==================== bh_bell_11.out ==================== PASS E1 outside half is exactly thermal and blind to the pair phase off-diagonal 0.0e+00, phase change 3.5e-18 r=0.1: CHSH = 2.0385848200 formula 2 sqrt(1+tanh(2r)^2) = 2.0385848200 r=0.3: CHSH = 2.2701737708 formula 2 sqrt(1+tanh(2r)^2) = 2.2701737708 r=0.7: CHSH = 2.6712151100 formula 2 sqrt(1+tanh(2r)^2) = 2.6712151100 r=1.2: CHSH = 2.8054348865 formula 2 sqrt(1+tanh(2r)^2) = 2.8054348865 PASS E2 CHSH above 2 at every squeezing, matching the formula omega/T_H = 0.5: squeezing r = 1.0423, CHSH = 2.785688 omega/T_H = 1: squeezing r = 0.7034, CHSH = 2.673161 omega/T_H = 2: squeezing r = 0.3860, CHSH = 2.383254 omega/T_H = 5: squeezing r = 0.0823, CHSH = 2.026418 omega/T_H = 10: squeezing r = 0.0067, CHSH = 2.000182 PASS E3 same local record, opposite futures imbalances +0.531186, -0.531186; sin(2gt) = 0.531186 PASS A1 anomaly flux equals the thermal flux (reproduction) flux = 1/(768*pi*M**2), T_H = 1/(8*pi*M) PASS A2 associator and compression defect identities max error 1.1e-14 PASS A3 one-way books close only with the flux a x counted dx/dt = P - kappa x - a x ALL PASS ==================== bh_core_02.out ==================== C1 matrix of A in E1..E5 (rows a, cols b), with w2 = 1 - w1 - w3: A[11] claim residual: 0 A[22] claim residual: 0 A[12] claim residual: 0 A[21] claim residual: 0 A[33] claim residual: 0 A[44] claim residual: 0 A[55] claim residual: 0 shape-shear coupling entries: {0} shear off-diagonal: {0} C2 A21 at w3=0: 0 C3 min discriminant over 20000 interior samples: 2.886e-04 C4 (A11 - A22) at w3=0 = -2*(t1*w1 - t1 - t2*w1 + t3) ; claim residual: 0 C5 joint point: w1-w2 = 0.3999999999999999, A12 at wall = 0.484974 C5a one loop in w3: swapped=True; two loops: returned=True C5b discriminant winding per loop: 1.000000 C5c ledger log det(cI+A_pair): start 2.561913986149+0.000000000000j, after loop 2.561913986149-0.000000000000j, drift 1.13e-20, max step 3.64e-08 C5d Riesz pair projector: Tr = 2.000000000000; drift after loop 2.45e-18; half-loop differs by 6.66e-16 (moves, but returns) C5e one loop in fold coordinate sigma3 (w3 = sigma3^2, encircles w3 twice): swapped=False sigma3=+0.02: eigenvalues [2.58174952 2.61841048], |cos angle between eigenvectors| = 0.997147 sigma3=-0.02: eigenvalues [2.58174952 2.61841048], |cos angle between eigenvectors| = 0.997147 gap / |sigma3| at sigma3 = 1e-2, 1e-3, 1e-4: [np.float64(1.83303464), np.float64(1.83303032), np.float64(1.83303028)] (constant => gap linear in sigma3, square-root in w3) C6 ledger table t=(2,1,1.3), w1-w2=0.4 (joint point): w3= 1e-02: pair [2.51032667 2.69367333], count 2, Tr P_pair 2.000000000000, ledger log det(1+A) 2.562330546753, ||P_single||*gap = 0.502311 w3= 1e-03: pair [2.57121656 2.62918344], count 2, Tr P_pair 2.000000000000, ledger log det(1+A) 2.561913986149, ||P_single||*gap = 0.486706 w3= 1e-04: pair [2.59085483 