Outputs of the certificate and diagnostic programs, one section per program. ===== d5/abelian_growth.out ===== (A) L=256, constant plaquette angle theta=9.587380e-05 scale j=0: side 256, plaquette angle min=9.587380e-05 max=9.587380e-05, 4^j theta=9.587380e-05 scale j=1: side 128, plaquette angle min=3.834952e-04 max=3.834952e-04, 4^j theta=3.834952e-04 scale j=2: side 64, plaquette angle min=1.533981e-03 max=1.533981e-03, 4^j theta=1.533981e-03 scale j=3: side 32, plaquette angle min=6.135923e-03 max=6.135923e-03, 4^j theta=6.135923e-03 scale j=4: side 16, plaquette angle min=2.454369e-02 max=2.454369e-02, 4^j theta=2.454369e-02 (B) L=32, random field with link angles ~1e-3: max |coarse plaquette - mean of four squares| = 1.332e-15 growth of the maximal plaquette angle in one step (random field): 6.479e-03 -> 5.119e-03 all checks passed ===== d5/bad_branch.out ===== Rational proof: cutoff is in its transition branch at q=0 Rational strict upper bound for f'(0): -91/3200 < 0 Numerical f(0)=0.183143917834; f'(0)=-0.361264822427 path angle q=-0.00642099318125, average angle f(q)=0.183143917834 path angle q=0, average angle f(q)=0.183143917834 path angle q=0.148576706208, average angle f(q)=0.183143917834 Three path-link solutions for the same seven non-path links and average. ===== d5/basin.out ===== basin radius r1 | plaquette level rho | beta for islands beaten by rarity 4.8e-18 | 1.2e-18 | 4.69e+41 4.7e-16 | 1.17e-16 | 4.54e+37 1.0e-8 | 2.5e-9 | 7.15e+22 0.0001 | 2.5e-5 | 5.58e+14 0.01 | 0.0025 | 4.78e+10 0.1 | 0.025 | 4.39e+8 chessboard-only coarse scales k (coarse blocks of side 2^k) at which the island condition still holds r1=4.8e-18 beta=1.0e+38: scales k = 0..-1 usable (0 scales) r1=4.8e-18 beta=1.0e+64: scales k = 0..18 usable (19 scales) r1=0.01 beta=1.0e+8: scales k = 0..-1 usable (0 scales) r1=0.01 beta=1.0e+20: scales k = 0..7 usable (8 scales) ===== d5/bootstrap.out ===== bootstrap radius R = 6.525e+5 blocks; window m = 2.9e+24 blocks, |L1| <= 2.35e+26 contraction factor at R: 0.5 (i) lam|L1|eps_loc <= 1e-3 needs beta >= 1.14e+64 (ii) torus: lam|L1| n_P q(rho_c) <= 1e-3 needs beta >= 2.11e+95 (rho_c = 3.59e-46) (iii) density bound: lam|L1| p_nc >= 1.3e+148 at beta = 1.0e+10 (iii) density bound: lam|L1| p_nc >= 1.09e+149 at beta = 1.0e+100 (iii) density bound: lam|L1| p_nc >= 1.07e+150 at beta = 1.0e+1000 (beta-independent up to log beta: the density bound can never make the floor small) ===== d5/chart_bounds.out ===== Rational upper bounds (displayed as decimals; comparisons use Fractions): complex contraction: 0.883105916531 fixed-point image radius: 0.00404360059569 normalised derivative error: 0.0648473322451 logdet residual: 0.201136446003 path acceleration bound: 3535488 Q Hessian lower bound: 0.0280870132923 Q Hessian absolute upper bound: 743472.00973 Sum logdet Hessian absolute bound: 1600000000 lambda >= 200000000000 implies Hess V_eff >= lambda/104 All rational inequalities passed. Weighted Hessian error / lambda <= 0.00324135 < 1/208 Restricted-chart covariance amplitude <= 3300000/lambda, rate 1e-14 ===== d5/chart_probe.out ===== scale=0.001 step=0.0001 detJ=0.00195318934373 detH=0.00195318934373 relerr=4.34e-13 scale=0.001 step=1e-05 detJ=0.0019531893438 detH=0.00195318934373 relerr=3.5e-11 scale=0.001 step=1e-06 detJ=0.00195318934343 detH=0.00195318934374 relerr=1.56e-10 fixed-point residual=8.23e-17, iterations=227 scale=0.0001 step=0.0001 detJ=0.00195312564353 detH=0.00195312564354 relerr=6.36e-12 scale=0.0001 step=1e-05 detJ=0.00195312564358 detH=0.00195312564354 relerr=1.93e-11 scale=0.0001 step=1e-06 detJ=0.00195312564396 detH=0.00195312564354 relerr=2.16e-10 fixed-point residual=7.9e-17, iterations=210 scale=1e-05 step=0.0001 detJ=0.00195312500644 detH=0.00195312500644 relerr=1.46e-12 scale=1e-05 step=1e-05 detJ=0.00195312500643 detH=0.00195312500644 relerr=3.37e-12 scale=1e-05 step=1e-06 detJ=0.00195312500669 detH=0.00195312500644 relerr=1.29e-10 fixed-point residual=7.66e-17, iterations=193 Nonabelian determinant diagnostic passed (floating computation only). ===== d5/check_labelcov.out ===== sigma(2 alpha_G) in [6.144e+15, 6.144e+15] PASS sigma(2 alpha_G) <= 6.145e15 (value in prop:islandmass) sigma(alpha_G) in [9.8304e+16, 9.8304e+16] PASS sigma(alpha_G) <= 9.831e16 PASS 11^4 = 14641 === s = 1 === b_* in [1.83837e+16, 1.83847e+16] nu_s = 658845 log10(b_* nu_s) in [22.0832, 22.0832] PASS s=1: beta* rho^2/16 > 8 (b_* eps^(1/2), b_* eps decrease) PASS s=1: 2 sqrt(eps) <= 1 lambda = -ln g_R at R = 2s+9 in [8105.11, 8105.11] PASS s=1: g_(2s+9) <= 1/2 log10 Phi_(2s+9) in [-7012.12, -7012.12] PASS s=1: (6s+34)(1-1e-4) >= 6s+33 (d >= t+6) log10 c_s = log10(11^4 b nu e^{2a(8s+41)}) in [26.2594, 26.2595] log10 2 Phi_R (relative floor) in [-7011.82, -7011.82] chain rate lambda/(1e4 R) in [0.0736828, 0.0736828] d where 10^{c_s} e^{-2a(1-1e-4)d} = 1 in [120941, 120941] log10 C_1 = log10((9/2) b nu e^{4a(2s+9)}) in [22.7412, 22.7412] PASS s=1: lambda/(1e4 R) >= 2 alpha_G PASS s=1: 10 <= (9/4) b nu PASS s=1: (2s+10)(1-1e-4) > 2s+9 (t < d) log10 (17/4) eps^{1/2} Phi_R (fibre floor) in [-14062.7, -14062.7] -ln g_R at R_max in [7623.95, 7623.95] PASS s=1: g(R_max) <= 1/2 PASS s=1: lambda_S/8002 > 1/2 log10 chain/e^{-sqrt d/2} at d=1e8 in [-1965.29, -1965.29] log10 main/e^{-sqrt d/2} at d=1e8 in [-8663.45, -8663.45] log10 far coefficient in [-14062.7, -14062.7] log10 C_S in [-1965.29, -1965.29] === s = 5 === b_* in [2.16535e+16, 2.16545e+16] nu_s = 5864445 log10(b_* nu_s) in [23.1038, 23.1038] PASS s=5: beta* rho^2/16 > 8 (b_* eps^(1/2), b_* eps decrease) PASS s=5: 2 sqrt(eps) <= 1 lambda = -ln g_R at R = 2s+9 in [9571.06, 9571.06] PASS s=5: g_(2s+9) <= 1/2 log10 Phi_(2s+9) in [-8286.25, -8286.25] PASS s=5: (6s+34)(1-1e-4) >= 6s+33 (d >= t+6) log10 c_s = log10(11^4 b nu e^{2a(8s+41)}) in [27.2869, 27.2869] log10 2 Phi_R (relative floor) in [-8285.95, -8285.95] chain rate lambda/(1e4 R) in [0.050374, 0.050374] d where 10^{c_s} e^{-2a(1-1e-4)d} = 1 in [125673, 125674] log10 C_1 = log10((9/2) b nu e^{4a(2s+9)}) in [23.7652, 23.7652] PASS s=5: lambda/(1e4 R) >= 2 alpha_G PASS s=5: 10 <= (9/4) b nu PASS s=5: (2s+10)(1-1e-4) > 2s+9 (t < d) log10 (17/4) eps^{1/2} Phi_R (fibre floor) in [-16612, -16612] -ln g_R at R_max in [9092.02, 9092.02] PASS s=5: g(R_max) <= 1/2 PASS s=5: lambda_S/8002 > 1/2 log10 chain/e^{-sqrt d/2} at d=1e8 in [-2762.06, -2762.06] log10 main/e^{-sqrt d/2} at d=1e8 in [-8662.43, -8662.43] log10 far coefficient in [-16612, -16612] log10 C_S in [-2762.06, -2762.06] === regime S, d-independent conditions === PASS d=1e8: d/2 - 1 >= 2000 sqrt d + 1 (t >= R) PASS d=1e8: d/2 >= sqrt d + 4001 (floor(t/R) >= sqrt d/8002) PASS d=1e8: alpha_G > 1/(4 sqrt d) (main ratio decreases) PASS d<=1e100: ceil(2000 sqrt d) <= 2000*10^50 + 1 = R_max (exact integers) === stated numbers === PASS s=1: log10(b_* nu) <= 22.084 PASS s=5: log10(b_* nu) <= 23.104 PASS s=1: -ln g per block >= 736 PASS s=5: -ln g per block >= 503 PASS s=1: c_1 <= 26.26 PASS s=5: c_5 <= 27.29 PASS s=1: 2 Phi_R <= 10^-7011.8 PASS s=5: 2 Phi_R <= 10^-8285.9 PASS s=1: lambda_1 >= 0.0736 PASS s=5: lambda_5 >= 0.0503 PASS s=1: d_0 <= 1.21e5 PASS s=5: d_0 <= 1.26e5 PASS s=1: a_1 >= log10 C_1 (22.75) PASS s=5: a_5 >= log10 