e-04 1.177e-01 8.438e-04 6.9e-16 50 4.4e-17 8.608e-05 5.752e-05 5.059e-02 1.481e-04 4.4e-16 100 5.1e-17 2.188e-05 1.450e-05 2.594e-02 3.824e-05 6.0e-16 log-log slope (Gamma_m 50->100): |edge - exact O(g^2)| ~ Gamma_m^-2.961 (expect 3: O(g^4/G^3)); rel. dev. from 2g^2/Gamma_m ~ Gamma_m^-0.963 (expect 1) g -> g/2 at Gamma_m = 20: max|edge - exact O(g^2)| 7.423e-06 -> 4.641e-07, ratio 15.993 (expect ~16 = O(g^4)) structure: max|G| on >=2-spin moves: 6.61e-03, 2.16e-03, 6.25e-04, 1.10e-04, 2.84e-05 | min off-diagonal G: -1.55e-05, -1.46e-06, -1.14e-07, -3.36e-09, -2.20e-10 pi-detailed-balance violation of G_Schur (rel.): 3.6e-17, 5.6e-17, 5.7e-17, 5.8e-17, 5.9e-17 | max Im G: 0.0e+00 Per-edge measured thermal contribution Gamma_eff := 2 g_i^2 / w - Gamma_m at Gamma_m = 100 (6 representative edges) edge (config bits, site) Gamma_R1 Gamma_loc R1: edge R2: edge R2: marginal ---------------------------------------------------------------------------------------------------------------------- ++++ , i=3 0.9204 0.8373 0.9220 0.9213 0.8394 -+++ , i=1 2.0000 1.9168 2.0021 2.0014 1.9188 -+++ , i=3 2.4168 1.5000 2.4190 2.4183 1.4962 +-++ , i=0 2.1709 1.2541 2.1725 2.1717 1.2541 ++-+ , i=0 2.6627 1.7459 2.6643 2.6636 1.7490 +++- , i=1 1.1663 1.0832 1.1685 1.1677 1.0854 reading: R1 edge -> Gamma_R1 (extensive sum over all live sites); R2 edge -> Gamma_R1 too at this order, but the R2 MARGINAL (edge + correlated 2-spin moves) -> the local Gamma_loc. Residuals are O(1/Gamma_m). Bonus diagnostic: slow-manifold population generator G_slow = (P V) diag(lambda_slow) (P V)^-1 (16 slowest modes) conv Gamma_m gap Re l17/Re l16 max G*|w_slow/w0-1| R1 pred. maxrel pi-DB violation ---------------------------------------------------------------------------------------------------------------------- R1 20 5.21 1.7680 6.49e-03 4.83e-04 R1 50 12.72 1.6860 1.02e-03 7.76e-05 R1 100 25.22 1.6609 2.53e-04 1.94e-05 R2 20 5.20 1.7635 n/a 4.83e-04 R2 50 12.70 1.6844 n/a 7.76e-05 R2 100 25.21 1.6601 n/a 1.94e-05 reading: the O(g^2 nu_th/Gamma_m^2) correction is definition dependent (zero-frequency Schur vs slow manifold); only the leading 2 g^2/Gamma_m is convention- and definition-free. ====================================================================================================================== CHECK SUMMARY ====================================================================================================================== [PASS] (a) Lindbladian from jumps reproduces D40 eq. 7.1 Bloch matrix expected 0 obtained 1.42e-14 [PASS] (a) exact slow eigenvalue matches closed form w=[D-sqrt(D^2-16g^2)]/4, D=Gamma_m-nu_th/2 expected rel diff < 1e-10 obtained 7.98e-13 [PASS] (a) next term: (w_eig - 2g^2/D)/(8g^4/D^3) -> 1 expected 1 (+O(8g^2/D^2)) obtained 1.03722, 1.00806, 1.00190, 1.00029, 1.00007 [PASS] (a) zero-frequency Schur rate equals 2g^2/(Gamma_m+nu_th/2) exactly expected rel diff < 1e-10 obtained 3.51e-14 [PASS] (b) G_Schur stationary on Gibbs pi (both conventions, all Gamma_m) expected < 1e-12 obtained 5.3e-17 [PASS] (b) analytic exact-O(g^2) rates (R1: 2g^2/(G+Gamma_R1); R2: Feynman-Kac resolvent) vs numeric O(g^2) Schur term expected rel diff < 1e-9 obtained R1 4.3e-16, R2 6.9e-16 [PASS] (b) R1: max rel err of tie-edge excess vs exact-O(g^2) formula, Gamma_m=5,10,20,50,100 expected -> 0 like Gamma_m^-2 (abs Gamma_m^-3) obtained 6.49e-03, 1.89e-03, 5.12e-04, 8.61e-05, 2.19e-05; abs slope 2.960 [PASS] (b) R1: max rel dev of tie-edge excess from universal 2g_i^2/Gamma_m expected -> 0 like Gamma_m^-1 obtained 3.497e-01, 2.112e-01, 1.178e-01, 5.062e-02, 2.595e-02; slope 0.964 [PASS] (b) R1: O(g^4) residual scaling under g -> g/2 at Gamma_m=20 expected ratio ~16 obtained 15.993 [PASS] (b) R2: max rel err of tie-edge excess vs exact-O(g^2) formula, Gamma_m=5,10,20,50,100 expected -> 0 like Gamma_m^-2 (abs Gamma_m^-3) obtained 6.51e-03, 1.89e-03, 5.12e-04, 8.61e-05, 2.19e-05; abs slope 2.961 [PASS] (b) R2: max rel dev of tie-edge excess from universal 2g_i^2/Gamma_m expected -> 0 like Gamma_m^-1 obtained 3.487e-01, 2.108e-01, 1.177e-01, 5.059e-02, 2.594e-02; slope 0.963 [PASS] (b) R2: O(g^4) residual scaling under g -> g/2 at Gamma_m=20 expected ratio ~16 obtained 15.993 [PASS] (b) R1: tie-edge excess vs 2g^2/(Gamma_m+Gamma_R1), Gamma_R1=(Lambda(s)+Lambda(s^i))/2 expected rel err O(g^2/Gamma_m^2) obtained 6.49e-03, 1.89e-03, 5.12e-04, 8.61e-05, 2.19e-05 [PASS] (b) R2: marginal flip-i rate vs 2g^2/(Gamma_m+Gamma_loc), local Gamma_loc expected rel err O(nu^2/Gamma_m^2) obtained 9.00e-03, 2.92e-03, 8.44e-04, 1.48e-04, 3.82e-05 [PASS] (b) G_Schur Gibbs detailed balance (diagnostic) expected report obtained R1 max 3.9e-17, R2 max 5.9e-17 15/15 checks passed