2.60918517], count 2, Tr P_pair 2.000000000000, ledger log det(1+A) 2.561872320543, ||P_single||*gap = 0.485147 w3= 1e-06: pair [2.59908368 2.60091672], count 2, Tr P_pair 2.000000000000, ledger log det(1+A) 2.561867737220, ||P_single||*gap = 0.484976 w3= 0e+00: pair [2.6 2.6], count 2, Tr P_pair 2.000000000000, ledger log det(1+A) 2.561867690924, ||P_single||*gap = nan |A12| at wall = 0.484974 (individual projector norm x gap -> |A12|) C7 T=G^-1 at w3=1e-6: relative asymmetry of G-metric * A = 1.10e-22 (0 => self-adjoint => diagonalisable, no Jordan block) w3=1e-02: |eigenvalues| = [ 4.2308 4.4211 101.6667 102.5641 198.0404] (x w3: [0.0423 0.0442 1.0167 1.0256 1.9804]) w3=1e-04: |eigenvalues| = [4.16730000e+00 4.33420000e+00 1.00016667e+04 1.00025006e+04 1.99980004e+04] (x w3: [4.0000e-04 4.0000e-04 1.0002e+00 1.0003e+00 1.9998e+00]) w3=1e-06: |eigenvalues| = [4.16670000e+00 4.33330000e+00 1.00000167e+06 1.00000250e+06 1.99999800e+06] (x w3: [0. 0. 1. 1. 2.]) C8 t=0.1: ||exp(-Jt)|| = 0.789975240034, formula 0.789975240034 C8 t=1.0: ||exp(-Jt)|| = 0.094436407453, formula 0.094436407453 C8 t=3.0: ||exp(-Jt)|| = 0.000804747797, formula 0.000804747797 growth ratio kappa = |mu|/b = 0.1865 (transient growth above 1 needs kappa > 2) RESULT PASS ==================== bh_drift_symmetrize_02.out ==================== exact drift: e^{-psi} L e^{psi} - [-Δ + (Q+|∇psi|^2-Δpsi)] = 0 rotational drift, first-order coefficients after conjugation: -2*y - 2*Derivative(psi(x, y), x) , 2*x - 2*Derivative(psi(x, y), y) curl of (-y,x) = 2 (nonzero => no psi can remove both) ==================== bh_ep_spinor_04.out ==================== H1 Q(sigma3) = (sigma3**2 + 21)/25 Q(0) = 21/25 A12 = 7*sqrt(3)/25 u.v - 2 sigma3^2 sqrt(Q) p = 0 residue of a at the wall = 1/2 H2 one w3 loop: M = [[0j, (-0-1.0594j)], [-0.9441j, 0j]] M[0,1]*M[1,0] = (-1.0001+0j) M1^2 = [[(-1.0001+0j), -0j], [-0j, (-1.0001+0j)]] two loops = [[(-1.0001-0j), -0j], [-0j, (-1.0001+0j)]] four loops = [[(1.0002-0j), (-0+0j)], [(-0+0j), (1.0002-0j)]] one sigma3 loop = [[(-1.0001+0j), -0j], [-0j, (-1.0001-0j)]] control (loop off the wall point) = [[(1+0j), 0j], [0j, (1-0j)]] off the joint condition t3=1.25: small sigma3 loop = [[(1-0j), 0j], [(-0+0j), (1+0j)]] IV B^2 = g1 g2: True (B^2)^2 = -I: True B^8 = I: True TR handed part Tr(P+[dP/dsigma3, dP/ddelta]) at t=(2,1,1.3), delta=0.4: sigma3=+-0.1: -27.257764 / +27.257764 sum 0.0e+00 symmetrised operator: 0.0e+00 sigma3=+-0.2: -6.799872 / +6.799872 sum 0.0e+00 symmetrised operator: 2.5e-12 sigma3=+-0.4: -1.685527 / +1.685527 sum 0.0e+00 symmetrised operator: 3.7e-13 RESULT PASS ==================== bh_figs_v10.out ==================== f19: thick layer max |R_A - R_B| = 0.11730296898876647 max ratio three-level = 0.9551096544990809 two-level dev from cos = 2.2426505097428162e-14 f20: measured growth per radian {np.float64(0.0): np.float64(0.999983), np.float64(0.3): np.float64(0.989505), np.float64(0.5): np.float64(0.968095), np.float64(0.7): np.float64(0.92624), np.float64(0.9): np.float64(0.831831)} ==================== bh_figs_v8_4.out ==================== max |U(s)-U(-s)| = 2.6645352591003757e-15 U/sigma3^2 near the wall = 1.8284211427042232 vs 1.8284271247461903 tetrahedron rise = 0.30838502114284516 (exact 2+4sqrt2-6sqrt(3/2) = 0.30838502114284694 ) gravity dU/dw3: min 1.2450012801816044 fraction > 0 1.0 ==================== bh_figs_v8_5.out ==================== max |U(s)-U(-s)| = 2.6645352591003757e-15 U/sigma3^2 near the wall = 1.8284211427042232 vs 1.8284271247461903 tetrahedron rise = 0.30838502114284516 (exact 2+4sqrt2-6sqrt(3/2) = 0.30838502114284694 ) rhombus root k* = 0.14455669481354932 value at k=1: 1.8284271247461903 ==================== bh_fold_parallel_01.out ==================== 3D: h_NN * N -> 4*m**2 (expect 4 m^2 = 1/(4 kappa^2), kappa=1/(4m)) 3D: h_lam_lam = 16*m**2/(lam**2 - 1)**4 | at lam=0: 16*m**2 (expect kappa^-2 = 16 m^2) 3D: even in lam: True 3D: proper distance ~ 4*lam*m + O(lam**2) 4D: h_NN * N -> m/2 (expect 1/(4 kappa^2) = m/2) 4D: h_lam_lam at 0 = 2*m even: True Bures: g_ww * w -> 1/4 ; in sigma: g_ss = 1 3D Kruskal Q(0) = sqrt(e): True 4D D*(0) = e/2: True ==================== bh_isco_10.out ==================== circular orbits: L^2 = r**2/(r - 3) V''/2 on the circular family = (r - 6)/(r**3*(r - 3)) zeros of V''/2: [6] PASS I1 simple sign change of V'' at r0 = 6 r0=12.00: V''/2 = +3.858025e-04 eigenvalues = 0.00000+0.01964j, 0.00000-0.01964j r0= 7.00: V''/2 = +7.288630e-04 eigenvalues = 0.00000+0.02700j, 0.00000-0.02700j r0= 6.10: V''/2 = +1.421179e-04 eigenvalues = 0.00000+0.01192j, 0.00000-0.01192j r0= 5.90: V''/2 = -1.678982e-04 eigenvalues = 0.01296+0.00000j, -0.01296+0.00000j r0= 5.00: V''/2 = -4.000000e-03 eigenvalues = 0.06325+0.00000j, -0.06325+0.00000j r0= 4.00: V''/2 = -3.125000e-02 eigenvalues = 0.17678+0.00000j, -0.17678+0.00000j PASS I2 Jordan block at the ISCO [[0.0, 1.0], [-0.0, 0.0]] side outside d=0.01: gap/sqrt(d) = 0.078241 eigenvalues imaginary side inside d=0.01: gap/sqrt(d) = 0.078896 eigenvalues real side outside d=0.001: gap/sqrt(d) = 0.078535 eigenvalues imaginary side inside d=0.001: gap/sqrt(d) = 0.078600 eigenvalues real side outside d=0.0001: gap/sqrt(d) = 0.078564 eigenvalues imaginary side inside d=0.0001: gap/sqrt(d) = 0.078571 eigenvalues real side outside d=1e-05: gap/sqrt(d) = 0.078567 eigenvalues imaginary side inside d=1e-05: gap/sqrt(d) = 0.078568 eigenvalues real PASS I3 square-root law, imaginary outside and real inside outside -> 0.078567, inside -> 0.078568 unstable circular orbit r_u = 4.697224, E_s^2 = 0.912543520, lam = 0.086060, Omega = 0.163414, Omega/lam = 1.898829 /home/claude/bh/levin/bh_isco_10.py:58: IntegrationWarning: The algorithm does not converge. Roundoff error is detected in the extrapolation table. It