C_1 (23.77) PASS s=1: fibre floor <= 10^-14062.6 PASS s=5: fibre floor <= 10^-16611.8 PASS s=1: C_S <= 10^-1965 PASS s=5: C_S <= 10^-2761 ALL PASS ===== d5/clusters.out ===== ================ s = 1 ================ PASS s=1: root bracket beta* in [1.12206202024e+37, 1.12206202024e+37] beta* rho^2/16 >= 33022.9865446 PASS s=1: beta* rho^2/16 >= 32481 log10 Mq* <= -14102.4456335 log10 A_* <= -14098.898734 --- vartheta = 0 kappa = 0.0005 R = 10000000 log10 c_1(beta*) <= 21.7370169078 sigma(kappa/2) <= 9.8304000256e+16 log10 tau <= -1047.00661666 PASS s=1 theta=0 R=10000000: tau <= 1e-1047 log10 l_{M_1} in [ 1704.44139693 , 1704.44139693 ] PASS s=1 theta=0: l_{M_1} <= 10^1705 log10 (N_l A^{1/2}) at l=1e1705 <= -228.24524701 nu <= 0.212917385368 log10 nu <= -0.671788875617 PASS s=1 theta=0: nu_l <= 0.213 for l <= l_{M_1} log10 prefactor A^{(H+1)/2H}/(1-nu) <= -7054.78477726 PASS s=1 theta=0: global bound <= |Lambda| 10^{-0.67 M_1} l_{m0} <= 10138155.1056 log10 near <= -28126.4353728 log10 tail <= -74233.6723389 log10 (near + tail) <= -28126.4353728 PASS s=1 theta=0: P(y bad, U(y) not an island) <= 1e-28126 + global PASS s=1 theta=0: same with the factor 1+1e-6 (beta_W > beta*) --- vartheta = 1/2 kappa = 0.00025 R = 20000000 log10 c_1(beta*) <= 21.735822598 sigma(kappa/2) <= 1.57286400102e+18 log10 tau <= -1045.80369099 PASS s=1 theta=1/2 R=20000000: tau <= 1e-1045 log10 l_{M_1} in [ 1704.74242693 , 1704.74242693 ] PASS s=1 theta=1/2: l_{M_1} <= 10^1705 log10 (N_l A^{1/2}) at l=1e1705 <= -228.24524701 nu <= 0.212917385368 log10 nu <= -0.671788875617 PASS s=1 theta=1/2: nu_l <= 0.213 for l <= l_{M_1} log10 prefactor A^{(H+1)/2H}/(1-nu) <= -7054.78477726 PASS s=1 theta=1/2: global bound <= |Lambda| 10^{-0.67 M_1} l_{m0} <= 20276310.2112 log10 near <= -28124.0271329 log10 tail <= -74233.6723389 log10 (near + tail) <= -28124.0271329 PASS s=1 theta=1/2: P(y bad, U(y) not an island) <= 1e-28124 + global PASS s=1 theta=1/2: same with the factor 1+1e-6 (beta_W > beta*) PASS s=1: (1+1e-7)^8 <= 1+1e-6 nu_l = 1 at log10 l about 1763.24464932 1763.24464932 N_l A_* = 1 at log10 l about 3524.4236535 3524.4236535 nu at log10 l = 1763.2 <= 0.734620029119 ; at 1763.3 >= 1.52337517749 PASS s=1: nu_l < 1 at log10 l = 1763.2 (A = eMHq*) PASS s=1: nu_l > 1 at log10 l = 1763.3 (A = eMHq*) log10 N_l A_* at log10 l = 3524.5 in 0.305385997092 0.305385997092 PASS s=1: N_l A_* >= 1 at log10 l = 3524.5 ================ s = 5 ================ PASS s=5: root bracket beta* in [1.32162904986e+37, 1.32162904986e+37] beta* rho^2/16 >= 38896.3689555 PASS s=5: beta* rho^2/16 >= 38354 log10 Mq* <= -16652.7966484 log10 A_* <= -16649.2497489 --- vartheta = 0 kappa = 0.0005 R = 10000000 log10 c_1(beta*) <= 22.914672447 sigma(kappa/2) <= 9.8304000256e+16 log10 tau <= -1045.82896112 PASS s=5 theta=0 R=10000000: tau <= 1e-1045 log10 l_{M_1} in [ 1704.44139693 , 1704.44139693 ] PASS s=5 theta=0: l_{M_1} <= 10^1705 log10 (N_l A^{1/2}) at l=1e1705 <= -1503.42075445 nu <= 0.0220942572173 log10 nu <= -1.65572059432 PASS s=5 theta=0: nu_l <= 0.0222 for l <= l_{M_1} log10 prefactor A^{(H+1)/2H}/(1-nu) <= -8331.03849309 PASS s=5 theta=0: global bound <= |Lambda| 10^{-1.65 M_1} l_{m0} <= 10138155.1056 log10 near <= -33227.1374026 log10 tail <= -173903.097925 log10 (near + tail) <= -33227.1374026 PASS s=5 theta=0: P(y bad, U(y) not an island) <= 1e-33227 + global PASS s=5 theta=0: same with the factor 1+1e-6 (beta_W > beta*) --- vartheta = 1/2 kappa = 0.00025 R = 20000000 log10 c_1(beta*) <= 22.9126095482 sigma(kappa/2) <= 1.57286400102e+18 log10 tau <= -1044.62690404 PASS s=5 theta=1/2 R=20000000: tau <= 1e-1044 log10 l_{M_1} in [ 1704.74242693 , 1704.74242693 ] PASS s=5 theta=1/2: l_{M_1} <= 10^1705 log10 (N_l A^{1/2}) at l=1e1705 <= -1503.42075445 nu <= 0.0220942572173 log10 nu <= -1.65572059432 PASS s=5 theta=1/2: nu_l <= 0.0222 for l <= l_{M_1} log10 prefactor A^{(H+1)/2H}/(1-nu) <= -8331.03849309 PASS s=5 theta=1/2: global bound <= |Lambda| 10^{-1.65 M_1} l_{m0} <= 20276310.2112 log10 near <= -33224.7291627 log10 tail <= -173903.097925 log10 (near + tail) <= -33224.7291627 PASS s=5 theta=1/2: P(y bad, U(y) not an island) <= 1e-33224 + global PASS s=5 theta=1/2: same with the factor 1+1e-6 (beta_W > beta*) PASS s=5: (1+1e-7)^8 <= 1+1e-6 nu_l = 1 at log10 l about 2082.30807828 2082.30807828 N_l A_* = 1 at log10 l about 4162.01140722 4162.01140722 all checks passed: True (33 checks) ===== d5/composed_threshold.out ===== check block scale (paper): s=1,5 at rho=1e-8 s=1: M=14641 H=1296 log q*=-21090.2 beta*=3.4267e+21 s=5: M=14641 H=1296 log q*=-33424.9 beta*=5.4007e+21 s = 5, rho = 1e-8 (label radii r=4.3e-7, eta=9.2e-7) j | side | M_j | H_j | |log q*_j| | rho_0 | beta*_j | 16^-j beta*_j | k_*(beta*_j) | k_*-j 0 | 11 | 14641 | 1296 | 33424.9 | 1.000e-08 | 5.4007e+21 | 5.4007e+21 | 8 | 8 1 | 30 | 810000 | 4096 | 105605.9 | 2.420e-09 | 2.8958e+23 | 1.8099e+22 | 9 | 8 2 | 64 | 16777216 | 4096 | 105608.9 | 6.049e-10 | 4.6342e+24 | 1.8102e+22 | 10 | 8 3 | 132 | 303595776 | 6561 | 169152.8 | 1.512e-10 | 1.1862e+26 | 2.8960e+22 | 11 | 8 4 | 268 | 5158686976 | 6561 | 169155.7 | 3.781e-11 | 1.8981e+27 | 2.8963e+22 | 12 | 8 5 | 540 | 85030560000 | 6561 | 169158.5 | 9.452e-12 | 3.0373e+28 | 2.8966e+22 | 13 | 8 6 | 1084 | 1380756603136 | 6561 | 169161.3 | 2.363e-12 | 4.8603e+29 | 2.8970e+22 | 14 | 8 7 | 2172 | 22255598797056 | 6561 | 169164.0 | 5.907e-13 | 7.7773e+30 | 2.8973e+22 | 15 | 8 8 | 4348 | 357402957250816 | 6561 | 169166.8 | 1.477e-13 | 1.2445e+32 | 2.8976e+22 | 16 | 8 9 | 8700 | 5728976100000000 | 6561 | 169169.6 | 3.692e-14 | 1.9915e+33 | 2.8980e+22 | 17 | 8 10 | 17404 | 91747935053414656 | 6561 | 169172.4 | 9.230e-15 | 3.1867e+34 | 2.8983e+22 | 18 | 8 11 | 34812 | 1468641849395118336 | 6561 | 169175.1 | 2.308e-15 | 5.0993e+35 | 2.8986e+22 | 19 | 8 12 | 69628 | 23503670094672691456 | 6561 | 169177.9 | 5.769e-16 | 8.1598e+36 | 2.8989e+22 | 20 | 8 s = 1, rho = 1e-8 (label radii r=4.3e-7, eta=9.2e-7) j | side | M_j | H_j | |log q*_j| | rho_0 | beta*_j | 16^-j beta*_j | k_*(beta*_j) | k_*-j 0 | 11 | 14641 | 1296 | 21090.2 | 1.000e-08 | 3.4267e+21 | 3.4267e+21 | 8 | 8 1 | 30 | 810000 | 4096 | 66622.2 | 2.420e-09 | 1.8303e+23 | 1.1440e+22 | 9 | 8 2 | 64 | 16777216 | 4096 | 66625.3 | 6.049e-10 | 2.9294e+24 | 1.1443e+22 | 10 | 8 3 | 132 | 303595776 | 6561 | 106708.5 | 1.512e-10 | 7.4928e+25 | 1.8293e+22 | 11 | 8 4 | 268 | 5158686976 | 6561 | 106711.4 | 3.781e-11 | 1.1991e+27 | 1.8296e+22 | 12 | 8 5 | 540 | 85030560000 | 6561 | 106714.2 | 9.452e-12 | 1.9189e+28 | 1.8300e+22 | 13 | 8 6 | 1084 | 1380756603136 | 6561 | 106716.9 | 2.363e-12 | 3.0707e+29 | 1.8303e+22 | 14 | 8 7 | 2172 | 22255598797056 | 6561 | 106719.7 | 5.907e-13 | 4.9141e+30 | 1.8306e+22 | 15 | 8 8 | 4348 | 357402957250816 | 6561 | 106722.5 | 1.477e-13 | 7.8639e+31 | 1.8310e+22 | 16 | 8 9 | 8700 | 5728976100000000 | 6561 | 106725.3 | 3.692e-14 | 1.2585e+33 | 1.8313e+22 | 17 | 8 10 | 17404 | 91747935053414656 | 6561 | 106728.1 | 9.230e-15 | 2.0139e+34 | 1.8316e+22 | 18 | 8 11 | 34812 | 1468641849395118336 | 6561 | 106730.8 | 2.308e-15 | 3.2228e+35 | 1.8320e+22 | 19 | 8 12 | 69628 | 23503670094672691456 | 6561 | 106733.6 | 5.769e-16 | 5.1574e+36 | 1.8323e+22 | 20 | 8 j_max(beta): largest j with beta >= beta*_j (s=5), and k_*(beta) beta=1e22: j_max= 0 a_jmax/a= 1 k_*= 8 a_k*/a= 256 ratio 2^(k_*-j_max)=256 (beta/2.9e22)^(1/4)=0.766 beta=1e24: j_max= 1 a_jmax/a= 2 k_*=10 a_k*/a= 1024 ratio 2^(k_*-j_max)=512 (beta/2.9e22)^(1/4)=2.42 beta=1e26: j_max= 2 a_jmax/a= 4 k_*=11 a_k*/a= 2048 ratio 2^(k_*-j_max)=512 (beta/2.9e22)^(1/4)=7.66 beta=1e28: j_max= 4 a_jmax/a= 16 k_*=13 a_k*/a= 8192 ratio 2^(k_*-j_max)=512 (beta/2.9e22)^(1/4)=24.2 beta=1e30: j_max= 6 a_jmax/a= 64 k_*=15 a_k*/a= 32768 ratio 2^(k_*-j_max)=512 (beta/2.9e22)^(1/4)=76.6 beta=1e32: j_max= 7 a_jmax/a= 128 k_*=16 a_k*/a= 65536 ratio 2^(k_*-j_max)=512 (beta/2.9e22)^(1/4)=242 beta=1e34: j_max= 9 a_jmax/a= 512 k_*=18 a_k*/a= 262144 ratio 2^(k_*-j_max)=512 (beta/2.9e22)^(1/4)=766 beta=1e36: j_max=11 a_jmax/a= 2048 k_*=19 a_k*/a= 524288 ratio 2^(k_*-j_max)=256 (beta/2.9e22)^(1/4)=2.42e+03 beta=1e38: j_max=12 a_jmax/a= 4096 k_*=21 a_k*/a= 2097152 ratio 2^(k_*-j_max)=512 (beta/2.9e22)^(1/4)=7.66e+03 beta=1e40: j_max=14 a_jmax/a= 16384 k_*=23 a_k*/a= 8388608 ratio 2^(k_*-j_max)=512 (beta/2.9e22)^(1/4)=2.42e+04 beta=1e42: j_max=16 a_jmax/a= 65536 k_*=24 a_k*/a=16777216 ratio 2^(k_*-j_max)=256 (beta/2.9e22)^(1/4)=7.66e+04 ===== d5/constants.out ===== N(2 alpha_G) <= 2.45883e+16 sqrt = 1.56807e+8 c_Q = 4.0 C_G <= 1.472e+13 n >= 17 C_G/alpha: 17C_G = 2.502e+14 c_Q = 8.0 C_G <= 2.944e+13 n >= 17 C_G/alpha: 17C_G = 5.004e+14 alpha = 0.5 n ~ 1.0e+15 |W| ~ 6.27e+58 beta_W = 4 lam >~ 2.29e+98 P(calm exterior) <= 9.86e-61 alpha = 0.9 n ~ 5.56e+14 |W| ~ 5.97e+57 beta_W = 4 lam >~ 2.16e+97 P(calm exterior) <= 3.3e-59 ===== d5/coverage_checks.out ===== radius 3: one-coordinate overlap <= 4; attained 4 four-coordinate M=2401, H=256 radius 5: one-coordinate overlap <= 6; attained 6 four-coordinate M=14641, H=1296 Exhaustive tiny coverage implication: passed ===== d5/domain_probe.out ===== Nonconvex physical-label pullback: floating diagnostic only r=0.1: endpoint path radii 0.0999999332053944, 0.0999999332053944; midpoint 0.100000066666667 max free radius=0.0144250444; max equation residual=0 r=0.03: endpoint path radii 0.0299999981996894, 0.0299999981996894; midpoint 0.0300000017999999 max free radius=0.00432751073; max equation residual=3.47e-18 r=0.01: endpoint path radii 0.00999999993333205, 0.00999999993333205; midpoint 0.0100000000666667 max free radius=0.0014425035; max equation residual=1.73e-18 r=0.003: endpoint path radii 0.00299999999819999, 0.00299999999819999; midpoint 0.0030000000018 max free radius=0.000432751048; max equation residual=0 At tested radii, endpoints are good and their coordinate midpoint is bad. ===== d5/eight_link.out ===== Part A1: near the point g_L*, q in QL = [-0.0182, -0.0146], |t| <= 1e-3 min over QL, t of (0.49 + min(t,0) - q) - (0.5 + q) >= 0.0182 m on QL x T: [0.7046958296, 0.7123629128] bracket sin(0.49 + t/2 - q) >= 0.48302 d/ds of 2 sqrt2 sin((0.49 + s - q)/2) at s = 0 >= 1.36863 t in [-0.001,0]: f(-0.0182) < gamma*, f(-0.0146) > gamma*, f' in [1.3558, 1.3728] t in [0,0.001]: f(-0.0182) < gamma*, f(-0.0146) > gamma*, f' in [1.3558, 1.3754] Part A2: near the point g*, q in [0.05, 0.1], |t| <= 1e-3 far margin (0.5 + q) - (0.49 + |t| - q) >= 0.109 m: [0.7680507221, 0.8358573684] chi_0: [0.3282852631, 0.4638985557] |chi_0''| <= 0.417929 ; |t chi_0''/16| <= 2.612e-05 f(0.05, t) >= 0.171164504 > gamma* for |t| <= 1e-3 f(0.1, t) >= 0.1713815269 > gamma* for |t| <= 1e-3 f_0'(0.0741) in [-9.50001e-05, -9.50001e-05] f_0'(0.0742) in [0.000389333, 0.000389333] Part B: the Kantorovich gap delta(g*, g_L*) >= 0.126812 0.1268 - 2*0.0034 = 0.12 >= 0.12 geodesic length of t -> e^{it}: sqrt2 |t|; sqrt2 = 1.4142136 Part C: constants of the chain law (1 + 6*1.1986)/sqrt2 <= 5.7923359 <= 5.793 6*0.5993/sqrt2 <= 2.5426146 <= 2.543 sqrt2*5.793 <= 8.1925392 sqrt2*64.94 <= 91.839029 sqrt2*2.543 <= 3.5963451 sqrt2*32.47 <= 45.919514 n = 24 reproduces 197 beta_W + 2206 and 87 beta_W + 1103 of thm:site Part D: exact enumeration of the chains on a line of blocks 1533 bad patterns (p = 7, 6, 4): chains disjoint, counts exact, n <= 23 n_b + 1, exterior links of chain constraints as stated, windows of collar s >= 2 disjoint from chains collar s = 1: 1920 (pattern, window) incidences, e.g. the end path link of a chain entering a window block at distance 2 Part E: the island threshold of thm:islandthreshold C0 = 4.90816232526 s=1: Delta=2400, log q* = -21090.194 s=2: Delta=14640, log q* = -25777.2785 s=5: Delta=279840, log q* = -33424.8721 beta*(1e-8, 1) in [3.426745e+21, 3.42674568e+21] beta*(1e-8, 2) in [4.17686847e+21, 4.17686931e+21] beta*(1e-8, 5) in [5.40073003e+21, 5.40073111e+21] beta*(1e-9, 1) in [3.4311672e+23, 3.43116788e+23] beta*(1e-9, 2) in [4.18129045e+23, 4.18129129e+23] beta*(1e-9, 5) in [5.40515177e+23, 5.40515286e+23] all checks passed ===== d5/fold_certificate.out ===== alpha = acos(3/4) = [0.7227342478134156105095, 0.7227342478134156114853] q_lo = 0.49 - alpha = [-0.2327342478134156117733, -0.2327342478134156102555] q_hi = -1/2 + alpha = [0.222734247813415610481, 0.2227342478134156115652] CHECK [q_lo, q_hi] subset of [-0.233, 0.223]: OK CHECK explicit f_0 agrees with the quaternion V on the torus: OK CHECK f_0'' > 0 on [-0.005, q_hi]: OK CHECK f_0'(0.0741) < 0 < f_0'(0.0742): OK, f_0'(0.0741) in [-0.00009500013762012930046419, -0.00009500013762011629003812] f_0'(0.0742) in [0.0003893334761051964534873, 0.0003893334761052103312751] gamma* = f_0(q*) in [0.1697826184473846117262, 0.1697826279473983767439] (by convexity: f_0(q) >= f_0(a) + f_0'(a)(q-a)) CHECK f_0'(-0.005) in [-0.3857865695075310824569, -0.3857865695075310711812] < 0; f_0'(q_hi) in [0.7086828497752134118151, 0.7086828497752134308971] > 0; f_0(-0.005) in [0.1850115389862948466967, 0.1850115389862948492988] > gamma*: OK CHECK chi(-0.005) in [0.6141787118748924349798, 0.6141787118748924419187] <= 0.6146, so f_0' > 0.46 on [q_lo,-0.005]: OK CHECK f_0(-0.0172) in [0.1686688411307782165059, 0.1686688411307782201921] < gamma*-1e-3 and f_0(-0.0156) in [0.1708518587336376431517, 0.1708518587336376481391] > gamma*+1e-3: OK so q_L(lambda) in [-0.0172, -0.0156] for every lambda in Lambda, and q* - q_L >= 0.0741+0.0156 = 0.0897 chordal distance delta(g*, g_L(lambda)) >= 2 sqrt2 sin(0.0897/2) >= 0.12681243222213579857626797 f_0'(q) on [-0.0172,-0.0156] in [1.360462103250522707468, 1.368310203825014581894] CHECK f_0'' > 4.6 on [0.05, 0.1]: OK (so |q_+- - q*| <= sqrt(2e-3/4.6) < 0.021 for lambda in Lambda) CHECK strip q in [-0.233,0.223], s in [0,0.02]: dV_k/ds > 0 on every box (3 boxes): OK on q in [-0.0172,-0.0156] (the atom g_L): dV_k/ds >= kappa = 0.51492711141699771572322053 on q in [0.05,0.1] (the atoms g_+-): dV_k/ds <= 0.73251359340960786083185585 , |dV_j/ds| <= 0.0047761798949506994570873841 CHECK region q in [-0.233,0.223], s in [0.02, alpha]: V_k > 0 on every box (97 boxes examined): OK Hence V^{-1}(T) = T: every path link g with V(g) in the torus lies in the torus. weight of the atom g_L: w_L <= 1/(f_0'(q_L) kappa^2) <= 2.7721838838451649293159426 (times e^{-beta S} <= 1) transverse factor at any torus point of the strip: |lambda_perp| = |dV/ds(q,0)| <= sqrt(LMAX^2+DJMAX^2) <= 0.73252916421412549256155181 done in 1.4 s ===== d5/fold_selftest.out ===== ok 0.0 0.0 [6.938893903907228e-18, 6.5052130349130266e-18, 0.0, 0.0] 2.3418766925686896e-17 ok 0.05 0.3 [1.6479873021779667e-17, 1.3877787807814457e-17, 8.239936510889834e-18, 1.3444106938820255e-17] 7.45931094670027e-17 ok -0.1 0.1 [1.6479873021779667e-17, 1.9949319973733282e-17, 3.686287386450715e-18, 4.87890977618477e-18] 5.204170427930421e-17 ok 0.2 0.6 [2.6020852139652106e-18, 6.505213034913027e-19, 0.0, 8.673617379884035e-19] 8.673617379884035e-19 ok -0.2 0.02 [1.3010426069826053e-17, 1.431146867680866e-17, 4.87890977618477e-19, 6.2341624917916505e-19] 3.122502256758253e-17 box ok; V_k box: [-0.01025808894511182218221, 0.01973136081901316445097] dV_k/ds box: [0.4652267954101105993867, 0.5210684409679629839826] ===== d5/goodside_constants.out ===== 16 r = 0.5 (<= 1/2: cut-off equals one at every branch) theta = 16r/sqrt2 = 0.353553390593273762200422181052 theta/sin theta = 1.02114120912651222454380753866 contraction constant (7/8) theta/sin theta = 0.89349855798569819647583159633 < 0.8936: True singular values of the path-link derivative in [1-k, 1+k] = [ 0.10650144201430180352416840367 , 1.89349855798569819647583159633 ] |det| in [ 0.00120799869272451686788868881937 , 6.78883004004240797722561311874 ] response to one non-path private link: (2/16) L /(1-k) = 1.19850631809917865885841255694 response to one inside link u_a or w_a: (1/16) L /(1-k) = 0.599253159049589329429206278472 pi/3 = 1.04719755119659774615421446109 (conversion d <= (pi/3) delta for delta <= 1) ===== d5/grid_check.out ===== beta in [1e22,1e42], 2001 log-spaced points: k_* - j_max ranges over 8 .. 9 16^-j beta*_j max over j<=12: 2.898946033871121e+22 ===== d5/groups_threshold.out ===== == (A) Haar small-ball constant C0(G) and the hypothesis beta_W >= K N == group d N kappa K N I_d(pi) C0(G) SU(2) 3 2 1 1 1/2*pi [4.3200138, 4.3200138] SU(3) 8 3 3/2 9/14 32/35 [10.44721, 10.44721] SU(4) 15 4 2 4/7 429/2048*pi [18.785167, 18.785167] SU(5) 24 5 5/2 25/46 1048576/2028117 [29.398314, 29.398314] SU(10) 99 10 5 25/49 3184701532372607112841303575/39614081257132168796771975168*pi [117.03274, 117.03274] SO(3) 3 3 1/4 3/8 1/2*pi [5.300843, 5.300843] SO(5) 10 5 3/4 5/12 256/315 [14.119813, 14.119813] SO(10) 45 10 2 5/11 263012370465/2199023255552*pi [56.108066, 56.108066] Sp(2) 10 4 3/2 2/3 256/315 [12.553134, 12.553134] Sp(3) 21 6 2 3/5 46189/262144*pi [25.258918, 25.258918] G2 14 7 1 7/13 2048/3003 [17.922554, 17.922554] F4 52 26 3/4 13/34 281474976710656/805867616040669 [67.511601, 67.511601] E6 78 27 1 27/77 18889465931478580854784/66341473743672640538025 [102.43901, 102.43901] E7 133 56 3/4 7/22 94347416541385238311148088072973180425/1361129467683753853853498429727072845824*pi [177.3511, 177.3511] E8 248 248 1/4 62/247 3533694129556768659166595001485837031654967793751237916243212402585239552/22178239322710820309383779481693762111479414123274215650411006904951614625 [348.14238, 348.14238] every K N <= 1, so Lemma (Wilson Z) applies at every beta_W >= 1; every C0(G) > 0 paper's SU(2) constant 1/2+(2/3)log(24 pi^3) = [4.9081623, 4.9081623]; C0(SU(2)) of the general lemma = [4.3200138, 4.3200138] == (B) log q* (group-free) == s=1: Delta=2400, |log q*|=[21090.194, 21090.194]; N_s=194480, |log q*|=[32481.6727, 32481.6727] s=5: Delta=279840, |log q*|=[33424.8721, 33424.8721]; N_s=1874160, |log q*|=[38354.0729, 38354.0729] == (C) consistency: SU(2) with the paper's C0 reproduces Theorem (island threshold) == rho=1.0e-8, s=1: beta* in 3.426745e+21 .. 3.426745e+21 rho=1.0e-8, s=5: beta* in 5.400731e+21 .. 5.400731e+21 rho=1.0e-9, s=1: beta* in 3.431168e+23 .. 3.431168e+23 rho=1.0e-9, s=5: beta* in 5.405152e+23 .. 5.405152e+23 rho=1e-8, s=5, Gamma=1e30: beta* in 1.97259e+22 .. 1.97259e+22 == (D) SU(3): f(b) = 16 log b + 6 C0 - b rho^2/24 == rho=1.0e-8, s=1: beta* in 5.268752e+21 .. 5.268752e+21; eps in [0.04092, 0.04092] rho=1.0e-8, s=5: beta* in 8.230788e+21 .. 8.230788e+21; eps in [0.02603, 0.02603] rho=1.0e-9, s=1: beta* in 5.286449e+23 .. 5.286449e+23; eps in [0.04441, 0.04441] rho=1.0e-9, s=5: beta* in 8.24848e+23 .. 8.24848e+23; eps in [0.02824, 0.02824] rho=1e-8, s=5, Gamma=1e30: beta* in 2.97216e+22 .. 2.97216e+22 ratio beta*_SU(3)/beta*_SU(2,paper C0) at rho=1e-8, s=1: 1.53754 .. 1.53754 == (E) SU(3): island threshold at scale j (s=5, rho=1e-8) == j=1: beta*_j in 4.3675e+23 .. 4.3675e+23 j=2: beta*_j in 6.9912e+24 .. 6.9912e+24 j=3: beta*_j in 1.786e+26 .. 1.786e+26 max_j<=12 16^-j beta*_j <= 4.3712e+22 == (F) SU(3): two islands, layer bound and the window == c3 = 1.5e4 sqrt3 in [25980.762, 25980.762] c6 = 6.81e5*8*c3^2 in [3.6774e+15, 3.6774e+15] hypothesis (a): rho <= 1/(2758*195*4e5*c3) in [1.7892e-16, 1.7892e-16] hypothesis (b): beta_W >= 480 N c6 = [5.29546e+18, 5.29546e+18] rho=1.78e-16, s=1, joining distance 2s+8: beta* in 2.569568e+37 .. 2.569568e+37 rho=1.78e-16, s=5, joining distance 2s+8: beta* in 3.014585e+37 .. 3.014585e+37 so hypothesis (b) holds at every beta_W >= beta* Lambda2bar(beta*) <= 1.69592e+39 c2 (s=1, theta=0) <= 10^22.134 check SU(2): beta*=1.12206e+37 .. 1.12206e+37, c2 <= 10^22.122 (paper: 10^22.13) == (G) the threshold in the other groups (rho = 1e-8 rho_G, s = 1,5) == group rho_G s=1: beta* eps s=5: beta* SU(2) 1 3.4262e+21 eps<0.0153 5.4002e+21 SU(3) 1 5.2688e+21 eps<0.0409 8.2308e+21 SU(4) 1 7.2682e+21 eps<0.0769 1.1219e+22 SU(5) 1 9.4782e+21 eps<0.124 1.442e+22 SU(10) 1 2.5607e+22 eps<0.518 3.5527e+22 SO(3) 1 5.1413e+21 eps<0.0157 8.1022e+21 SO(5) 1 8.8743e+21 eps<0.0519 1.3812e+22 SO(10) 1 2.0842e+22 eps<0.235 3.0737e+22 Sp(2) 1 7.095e+21 eps<0.0513 1.1045e+22 Sp(3) 1 1.122e+22 eps<0.108 1.7149e+22 G2 0.99 1.2926e+22 eps<0.0727 1.9981e+22 F4 0.99 5.7183e+22 eps<0.278 8.3444e+22 E6 0.99 6.5898e+22 eps<0.418 9.3202e+22 E7 0.76 2.8325e+23 eps<0.732 3.7952e+23 E8 0.99 1.0243e+24 eps<1.4 1.2762e+24 == (H) the general formula == beta*_G = 8 N |log q*| rho^-2 (1+eps), eps = (2 d log beta*_G + 6 C0(G))/|log q*|, an identity at the root; table (G) gives the values of eps. == (I) group constants of the layer bound, the mixed derivatives and the site == c3 = max(1.5e4 sqrt N, d), c6 = 6.81e5 d c3^2; (b): beta_W >= 480 N c6; (a) at r=43 rho, eta=92 rho: rho <= 1/(2758*7.8e7*c3); Lambda2 <= (2/N) 99.0001 beta_W + (d/3) 4.65e6; site: C <= 197 beta_W + a_G (non-path), 87 beta_W + b_G (inside), a_G = 24 sqrt N * 64.94 (d/3) sqrt(kappa) + 1.1986, b_G = 24 sqrt N * 32.47 (d/3) sqrt(kappa) + 0.5993; delta_G = kappa/3 group c3 (b) beta_W >= (a) rho <= cW coeff cpsi a_G b_G SU(2) 21213.2 8.826e+17 2.191e-16 99.0001 4.65e+6 2205.34 1102.67 SU(3) 25980.8 5.295e+18 1.789e-16 66.00007 1.24e+7 8817.75 4408.87 SU(4) 30000.0 1.765e+19 1.549e-16 49.50005 2.325e+7 22042.6 11021.3 SU(5) 33541.0 4.413e+19 1.386e-16 39.60004 3.72e+7 44083.9 22042.0 SU(10) 47434.2 7.281e+20 9.8e-17 19.80002 1.534e+8 363684.0 181842.0 SO(3) 25980.8 1.986e+18 1.789e-16 66.00007 4.65e+6 1350.95 675.476 SO(5) 33541.0 1.839e+19 1.386e-16 39.60004 1.55e+7 10061.7 5030.83 SO(10) 47434.2 3.31e+20 9.8e-17 19.80002 6.975e+7 104553.0 52276.3 Sp(2) 30000.0 1.177e+19 1.549e-16 49.50005 1.55e+7 12726.8 6363.39 Sp(3) 36742.3 5.56e+19 1.265e-16 33.00003 3.255e+7 37794.3 18897.1 G2 39686.3 5.045e+19 1.171e-16 28.28574 2.17e+7 19244.5 9622.24 F4 76485.3 2.585e+21 6.078e-17 7.615392 8.06e+7 