is assumed that the requested tolerance cannot be achieved, and that the returned result (if full_output = 1) is the best which can be obtained. I,_=quad(g,rp,ra,limit=2000,epsabs=1e-12,epsrel=1e-12) dE=0.001: periapsis 5.126872, apoapsis 13.1100, azimuth per radial period 16.032207 (whirls ~ 2.552) dE=0.0001: periapsis 4.818807, apoapsis 13.4378, azimuth per radial period 20.332523 (whirls ~ 3.236) dE=1e-05: periapsis 4.734487, apoapsis 13.4704, azimuth per radial period 24.692552 (whirls ~ 3.930) dE=1e-06: periapsis 4.708895, apoapsis 13.4737, azimuth per radial period 29.063022 (whirls ~ 4.626) dE=1e-07: periapsis 4.700904, apoapsis 13.4740, azimuth per radial period 33.435010 (whirls ~ 5.321) dE=1e-08: periapsis 4.698387, apoapsis 13.4741, azimuth per radial period 37.807195 (whirls ~ 6.017) PASS I4 whirl law: slope of azimuth vs ln(1/dE) equals Omega/lam slope 1.898773 vs Omega/lam 1.898829 ALL PASS ==================== bh_isco_cube_17.out ==================== PASS C1 omega_r^2/Omega^2 = 1 - 6M/r, so p = 2 + 6M/r ratio = (r - 6)/r, p = 2 + 6/r, p(6M) = 3 PASS C2 R = -2 N''/N; linear lapse is flat R = -2*Derivative(N(x), (x, 2))/N(x) PASS C2 Schwarzschild lapse = rho/(4M) near the horizon lapse ~ sqrt(2)/(2*sqrt(M)) sqrt(eps), rho ~ 2*sqrt(2)*sqrt(M) sqrt(eps) PASS C3 extremal RN: proper distance diverges like -M log(eps); lapse ~ eps/M ~ exp(-rho/M) distance ~ M * log(1/eps) PASS C4 Lamb: psi'' + (omega^2 - omega_ac^2)/c^2 psi = 0, omega_ac = c/2H; generator nilpotent at the cutoff ALL PASS ==================== bh_knot_08.out ==================== N= 50: (2pi/N) log<4_1> = 2.734201 corrected = 2.031316 (Vol = 2.029883) N= 100: (2pi/N) log<4_1> = 2.447006 corrected = 2.030236 (Vol = 2.029883) N= 200: (2pi/N) log<4_1> = 2.271020 corrected = 2.029971 (Vol = 2.029883) N= 400: (2pi/N) log<4_1> = 2.166761 corrected = 2.029905 (Vol = 2.029883) N= 800: (2pi/N) log<4_1> = 2.106483 corrected = 2.029889 (Vol = 2.029883) PASS K1 volume conjecture, figure-eight error at N=800: 5.45e-06 N= 100: |F|/N^1.5 = 1.000976 arg F = -0.084103 pi N= 200: |F|/N^1.5 = 1.000102 arg F = -0.416976 pi N= 400: |F|/N^1.5 = 0.999879 arg F = +0.916449 pi N= 800: |F|/N^1.5 = 1.000012 arg F = -0.416757 pi PASS K2 trefoil Kontsevich-Zagier growth N^{3/2} 0.999879 -> 1.000012 PASS K3 4_1 invariant is real (amphichiral) Im/Re = 1.0e-13 PASS K3 trefoil: mirror conjugates, invariant not real F = 729.4725-2733.0380j, mirror = 729.4725+2733.0380j PASS K3 signatures sigma(4_1)=0, mirror 0; sigma(3_1)=-2, mirror 2 ALL PASS ==================== bh_knot_14.out ==================== PASS K1 formula checks: unknot, Jones N=2, trefoil vs Kontsevich-Zagier, h-stability unknot 3.1e-51; Jones 3_1 5.0e-51, 8_19 3.8e-51 (formula gives 8_19 in variable 1/q); |J|/|F| trefoil 3.8e-30; h-stability 6.7e-17 N= 100: |J_N(8_19)|/N^1.5 = 2.061937 (2pi/N) log|J_N| = 0.479495 arg = +0.373410 pi N= 200: |J_N(8_19)|/N^1.5 = 2.914559 (2pi/N) log|J_N| = 0.283284 arg = -0.058535 pi N= 400: |J_N(8_19)|/N^1.5 = 0.085823 (2pi/N) log|J_N| = 0.102600 arg = -0.403829 pi N= 800: |J_N(8_19)|/N^1.5 = 0.085744 (2pi/N) log|J_N| = 0.059459 arg = +0.922825 pi PASS K2 8_19 grows like N^{3/2}, no volume 0.085823 -> 0.085744 PASS K3 mirror conjugates 8_19; invariant not real J = 8104.6272-1507.4055j, mirror = 8104.6272+1507.4055j PASS K4 Alexander, genus, Moser surgeries Delta = t**6 - t**5 + t**3 - t + 1, genus 3; Sigma(2,3,5), Sigma(2,3,7), Sigma(3,4,11), Sigma(3,4,13) period test: max|c(N)-c(N+P)| = {8: 2.828468, 12: 1.975873, 16: 0.000104} constants by N mod 16 (N = 640..655): [0.0858, 0.9713, 1.1181, 1.3073, 2.0616, 1.6707, 2.6926, 1.8859, 2.9142, 1.8858, 2.6925, 1.6706, 2.0615, 1.3072, 1.118, 0.9714] PASS K5 the N^{3/2} constant of 8_19 has period 16 in N range 0.0858 to 2.9142 ALL PASS ==================== bh_lift_05.out ==================== PASS K1 g A - A^T g = 0 (5x5, symbolic) PASS K2 trial 0: g(W,H)=TrH/4 and A=P_W D_T 1.4e-17 PASS K2 trial 1: g(W,H)=TrH/4 and A=P_W D_T 0.0e+00 PASS K2 trial 2: g(W,H)=TrH/4 and A=P_W D_T 0.0e+00 PASS K3 closed form of B at s=0.3 0.0e+00 PASS K3 closed form of B at s=0.05 1.4e-17 PASS K3 closed form of B at s=-0.2 5.6e-17 PASS K3 B(0) = lambda* I on the joint condition [[2.6, -0.0], [-0.0, 2.6]] PASS K4 complex s-loop, Gram frame holonomy -1.00012+0.00000i PASS K4 complex s-loop, regular frame holonomy +1.00000-0.00000i PASS K5 folded weight u2v2/(u.v), branch 0 0.500000000000 PASS K5 folded weight u2v2/(u.v), branch 1 0.500000000000 PASS K6 s=0.1: kinetic norm 1, Euclidean norm*gap kinetic 1.000000000000 Euclid 2.74 Euclid*gap 0.502311 PASS K6 s=0.01: kinetic norm 1, Euclidean norm*gap kinetic 1.000000000000 Euclid 26.47 Euclid*gap 0.485147 PASS K6 s=0.001: kinetic norm 1, Euclidean norm*gap kinetic 1.000000000000 Euclid 264.58 Euclid*gap 0.484976 |A12| = 0.4849742261192857 PASS K7 real (s,t3) loop around the joint point, regular frame, eigenline orientation -1.000000000+0.000000000j (biorthogonal -1.000248) PASS K7 real (s,t3) loop around the joint point, Gram frame, eigenline orientation +1.000000000+0.000000000j (biorthogonal +1.000140) PASS K7 real loop not enclosing it, regular frame, eigenline orientation +1.000000000+0.000000000j (biorthogonal +1.000473) PASS K7 real loop not enclosing it, Gram frame, eigenline orientation +1.000000000+0.000000000j (biorthogonal +1.110969) cone: s=0.001, dt3=0: gap = 1.833030e-03 cone: s=0, dt3=0.001: gap = 2.000000e-03 PASS K8 real paths: eigenvalues real on both sides of the wall max |Im| = 0.0e+00 ALL PASS ==================== bh_matched_12.out ==================== R_H = 376.7303135 ohm, Z_0 = mu_0 c = 376.7303135 ohm PASS M1 horizon resistivity equals the vacuum impedance ratio 1.000000000 PASS M2 zero reflection exactly at the matched value S(1) = 0.0, S(1/2) = -0.3333, S(2) = 