119296.0 59648.2 E6 77942.3 4.182e+21 5.964e-17 7.333341 1.209e+8 210563.0 105281.0 E7 112250.0 3.068e+22 4.141e-17 3.535718 2.062e+8 447795.0 223898.0 E8 236220.0 1.122e+24 1.968e-17 0.7983879 3.844e+8 1.0145e+6 507248.0 check SU(2): a_G < 2206 - 0 and b_G < 1103 (the constants of Theorem (site)) a_SU(2) <= 2205.335, b_SU(2) <= 1102.668 Haar part of the layer bound: kappa_G/3 * 1e-3 * (1+1e-5) <= 0.84 needs kappa_G <= 2519; holds for every group of the table, and for SU(N) with N <= 5038. ===== d5/island_radius.out ===== paper check beta*(1e-8,1) in [3.42675e+21,3.42675e+21] paper check beta*(1e-8,5) in [5.40073e+21,5.40073e+21] |log q*| s=1: [21090.1940153168026952583708448008918305211144, 21090.1940153168026952583708448008918305211144] s=5: [33424.8721333080907686042244155275463547814527, 33424.8721333080907686042244155275463547814527] === paper: rho=1e-08, s=5 one scale beta*(1e-08,5) in [5.40073e+21, 5.40073e+21] j=1: beta*_j in [2.89584e+23, 2.89584e+23], 16^-j beta*_j <= 1.8099e+22 j=2: beta*_j in [4.6342e+24, 4.6342e+24], 16^-j beta*_j <= 1.81023e+22 j=3: beta*_j in [1.18619e+26, 1.18619e+26], 16^-j beta*_j <= 2.89596e+22 j=4: beta*_j in [1.89811e+27, 1.89811e+27], 16^-j beta*_j <= 2.89629e+22 j=20: beta*_j in [3.50782e+46, 3.50782e+46], 16^-j beta*_j <= 2.9016e+22 max_j<= 20 of 16^-j beta*_j <= 2.9016e+22 k*-j_max over 2001 couplings 10^22..10^42 (skipping beta 0 > g(beta_hi), beta_lo > 96/rho^2 beta* in [ 1.1220620202447e+37 , 1.1220620202447e+37 ] PASS s=1: beta* > 112/rho^2 (monotonicity of Lambda^2 e^f / beta) log10(M q*) <= -14102.445633469 log10 A* <= -14098.898733986 x = N^2 A*^{1/H}: log x in -0.69314718055995 -0.69314718055995 (log 1/2 = -0.69314718056 ) PASS s=1: x <= 1/2 at beta* log10 G_0 <= -14099.333028467 --- vartheta = 0 kappa = 0.0005 sigma(kappa) in [ 6.144000064e+15 , 6.144000064e+15 ] sigma(kappa/2) <= 9.8304000256e+16 prop:islandmass(ii) at Gamma=1 per |J|: 324886.23820828 324886.23820828 new mean bound G_0 sigma(kappa): log10 <= -14083.544577256 log10 c_2(beta*) <= 22.121873696523 PASS s=1 theta=0: log10 c_2 <= 22.13 PASS s=1 theta=0: log10 G_0 sigma(kappa) <= -14082.3 R = 15: log10 near/|W| <= -28194.760249422 ; log10 far/|W| <= -28163.868337028 - log10(sigma) sigma = 1e-100: log10 eps/|W| <= -28063.868337028 sigma = 1e-1000: log10 eps/|W| <= -27163.868337028 R = 200000000: log10 near/|W| <= -28166.317456228 ; log10 far/|W| <= -49877.388312212 - log10(sigma) sigma = 1e-100: log10 eps/|W| <= -28166.317456228 PASS s=1 theta=0 R=200000000 sigma=1e-100: far <= near sigma = 1e-1000: log10 eps/|W| <= -28166.317456228 PASS s=1 theta=0 R=200000000 sigma=1e-1000: far <= near --- vartheta = 1/2 kappa = 0.00025 sigma(kappa) in [ 9.8304000256e+16 , 9.8304000256e+16 ] sigma(kappa/2) <= 1.572864001024e+18 prop:islandmass(ii) at Gamma=1 per |J|: 5198179.7707217 5198179.7707217 new mean bound G_0 sigma(kappa): log10 <= -14082.340457277 log10 c_2(beta*) <= 22.120136518595 PASS s=1 theta=1/2: log10 c_2 <= 22.13 PASS s=1 theta=1/2: log10 G_0 sigma(kappa) <= -14082.3 R = 15: log10 near/|W| <= -28194.760249422 ; log10 far/|W| <= -28162.665954227 - log10(sigma) sigma = 1e-100: log10 eps/|W| <= -28062.665954227 sigma = 1e-1000: log10 eps/|W| <= -27162.665954227 R = 400000000: log10 near/|W| <= -28165.113336247 ; log10 far/|W| <= -49876.185929408 - log10(sigma) sigma = 1e-100: log10 eps/|W| <= -28165.113336247 PASS s=1 theta=1/2 R=400000000 sigma=1e-100: far <= near sigma = 1e-1000: log10 eps/|W| <= -28165.113336247 PASS s=1 theta=1/2 R=400000000 sigma=1e-1000: far <= near PASS s=1: log10(M q*) <= -14102.4 PASS s=1: log10 A* <= -14098.8 PASS s=1: log10 G_0 <= -14099.3 ================ s = 5 ================ N_s = 1874160 |log q*| in [ 38354.07290025 , 38354.07290025 ] PASS s=5: g(beta_lo) > 0 > g(beta_hi), beta_lo > 96/rho^2 beta* in [ 1.3216290498578e+37 , 1.3216290498578e+37 ] PASS s=5: beta* > 112/rho^2 (monotonicity of Lambda^2 e^f / beta) log10(M q*) <= -16652.796648353 log10 A* <= -16649.24974887 x = N^2 A*^{1/H}: log x in -0.69314718055995 -0.69314718055995 (log 1/2 = -0.69314718056 ) PASS s=5: x <= 1/2 at beta* log10 G_0 <= -16649.684043351 --- vartheta = 0 kappa = 0.0005 sigma(kappa) in [ 6.144000064e+15 , 6.144000064e+15 ] sigma(kappa/2) <= 9.8304000256e+16 prop:islandmass(ii) at Gamma=1 per |J|: 3498.3872225533 3498.3872225533 new mean bound G_0 sigma(kappa): log10 <= -16633.89559214 log10 c_2(beta*) <= 23.139705702098 PASS s=5 theta=0: log10 c_2 <= 23.14 PASS s=5 theta=0: log10 G_0 sigma(kappa) <= -16632.6 R = 23: log10 near/|W| <= -33294.739334534 ; log10 far/|W| <= -33263.552534791 - log10(sigma) sigma = 1e-100: log10 eps/|W| <= -33163.552534791 sigma = 1e-1000: log10 eps/|W| <= -32263.552534791 R = 200000000: log10 near/|W| <= -33267.019485996 ; log10 far/|W| <= -54977.072509974 - log10(sigma) sigma = 1e-100: log10 eps/|W| <= -33267.019485996 PASS s=5 theta=0 R=200000000 sigma=1e-100: far <= near sigma = 1e-1000: log10 eps/|W| <= -33267.019485996 PASS s=5 theta=0 R=200000000 sigma=1e-1000: far <= near --- vartheta = 1/2 kappa = 0.00025 sigma(kappa) in [ 9.8304000256e+16 , 9.8304000256e+16 ] sigma(kappa/2) <= 1.572864001024e+18 prop:islandmass(ii) at Gamma=1 per |J|: 55974.195123555 55974.195123555 new mean bound G_0 sigma(kappa): log10 <= -16632.691472161 log10 c_2(beta*) <= 23.137099935207 PASS s=5 theta=1/2: log10 c_2 <= 23.14 PASS s=5 theta=1/2: log10 G_0 sigma(kappa) <= -16632.6 R = 23: log10 near/|W| <= -33294.739334534 ; log10 far/|W| <= -33262.351020578 - log10(sigma) sigma = 1e-100: log10 eps/|W| <= -33162.351020578 sigma = 1e-1000: log10 eps/|W| <= -32262.351020578 R = 400000000: log10 near/|W| <= -33265.815366015 ; log10 far/|W| <= -54975.870995759 - log10(sigma) sigma = 1e-100: log10 eps/|W| <= -33265.815366015 PASS s=5 theta=1/2 R=400000000 sigma=1e-100: far <= near sigma = 1e-1000: log10 eps/|W| <= -33265.815366015 PASS s=5 theta=1/2 R=400000000 sigma=1e-1000: far <= near PASS s=5: log10(M q*) <= -16652.79 PASS s=5: log10 G_0 <= -16649.6 ================ statements of the theorem ================ s=1: beta* in [1.1220620202447e+37, 1.1220620202447e+37] s=5: beta* in [1.3216290498578e+37, 1.3216290498578e+37] PASS s=1: beta* in [1.1220e37, 1.1221e37] PASS s=5: beta* in [1.3216e37, 1.3217e37] PASS s=1, theta=0, R=15: near <= 1e-28194 |W| PASS s=1, theta=0, R=15: far <= 1e-28163.8 |W| / sigma PASS s=1, theta=0, R=2e8, sigma>=1e-1000: eps <= 1e-28166 |W| PASS s=1, theta=1/2, R=4e8, sigma>=1e-1000: eps <= 1e-28165 |W| PASS s=5, theta=0, R=2e8, sigma>=1e-1000: eps <= 1e-33267 |W| PASS s=1: G_0 sigma(2 alpha_G) <= 1e-14083.5 PASS s=5: G_0 sigma(2 alpha_G) <= 1e-16633.8 PASS |W| <= 1e28000 gives eps <= 1e-166 (s=1, theta=0) rho=2.16e-16, s=1: g(1.124e+37) = 236.95276661755, g(1.136e+37) = -92.495232882432 rho=2.16e-16, s=5: g(1.323e+37) = 307.49076324862, g(1.337e+37) = -80.277236251374 sigma(5e-4) (paper: 6.144e15): 6.144000064e+15 6.144000064e+15 all checks passed: True (39 checks) ===== d5/islands.out ===== alpha=0.9: n0=5.67e+14 m=6.44e+57 beta for small off-chart error >= 3.79e+97 rho_c=1.62e-79 beta for calm-window islands >= 8.28e+163 alpha=0.5: n0=1.02e+15 m=6.77e+58 beta for small off-chart error >= 4.02e+98 rho_c=1.54e-80 beta for calm-window islands >= 9.22e+165 ===== d5/lambda2_constants.out ===== Part 1: combinatorics (enumeration over all link types) non-path active links per block: 45 (private 28 , inside 17 ) each pair: 21 arguments, 7 private non-path, 14 inside (tree ones included) private: one pair, two paths; inside: two pairs (parallel path links), one path each largest number of path links on one plaquette: 2 each path link: six plaquettes, three of them with a second path link, at most two non-path links on each largest number of path links on the six plaquettes of a non-path link: 3 exact value of the Wilson row sum at gamma_P = 1 (not used in the proof): 83.0 Part 2: constants, general lemma: r <= 8e-5, kappa <= 1e-8 r 8.0e-5 kappa 1.0e-8 m_1 1.00000013653 e_Z 0.000905233336841 e_X 0.000792064128259 e_A 0.00022627416998 eps_T 0.00192357163508 m = 8/(1-7eps_T) 8.10919025855 gamma_P 1.01364892072 M_G 1.04617527008 B_2 16.8248372246 B_3 227.718261281 X_2 165.764162683 C_2 166.792798245 M_1 17.0257729853 M_2 2717.00444476 L 123284.143151 9+27g+63g^2 101.100021331 kappa term 0.000200151357894 c_W 101.100221482 c_psi 4931365.72603 Part 2: constants, hypothesis (a) of thm:layerbound: r <= 6.0e-13/62, kappa <= 6.0e-13 r 9.67741935484e-15 kappa 6.0e-13 m_1 1.0 e_Z 1.09487501603e-13 e_X 9.58015639027e-14 e_A 2.73718754008e-14 eps_T 2.32660940907e-13 m = 8/(1-7eps_T) 8.00000000001 gamma_P 1.0 M_G 1.04617527008 B_2 16.7388043213 B_3 225.973858337 X_2 159.018641053 C_2 160.018641053 M_1 16.7388043213 M_2 2597.40313329 L 116133.68866 9+27g+63g^2 99.0000000002 kappa term 1.15213421558e-8 c_W 99.0000000118 c_psi 4645347.54638 Part 3: Delta of cor:twoislands, at the threshold of cor:twoislands, s = 1 beta_W = 1.13e+37 s = 1 rho = 2.17e-16 r = 43 rho = 9.331e-15 eta+30r = 2.99894e-13 single-block islands: N_{s+1},N_{s+2},N_{s+3} = 544 1776 4160 n_x = n_y = 175120 Lambda_2 <= 1.1187e+39 beta_W r^2 = 9.8386344e+8 Delta <= 3.24222e+21 * exp(-2 alpha_G (w-2s-8)) ln(prefactor) = 49.530545 Delta <= 1 for w >= 99071.089 i.e. w >= 99072 each factor e in beta_W/beta* adds 2000.0 layers; islands of a and b blocks add 1000 ln(ab) layers at most Part 3: Delta of cor:twoislands, at the threshold of cor:twoislands, s = 5 beta_W = 1.33e+37 s = 5 rho = 2.17e-16 r = 43 rho = 9.331e-15 eta+30r = 2.99894e-13 single-block islands: N_{s+1},N_{s+2},N_{s+3} = 13920 22064 32896 n_x = n_y = 2178512 Lambda_2 <= 1.3167e+39 beta_W r^2 = 1.1579986e+9 Delta <= 4.74722e+22 * exp(-2 alpha_G (w-2s-8)) ln(prefactor) = 52.214432 Delta <= 1 for w >= 104446.86 i.e. w >= 104447 each factor e in beta_W/beta* adds 2000.0 layers; islands of a and b blocks add 1000 ln(ab) layers at most ===== d5/largefield_constants.out ===== === A. basic constants === alpha in [0.722734247813, 0.722734247813] Lambda in [1.0926714764, 1.0926714764] 1-7Lambda/8 in [0.0439124581482, 0.0439124581482] M in [23620.2851326, 23620.2851326] log M in [10.0698611613, 10.0698611613] 8 log M in [80.5588892907, 80.5588892907] kappa_c in [0.0837613327502, 0.0837613327502] C_E = 8/kappa_c^2 in [1140.25721565, 1140.25721565] 16.003^2 = [256.0960090000000000000000000000000000000000000000000000000000000000000000000000006225, 256.0960090000000000000000000000000000000000000000000000000000000000000000000000006225] < C_E: OK === B. lattice geometry (exact enumeration) === links based in a block: {'tree': 15, 'inside': 17, 'private': 32} owned plaquettes per block: 96 owned plaquettes containing an active link based in the block: 96 owned plaquettes whose links are based in {y, y+e_mu, y+e_nu}: 96 owned plaquettes inside the block: 24 blocks in a witness region: 144 N_ov = max_p #{P : p in N(P)} = 1441 at ((0, 0, 1, 1), 2, 3) C_* = C_E N_ov in [1643110.64775, 1643110.64775] coarse plaquettes whose witness region meets Pi(y): 2310 === C. c_1(lambda, eta) and a(lambda, kappa) === 12 log 1.957 + 45 log 14 in [126.814532094, 126.814532094] rho at lambda=100, eta=1 in [0.000658807845868, 0.000658807845868] c_1(1e3,1) in [1338.62464367, 1338.62464367] c_1(1e6,1) in [1804.89812501, 1804.89812501] c_1(1e12,1) in [2737.44508767, 2737.44508767] c_1(1e20,1) in [3980.84103789, 3980.84103789] 14*90.6^3*384^3 (thm LF2(ii)) in [5.89530531361e+14, 5.89530531361e+14] === D. Kotecky-Preiss rate === u in [0.000136388926482, 0.000136388926482] 81u/(1-6400u) in [0.0869123388131, 0.0869123388131] optimal b = 0.1063497129541231 least a = 8.982843740441535 === E. thresholds lambda*(kappa): a(lambda,kappa) = 9 === kappa=0.08: lambda* in [282683.7, 282683.71], beta*=4lambda* in [1130735.0, 1130735.0], lambda* kappa^2 = 1809.18 kappa=1e-2: lambda* in [20999610.0, 20999610.0], beta*=4lambda* in [8.399844e+7, 8.399844e+7], lambda* kappa^2 = 2099.96 kappa=1e-4: lambda* in [2.739607e+11, 2.739607e+11], beta*=4lambda* in [1.095843e+12, 1.095843e+12], lambda* kappa^2 = 2739.61 kappa=1e-8: lambda* in [4.0086969e+19, 4.0086969e+19], beta*=4lambda* in [1.603479e+20, 1.603479e+20], lambda* kappa^2 = 4008.7 === F. inherited count at scale j === closed-form counts agree with enumeration for j<=10 j=1: m_j=15^4, H'_j=8^4, a_0 needed in [36884.15,36884.15], beta^pw_j in [2.732095e+14, 2.732095e+14], 16^-j beta^pw_j <= 1.7076e+13 j=2: m_j=32^4, H'_j=8^4, a_0 needed in [36887.1807,36887.1807], beta^pw_j in [4.392177e+15, 4.392177e+15], 16^-j beta^pw_j <= 1.7157e+13 j=3: m_j=66^4, H'_j=9^4, a_0 needed in [59075.5475,59075.5475], beta^pw_j in [1.094641e+17, 1.094641e+17], 16^-j beta^pw_j <= 2.6725e+13 j=4: m_j=134^4, H'_j=9^4, a_0 needed in [59078.3803,59078.3803], beta^pw_j in [1.756748e+18, 1.756748e+18], 16^-j beta^pw_j <= 2.6806e+13 j=5: m_j=270^4, H'_j=9^4, a_0 needed in [59081.1826,59081.1826], beta^pw_j in [2.819312e+19, 2.819312e+19], 16^-j beta^pw_j <= 2.6887e+13 j=6: m_j=542^4, H'_j=9^4, a_0 needed in [59083.97,59083.97], beta^pw_j in [4.524521e+20, 4.524521e+20], 16^-j beta^pw_j <= 2.6968e+13 j=10: m_j=8702^4, H'_j=9^4, a_0 needed in [59095.0741,59095.0741], beta^pw_j in [3.000898e+25, 3.000898e+25], 16^-j beta^pw_j <= 2.7293e+13 j=15: m_j=278526^4, H'_j=9^4, a_0 needed in [59108.938,59108.938], beta^pw_j in [3.19347e+31, 3.19347e+31], 16^-j beta^pw_j <= 2.7699e+13 j=20: m_j=8912894^4, H'_j=9^4, a_0 needed in [59122.8009,59122.8009], beta^pw_j in [3.39767e+37, 3.39767e+37], 16^-j beta^pw_j <= 2.8105e+13 j=25: m_j=285212670^4, H'_j=9^4, a_0 needed in [59136.6639,59136.6639], beta^pw_j in [3.614172e+43, 3.614172e+43], 16^-j beta^pw_j <= 2.8511e+13 j=30: m_j=9126805502^4, H'_j=9^4, a_0 needed in [59150.5268,59150.5268], beta^pw_j in [3.843689e+49, 3.843689e+49], 16^-j beta^pw_j <= 2.8917e+13 max_j 16^-j beta^pw_j (1<=j<=30): 2.89167e+13 min_j 16^-j beta^pw_j (1<=j<=30): 1.70756e+13 beta^pw_j increasing in j: OK j=1: beta*_j(islandswide)/beta^pw_j = 10.588 to 10.588 j=2: beta*_j(islandswide)/beta^pw_j = 10.54 to 10.54 j=3: beta*_j(islandswide)/beta^pw_j = 10.829 to 10.829 k_* - j^pw_max over 2001 couplings in [1e22,1e42]: {1: 1435, 2: 566} real 0m9.435s user 0m8.786s sys 0m0.533s ===== d5/layer_constants.out ===== === Part 1: the chain of prop:explicitdecay at theta = 0.197 (paper: 0.862, 17.37, 14.96, 5.80, 7.41, 1966) b <= 0.86117 g0 <= 17.368 h0 <= 17.368 g1 <= 17.369 y <= 14.957 sbar <= 5.7988 pbar <= 5.7989 g2 <= 7.4046 k1 <= 1965.6 x1 <= 3391.2 === Part 2: constants