0.3333 ALL PASS ==================== bh_nonnormal_04.out ==================== both anisotropic ||[A,A^T]||_F = 0.231425 T isotropic ||[A,A^T]||_F = 0.000000 W isotropic ||[A,A^T]||_F = 0.000000 at the wall, joint point ||[A,A^T]||_F = 0.332623 ==================== bh_offset_15.out ==================== PASS K1 one pair: outside record Fermi-Dirac, tau = w/T for every phase max off-diagonal 0.0e+00, max |tau - w/T| 5.6e-16 PASS K2 four pairs: tau from covariance = log P(empty)/P(full) = sum w_k/T tau_C = 13.260000000000, tau_Fock = 13.260000000000, sum w/T = 13.260000000000; outside Fock off-diagonal 0.0e+00 PASS K3 the offset is the ratio w/T; a common rescaling leaves it unchanged max change 0.0e+00 ALL PASS ==================== bh_petrov_06.out ==================== PASS P1 map Q -> quartic invertible det = 3072 PASS P2 type I (1111): Jordan structure [(np.complex128(-0.406667-0j), [1]), (np.complex128(0.358333-0j), [1]), (np.complex128(0.048333+0j), [1])] PASS P2 type II (211): Jordan structure [(np.complex128(-0.425+0j), [1]), (np.complex128(0.2125+0j), [2])] PASS P2 type D (22): Jordan structure [(np.complex128(0.666667+0j), [1]), (np.complex128(-0.333333+0j), [1, 1])] eigenvalue ratio (-2-0j) PASS P2 type III (31): Jordan structure [(np.complex128(-0j), [3])] PASS P2 type N (4): Jordan structure [(np.complex128(-0+0j), [2, 1])] P3 split a double star of type II by delta delta=0.1: gap=8.000000e-02 gap/delta=0.800000 gap/sqrt(eps)=1.600000 delta=0.01: gap=8.000000e-03 gap/delta=0.800000 gap/sqrt(eps)=1.600000 delta=0.001: gap=8.000000e-04 gap/delta=0.800000 gap/sqrt(eps)=1.600000 delta=0.0001: gap=8.000000e-05 gap/delta=0.800000 gap/sqrt(eps)=1.600000 PASS P3 gap linear in star separation (II) gap/delta -> 0.800000 P4 split both double stars of type D by delta delta=0.1: gap=2.500000e-03 gap/delta^2=0.250000 gap/delta=0.025000 delta=0.01: gap=2.500000e-05 gap/delta^2=0.250000 gap/delta=0.002500 delta=0.001: gap=2.500000e-07 gap/delta^2=0.250000 gap/delta=0.000250 PASS P4 gap quadratic in star separation (D) gap/delta^2 -> 0.250000 PASS P5 Kerr r=2.5, th=0.4: type D, ratio (1,1,-2) [(np.complex128(0.021115+0.019957j), [1, 1]), (np.complex128(-0.042231-0.039913j), [1])] PASS P5 Kerr r=1.9, th=1.2: type D, ratio (1,1,-2) [(np.complex128(0.065438+0.027524j), [1, 1]), (np.complex128(-0.130876-0.055048j), [1])] PASS P5 Kerr r=6.0, th=2.8: type D, ratio (1,1,-2) [(np.complex128(0.002152-0.000733j), [1, 1]), (np.complex128(-0.004304+0.001467j), [1])] ALL PASS ==================== bh_qnm_13.out ==================== eta=1e-01: gap=9.165e-01 |B+|=1.569e+00 |B-|=1.032e+00 |B+|*gap=1.437879 B++B-=0.560000-0.050000j arg(B+/B-)=+0.957974 pi eta=1e-02: gap=2.835e-01 |B+|=4.200e+00 |B-|=3.664e+00 |B+|*gap=1.190892 B++B-=0.560000-0.050000j arg(B+/B-)=+0.986201 pi eta=1e-03: gap=8.947e-02 |B+|=1.264e+01 |B-|=1.211e+01 |B+|*gap=1.131171 B++B-=0.560000-0.050000j arg(B+/B-)=+0.995620 pi eta=1e-04: gap=2.828e-02 |B+|=3.939e+01 |B-|=3.885e+01 |B+|*gap=1.114009 B++B-=0.560000-0.050000j arg(B+/B-)=+0.998614 pi eta=1e-05: gap=8.944e-03 |B+|=1.240e+02 |B-|=1.234e+02 |B+|*gap=1.108750 B++B-=0.560000-0.050000j arg(B+/B-)=+0.999562 pi eta=1e-06: gap=2.828e-03 |B+|=3.914e+02 |B-|=3.909e+02 |B+|*gap=1.107104 B++B-=0.560000-0.050000j arg(B+/B-)=+0.999861 pi PASS K1 single excitation factors grow like |coupling|/gap |B+|*gap -> 1.107104 against |u.N v| = 1.106345; |B+| = 3.914e+02 at eta=1e-6 PASS K2 B+ + B- = u.v (pair projector) and opposite phases u.v = 0.560000-0.050000j; arg(B+/B-) = +0.999861 pi nilpotent check: |N0^2| = 0.0e+00, u.N0.v = -0.240000+1.080000j eta=1e-02: max|h| = 1.058211 max|single mode term| = 4.200e+00 eta=1e-04: max|h| = 1.040295 max|single mode term| = 3.939e+01 eta=1e-06: max|h| = 1.040120 max|single mode term| = 3.914e+02 two-mode sum vs Jordan: 5.07e-04 at eta=1e-4, 5.06e-06 at eta=1e-6 (gap ratio 10) PASS K3 pair sum bounded through the EP; Jordan form t e^{-i w0 t} at the EP max|h| = 1.0582; exact-vs-Jordan 1.8e-15; modes(eta=1e-6)-vs-Jordan 5.1e-06 PASS K4 contour integral equals the waveform, at and beside the EP max rel err 2.8e-13 PASS K5 one loop exchanges the pair, two restore it; ledger single-valued after 1 turn |w-w_-| = 0.0e+00, after 2 |w-w_+| = 0.0e+00; ledger change 0.0e+00, 0.0e+00 ALL PASS ==================== bh_qutrit_18.out ==================== PASS Q1 F_Q = 4 g (so F_Q = 8 x shape kinetic energy at unit inertia) F_Q/g over 200 random curves: 3.999999999988 .. 4.000000000002 PASS Q2 wall distance arcsin sqrt(eps) = Bures angle arccos sqrt(F) max difference 3.0e-14 PASS Q3 mirror pair: fidelity 1, cover distance 2 arcsin|sigma3| > 0 F = 0.999999999999996, sign det X = -1 vs +1, cover distance on the fold path 2 arcsin|sigma3| = 0.2327 PASS P1 disk potential: min over orientation of sum m z^2 = w3 = sigma3^2 (frequency nu exactly) max |min - w3| 2.2e-16; Jacobi identity 0.0e+00 grav: dV/dw3 at the wall over 400 planar configurations: min +1.8926, max +53472.1508, fraction stiffening (>0) 1.000 coul: dV/dw3 at the wall over 400 planar configurations: min -53472.1508, max -1.8926, fraction stiffening (>0) 0.000 PASS P2 measured: equal-mass gravity stiffens toward coplanarity at every sampled planar configuration; like charges do the opposite square control dU/dw3 = 1.82842699 vs 2 sqrt2 - 1 = 1.82842712; transverse omega = sqrt(2 dU/dw3) = 1.912290 (G = m = I = 1) PASS C1 cone slope closed form 2|t1-t2| sqrt(1-delta^2) 1.833030278 vs measured 1.833030278 ALL PASS ==================== bh_shadow_07.out ==================== zeta = -1: L(zeta) = (4.0 + 0.0j) 1-|q| = 1e-2: |f| = 1.04112e+19 |b| = 1.04112e+19 f-(-1)^k b = (3.96113346 + 0.0j) 1-|q| = 3e-3: |f| = 8.96272e+60 |b| = 8.96272e+60 f-(-1)^k b = (3.988106103 + 0.0j) 1-|q| = 1e-3: |f| = 1.80128e+180 |b| = 