of the zeta-coordinate Hessian (thm:layerbound, Step 4) ||S||_1->1 <= 3.00255, ||Pbar||_1->1 = q1 <= 3.00257, ||Dg||_1->1 <= 1.94e-05, j1 <= 1.0000194 theta = 0: ||Dg|| <= 3.57e-05, j0 = ||J|| <= 1.0000357, y0 <= 11.317, pbar0 <= 4.796, x1(0) <= 1564.3 n0 <= 296.5 (x c_3), 12 q1 (c3 kappa') <= 2.775e-06 (x lambda / c_3) middle term: 0.004341 lambda + 463800.0 c_3^2 third term : W3a..W3e = 406600.0, 169600.0, 7491.0, 509500.0, 1530000.0 sum = 2623000.0 c_3^2 lambda-independent total (besides c_6): 3087000.0 c_3^2 = 1.511 c_6 C_lambda = 60: m'/lambda >= -0.01285 (1/60 = 0.01667) -- C_lambda = 120: m'/lambda >= +0.00807 (1/60 = 0.01667) -- C_lambda = 180: m'/lambda >= +0.01504 (1/60 = 0.01667) -- C_lambda = 200: m'/lambda >= +0.01644 (1/60 = 0.01667) -- C_lambda = 220: m'/lambda >= +0.01758 (1/60 = 0.01667) OK C_lambda = 240: m'/lambda >= +0.01853 (1/60 = 0.01667) OK C_lambda = 300: m'/lambda >= +0.02062 (1/60 = 0.01667) OK C_lambda = 400: m'/lambda >= +0.02271 (1/60 = 0.01667) OK least integer C_lambda with m' >= lambda/60: 204 theta = 0.197: ||Dg||_(th) <= 6.94e-05, j_th <= 1.0000694 middle term: 0.00941 lambda + 1228000.0 c_3^2 third term : W3a..W3e = 2742000.0, 945500.0, 25960.0, 1766000.0, 5300000.0 sum = 10780000.0 c_3^2 with lambda >= 240 c_6: K'/lambda <= 2068.4 admissible alpha <= theta'(lambda/(240 K'))^(1/2) = 0.0002797 (need >= 2.5e-4: OK) with lambda >= 60 c_6 instead: K'/lambda <= 2068.4 === Part 2b: chart inclusion (thm:layerbound, Step 2) and the size of h c_3 r <= 4.04e-10; max_k||z_k|| <= 8.17e-05 r; ||eta||_inf <= 31.5001 r < 32 r; ||h||_inf <= 7.19e-05 r === Part 3: island threshold arithmetic (cor:twoislands) check paper: rho=1e-08 s=1: |log q*|=21090.2 beta*=3.4267e+21 check paper: rho=1e-08 s=5: |log q*|=33424.9 beta*=5.4007e+21 check paper: rho=1e-09 s=1: |log q*|=21090.2 beta*=3.4312e+23 kappa_0 + s <= 2758 rho; SU(2) chart hypothesis kappa_0 + s <= 6.0e-13 => rho <= 2.17549e-16 rho=2.170e-16 s=1 join=10: |log q*|=32481.7 beta*=1.1221e+37 rho=2.170e-16 s=1 join=12: |log q*|=34289.4 beta*=1.1835e+37 rho=2.170e-16 s=5 join=18: |log q*|=38354.1 beta*=1.3216e+37 rho=2.170e-16 s=5 join=20: |log q*|=39418.4 beta*=1.3578e+37 beta_W >= 4 C_lambda c_6 = 8.928e+17 (C_lambda = 240, c_6 <= 9.3e14 for SU(2)) ===== d5/loop_filling.out ===== PASS (A) product inequality on 2000 random products near and far from I, max (lhs-rhs)/rhs = 0.00e+00 PASS (B) ordered filling identity on 240 random closed paths, lengths 4-14, max error 4.9e-14 PASS (B) number of plaquettes = number of inversions <= l(l-1)/2 PASS (B) every plaquette of the filling lies in the bounding box of the path PASS (C) flat pure-gauge backgrounds: d2 s(V_C) <= (A_C/N) sum ||D_p w||^2, max excess / (bound + 1e-3) 8.7e-08 PASS (D) R_m'' = -eps m^2 at pi/2 for m = 4, 8, 12, 40, and |R_m| <= eps s 2 c_* = 0.29131713 n = 2: eps'' = 5.59528e-9 n = 1024: eps'' = 5.59528e-9 n = 1000000: eps'' = 5.59528e-9 n = 10000000000000000000000000000000000000000: eps'' = 5.59528e-9 PASS (E) eps'' <= 5.60e-9 for n <= 1e40 largest n with eps'' <= 0.1: 9.28739e+49 PASS (E) eps'' <= 0.1 for n <= 9.2e49 0 failures ===== d5/relstep_constants.out ===== == constants of the window lemma c_* in [0.145659, 0.145659] (claim c_* > 0.1456) 7/256 * 8 = 0.21875 (sum |v'|^2 <= (7/32) Q); 12*7/32 = 2.625 == C_n (upper, proved) and C_n^- (lower, test field); asymptotics n^2/pi^2 and n^2/(3 pi^2) n C_n<= 1+n^2/4 C_n^->= (n+1)^2/(3pi^2)<= ratio C_n/C_n^- 1 1.25 1.25 0.166667 0.135095 7.5000 2 1.5 2 0.333333 0.303964 4.5000 3 2 3.25 0.569036 0.54038 3.5147 4 2.70711 5 0.872678 0.844343 3.1021 5 3.61803 7.25 1.24402 1.21585 2.9083 6 4.73205 10 1.68297 1.65491 2.8117 8 7.56854 17 2.76362 2.73567 2.7386 16 27.0217 65 9.78843 9.76061 2.7606 32 104.836 257 36.8074 36.7796 2.8482 64 416.095 1025 142.722 142.694 2.9154 128 1661.13 4097 562.056 562.029 2.9555 1024 106244 262145 35483.6 35483.5 2.9942 10000 1.01321e+07 2.5e+07 3.37805e+06 3.37805e+06 2.9994 == relative constant eps' = (eps_1 + delta_G/lambda) C_n / (2 c_*) n= 2: eps' <= 8.393e-09 n= 4: eps' <= 1.515e-08 n= 8: eps' <= 4.235e-08 n= 16: eps' <= 1.512e-07 n= 64: eps' <= 2.328e-06 n= 256: eps' <= 3.716e-05 n= 1024: eps' <= 0.0005945 n= 4096: eps' <= 0.009511 eps' <= 0.1 for every n <= 8455 (checked through C_n <= 1 + n^2/4) == across scales: windows of side n_k = n_1 2^(1-k), k = 1..K, n_K >= 1 sum_k C_{n_k} <= K + n_1^2/3 (from C_n <= 1 + n^2/4 and sum 4^(1-k)/4 <= 1/3) n_1=2^ 1, K= 2: sum C <= 2.75 <= K + n_1^2/3 = 3.33333; sum eps'_k <= 1.539e-08 (constant eps_1) n_1=2^ 4, K= 5: sum C <= 40.0474 <= K + n_1^2/3 = 90.3333; sum eps'_k <= 2.241e-07 (constant eps_1) n_1=2^ 8, K= 9: sum C <= 8863.38 <= K + n_1^2/3 = 21854.3; sum eps'_k <= 4.959e-05 (constant eps_1) n_1=2^12, K=13: sum C <= 2.26653e+06 <= K + n_1^2/3 = 5.59242e+06; sum eps'_k <= 0.01268 (constant eps_1) the same errors kept absolute (n_1=2^12, K=13): the error of scale 1 at the last scale, 4^(K-1) eps_1 <= 0.02735; all K errors, sum_k 4^(K-k) eps_1 <= 0.03646 ===== d5/rough_certificate.out ===== Exact four-link certificate on fine torus side 4 All 64 average numerators: zero Nonzero plaquettes by (central sign, curl squared): (-1, 1): 10 (-1, 4): 1 (+1, 1): 12 Positive sum: 12; negative sum: 14; signed sum: -2 Q second derivative for HS-unit generator: -4 T(xi): 4; Q second derivative / T(xi): -1 Wilson action beta_W*Q/4 second derivative: -beta_W ===== d5/scales.out ===== beta_W = 0.3715 * log(mu/Lambda) m/Lambda= 1.0 n0=chart C_G beta_top <= 13.7 m/Lambda= 1.0 n0=Gaussian constrained beta_top <= 14.2 m/Lambda= 1.0 n0=Gaussian all scales beta_top <= 708.6 m/Lambda= 10.0 n0=chart C_G beta_top <= 14.6 m/Lambda= 10.0 n0=Gaussian constrained beta_top <= 15.1 m/Lambda= 10.0 n0=Gaussian all scales beta_top <= 709.4 m/Lambda= 100.0 n0=chart C_G beta_top <= 15.4 m/Lambda= 100.0 n0=Gaussian constrained beta_top <= 15.9 m/Lambda= 100.0 n0=Gaussian all scales beta_top <= 710.3 threshold chartdecay/fibrechart beta >= 3.8e+38: needs log(n0/(ell Lambda)) >= 1.02e+39 threshold off-chart window beta >= 3.8e+97: needs log(n0/(ell Lambda)) >= 1.02e+98 threshold islands beta >= 8.3e+163: needs log(n0/(ell Lambda)) >= 2.23e+164 ===== d5/sf_constants.out ===== 2*216*8100/219 < 15978.08219178082 <= 15979 8 log M < 80.55913140686165 c_2(10^12,1) < 4479.615373035101 kappa=0.08: a_2 >= 9 for lam >= 915476.5934; beta** = 4 lam <= 3661906; lam kappa^2 <= 5859.05 kappa=1e-2: a_2 >= 9 for lam >= 67650935.55; beta** = 4 lam <= 2.706037e+08; lam kappa^2 <= 6765.094 kappa=1e-4: a_2 >= 9 for lam >= 8.758971211e+11; beta** = 4 lam <= 3.503588e+12; lam kappa^2 <= 8758.971 kappa=1e-8: a_2 >= 9 for lam >= 1.271644831e+20; beta** = 4 lam <= 5.086579e+20; lam kappa^2 <= 12716.45 8100/11520 = 0.703125 ===== d5/sf_lattice.out ===== (1) f_0 depends exactly on I_0, P_{0,mu}, I_{e_nu}, P_{e_nu,mu} (mu!=nu): 21 groups active links per block: inside 17 private 32 (2) V_{y,mu} depends only on P_{y,mu}, I_y, I_{y+e_mu} (3) X={0}: |X^|=11, |d^-X|=4, |d^=X|=6, pairs into X=4, moved plaquettes=216 300 random l_inf-connected sets of 1..7 blocks: kernel factor set = X u d^-X u d^=X, moved plaquettes per block <= 216.000 <= 384 (4) components at l_inf distance >= 2: disjoint factor sets and variables (300 trials) all checks passed ===== d5/sf_schur.out ===== Row of n blocks along e_0; A=(0,1), B=((n-1)e_0,1); exact rationals, rounded: n=1: |A_W|=49, n_P=164, M_AA=2.0411089e+01, M_AB=2.0411089e+01 n=2: |A_W|=98, n_P=311, M_AA=2.0083269e+01, M_AB=-3.3214163e+00 n=3: |A_W|=147, n_P=458, M_AA=2.0079608e+01, M_AB=3.3523899e-01 n=4: |A_W|=196, n_P=605, M_AA=2.0079559e+01, M_AB=-3.5264626e-02 n=4 exact M_AB = -341079344760135289665457367170710450110856475415016913390426954350805829491654383077774430047938656670991991754832452422357541308771039340002359472896/9671996784486519612172861230862809894803420060577524391651881615452399566993289160705293552776228810229477091986914777969733094976324693228942777958177 n=5: |A_W|=245, n_P=752, M_AA=2.0079558e+01, M_AB=3.8785828e-03 n=6: |A_W|=294, n_P=899, M_AA=2.0079558e+01, M_AB=-4.4728020e-04 n=7: |A_W|=343, n_P=1046, M_AA=2.0079558e+01, M_AB=5.3953951e-05 ===== d5/slab_energy_constants.out ===== K: [2.9717829769163, 2.9717829769163] A: [3.71472872114537, 3.71472872114537] B: [0.469833424547118, 0.469833424547118] D: [0.148376937984496, 0.148376937984496] (1/2)log(2/pi) - (3/2)log(lambda) [must be >= 0]: [2.18836551600642, 2.18836551600642] delta^2 at beta=1 = c [must be <= pi^2/4]: [0.625, 0.625] pi^2/4: [2.46740110027234, 2.46740110027234] certified: A < 3.7148, B < 0.4699, D < 0.149 grid minimum of A over (lambda, c), floating point: (mpf('3.71472872114536912683435752211769'), 0.2, 0.625) ===== d5/touching_checks.out ===== a = 0.956087541852 46a = 43.980026925183 < 44 2a/(1-a) = 43.545161540508 < 44 M = Dsep = 2401; 6M = 14406 Offset enumeration and periodic separation (sides 1..64): passed No Hessian, label-domain, or island-threshold claim is checked here. ===== d5/witness_geometry.out ===== j=1: corner range [-14,15], side=30 (formula 30), per-coordinate overlap 8 -> H_j=4096 (formula 4096) j=2: corner range [-30,33], side=64 (formula 64), per-coordinate overlap 8 -> H_j=4096 (formula 4096) j=3: corner range [-62,69], side=132 (formula 132), per-coordinate overlap 9 -> H_j=6561 (formula 6561) j=4: corner range [-126,141], side=268 (formula 268), per-coordinate overlap 9 -> H_j=6561 (formula 6561) j=5: corner range [-254,285], side=540 (formula 540), per-coordinate overlap 9 -> H_j=6561 (formula 6561) j=6: corner range [-510,573], side=1084 (formula 1084), per-coordinate overlap 9 -> H_j=6561 (formula 6561) chart-calm labels rho=1e-10 (eta=9.2e-9<=1e-8), s=5: j=0: beta*_j=5.4096e+25 16^-j beta*_j=5.4096e+25 k_*(beta*_j)=11 k_*-j=11 j=1: beta*_j=2.8973e+27 16^-j beta*_j=1.8108e+26 k_*(beta*_j)=12 k_*-j=11 j=2: beta*_j=4.6366e+28 16^-j beta*_j=1.8112e+26 k_*(beta*_j)=13 k_*-j=11 j=3: beta*_j=1.1866e+30 16^-j beta*_j=2.8969e+26 k_*(beta*_j)=15 k_*-j=12 j=4: beta*_j=1.8987e+31 16^-j beta*_j=2.8972e+26 k_*(beta*_j)=16 k_*-j=12 j=5: beta*_j=3.0383e+32 16^-j beta*_j=2.8976e+26 k_*(beta*_j)=17 k_*-j=12 j=6: beta*_j=4.8619e+33 16^-j beta*_j=2.8979e+26 k_*(beta*_j)=18 k_*-j=12 j=7: beta*_j=7.7799e+34 16^-j beta*_j=2.8982e+26 k_*(beta*_j)=19 k_*-j=12 j=8: beta*_j=1.2449e+36 16^-j beta*_j=2.8986e+26 k_*(beta*_j)=20 k_*-j=12 ===== d5/witness_radius.out ===== psi_c: [0.05923686695, 0.05923686695] kappa_c: [0.08376133275, 0.08376133275] sup_{kappa<=kappa_c} K_iv(kappa)/kappa <=: [28.03743724, 28.03743724] limit coefficient as kappa->0 (G(0+)/sqrt2): [24.21861727, 24.21991736] p/kappa at kappa_c: [0.7072101769, 0.7072101769] link witness 12*sqrt2*p/kappa: [12.00175468, 12.00175468] coarse plaquette 16*sqrt2*p/kappa: [16.00233958, 16.00233958] example kappa: [0.08676144535, 0.08676144535] example max delta(P_x,v): [0.517714984, 0.517714984] example chi: [0.964570032, 0.964570032] example 1/16 sum of angles of w_x = 3*phi (angle of M): [0.1840776945, 0.1840776945] branchbound contraction at r=1/32: [0.893498558, 0.893498558] a = [0.00000099999999999999999999999999999992885, 0.00000099999999999999999999999999999992885] m=1: m! S(R)/R^m: [4.063427932, 4.063427932] m=2: m! S(R)/R^m: [306.7588679, 306.7588679] m=3: m! S(R)/R^m: [14810.24442, 14810.24442] |z|<0.1: e0+psi: [0.3528598076, 0.3528598076] gamma0+eta: [0.9016715955, 0.9016715955] gamma0: [0.004009030099, 0.004009030099] a = [0.00059999999999999999999999999999998891, 0.00059999999999999999999999999999998891] m=1: m! S(R)/R^m: [4.065287634, 4.065287634] m=2: m! S(R)/R^m: [307.0325579, 307.0325579] m=3: m! S(R)/R^m: [14831.28506, 14831.28506] |z|<0.1: e0+psi: [0.3534588076, 0.3534588076] gamma0+eta: [0.9026241248, 0.9026241248] gamma0: [0.00401083902, 0.00401083902] c_2 = sqrt2*307: [434.1635636, 434.1635636] R_iv (kappa+6s <=): [1.292064771e-9, 1.292064771e-9] paper: kappa+s <=: [6.019020104e-13, 6.019020104e-13] smallfibre(i) literal 52(1+1.4e6 k) holds (paper's proof) up to k = 4.90719e-8 flat second derivative lower bound: 1/16 = 0.0625 (exact, prop:flatsecond); route bound kappa' <= [0.0000092098407885613040618805518647177469, 0.0000092098407885613040618805518647177469] ===== slice/composite_free.out ===== == magnetic (the paper's slice field) t_f = 0.05: continuum D(.5)/D(.25), D(1)/D(.5) = 0.27199, 0.12736 (infinite volume); 0.27152, 0.12480 (side 6.0) 1/a= 8 L= 48: D(.5)/D(.25)=0.27341 D(1)/D(.5)=0.12524 D(a)/D(0)=0.4512 D(.25)/D(0)=2.169e-01 1/a= 12 L= 72: D(.5)/D(.25)=0.27239 D(1)/D(.5)=0.12500 D(a)/D(0)=0.5818 D(.25)/D(0)=2.151e-01 1/a= 16 L= 96: D(.5)/D(.25)=0.27201 D(1)/D(.5)=0.12491 D(a)/D(0)=0.6636 D(.25)/D(0)=2.144e-01 1/a= 24 L=144: D(.5)/D(.25)=0.27174 D(1)/D(.5)=0.12485 D(a)/D(0)=0.7589 D(.25)/D(0)=2.139e-01 t_f = 0.0: continuum D(.5)/D(.25), D(1)/D(.5) = 0.03125, 0.03125 (infinite volume); 0.03125, 0.03102 (side 6.0) 1/a= 8 L= 48: D(.5)/D(.25)=0.02238 D(1)/D(.5)=0.02875 D(a)/D(0)=0.0198 D(.25)/D(0)=7.990e-04 1/a= 12 L= 72: D(.5)/D(.25)=0.02652 D(1)/D(.5)=0.03015 D(a)/D(0)=0.0198 D(.25)/D(0)=8.369e-05 1/a= 16 L= 96: D(.5)/D(.25)=0.02895 D(1)/D(.5)=0.03055 D(a)/D(0)=0.0198 D(.25)/D(0)=1.788e-05 1/a= 24 L=144: D(.5)/D(.25)=0.03036 D(1)/D(.5)=0.03082 D(a)/D(0)=0.0198 D(.25)/D(0)=2.220e-06 == electric + magnetic t_f = 0.05: continuum D(.5)/D(.25), D(1)/D(.5) = 0.27199, 0.12736 (infinite volume); 0.27152, 0.12480 (side 6.0) 1/a= 8 L= 48: D(.5)/D(.25)=0.27092 D(1)/D(.5)=0.12390 D(a)/D(0)=0.0422 D(.25)/D(0)=2.018e-02 1/a= 12 L= 72: D(.5)/D(.25)=0.27126 D(1)/D(.5)=0.12440 D(a)/D(0)=0.0232 D(.25)/D(0)=8.545e-03 1/a= 16 L= 96: D(.5)/D(.25)=0.27137 D(1)/D(.5)=0.12458 D(a)/D(0)=0.0145 D(.25)/D(0)=4.677e-03 t_f = 0.0: continuum D(.5)/D(.25), D(1)/D(.5) = 0.03125, 0.03125 (infinite volume); 0.03125, 0.03102 (side 6.0) 1/a= 8 L= 48: D(.5)/D(.25)=0.01519 D(1)/D(.5)=0.02594 D(a)/D(0)=0.0773 D(.25)/D(0)=2.376e-03 1/a= 12 L= 72: D(.5)/D(.25)=0.02158 D(1)/D(.5)=0.02893 D(a)/D(0)=0.0773 D(.25)/D(0)=1.919e-04 1/a= 16 L= 96: D(.5)/D(.25)=0.02611 D(1)/D(.5)=0.02987 D(a)/D(0)=0.0773 D(.25)/D(0)=3.610e-05 ===== slice/normal_order.out ===== [((0, 0), 40), ((0, 2), -34), ((0, 4), 12), ((0, 6), -1), ((1, 1), -48), ((1, 3), 12), ((2, 0), 34), ((2, 2), -24), ((2, 4), 3), ((3, 1), -12), ((4, 0), 12), ((4, 2), -3), ((6, 0), 1)] lambda = 236 max a, max b = 6 6 sum bound = [0.2591937887450115605980624923176737, 0.25919378874501156059806249231782162] pi*sum bound = [0.81428130257743315322826002792259351, 0.81428130257743315322826002792318516]