1.80128e+180 f-(-1)^k b = (3.996011929 + 0.0j) 1-|q| = 3e-4: |f| = 1.74648e+597 |b| = 1.74648e+597 f-(-1)^k b = (3.998801078 + 0.0j) PASS S1 zeta = -1: f and b blow up |f| 1.04e+19 -> 1.75e+597 PASS S2 zeta = -1: f-(-1)^k b -> L(zeta) Richardson (3.9999999 + 0.0j); raw error 0.00399 -> 0.0012 zeta = i: L(zeta) = (0.0 + 4.0j) 1-|q| = 1e-2: |f| = 346204.0 |b| = 346208.0 f-(-1)^k b = (0.07052610774 + 3.82286692j) 1-|q| = 3e-3: |f| = 1.6589e+16 |b| = 1.6589e+16 f-(-1)^k b = (0.02302477682 + 3.942353953j) 1-|q| = 1e-3: |f| = 1.67771e+46 |b| = 1.67771e+46 f-(-1)^k b = (0.007886708938 + 3.98027346j) 1-|q| = 3e-4: |f| = 4.65059e+150 |b| = 4.65059e+150 f-(-1)^k b = (0.00238963484 + 3.994025019j) PASS S1 zeta = i: f and b blow up |f| 3.46e+5 -> 4.65e+150 PASS S2 zeta = i: f-(-1)^k b -> L(zeta) Richardson (2.1982687e-6 + 3.9999947j); raw error 0.0212 -> 0.00644 zeta = e^(i pi/3): L(zeta) = (-2.0 - 3.464101615j) 1-|q| = 1e-2: |f| = 958.595 |b| = 962.074 f-(-1)^k b = (-1.703124438 - 3.112779117j) 1-|q| = 3e-3: |f| = 7.51328e+7 |b| = 7.51328e+7 f-(-1)^k b = (-1.895583573 - 3.340418061j) 1-|q| = 1e-3: |f| = 2.20807e+21 |b| = 2.20807e+21 f-(-1)^k b = (-1.963167616 - 3.420459263j) 1-|q| = 3e-4: |f| = 8.09702e+67 |b| = 8.09702e+67 f-(-1)^k b = (-1.98870893 - 3.450721387j) PASS S1 zeta = e^(i pi/3): f and b blow up |f| 959.0 -> 8.1e+67 PASS S2 zeta = e^(i pi/3): f-(-1)^k b -> L(zeta) Richardson (-1.9999547 - 3.4640476j); raw error 0.0571 -> 0.0175 ALL PASS ==================== bh_spine_09.out ==================== PASS S1 C - C^2 = BB^dagger and Tr(C^2 - C) = -Tr BB^dagger max error 7.8e-16 PASS S2 K(0) = BB^dagger = PH^2P - (PHP)^2 = PHQHP max error 1.4e-14 PASS S3 A = P_W D_T P_W; P D_T^2 P - A^2 = P D_T Q D_T P, rank one, kinetic-positive error 4.9e-15, ranks {np.int64(1)}, min eigenvalue -1.4e-16 PASS S4 G*JG = I - S*S = 4(WV)* Herm L (WV), ||S|| <= 1 error 2.4e-15, max ||S|| 0.9011 PASS S5 Q = g + (i/2) Omega >= 0, |Omega_12| <= 2 sqrt(det g) min slack 1.02e-03 PASS S6 exterior signal independent of hidden rate and state; P(X+QYQ)P = PXP spread 9.5e-15 PASS S7 retention: escaped fraction 1, mean residence (1+a/b)/kappa; b=0 keeps a/(kappa+a) mean = (a + b)/(b*kappa), retained(b=0) = E0*a/(a + kappa) ALL PASS ==================== bh_stiff_19.out ==================== PASS S1 closed form = finite-difference lift max rel diff 4.0e-06 PASS S2 square 1.828427124746189 vs 1.828427124746190 PASS S3 rhombus sign change k* = 0.144557 (1/k* = 6.9177); stiffness at k=1 +1.828427, k=0.2 +0.6651, k=0.1 -1.0801 exact rhombus formula at k* : -1.7763568394002505e-15 and at k=0.5: 1.642573002183047 1.6425730021830471 PASS S4 negative stiffness exists away from collisions min found -7.5253 with min pair distance 0.0500 PASS S5 rhombus root from (1+k^2)(1+k)^2 = 64k^2 k* = 0.144556694814 fraction of 20000 Gaussian flat shapes that stiffen: 1.0000 ALL PASS ==================== bh_sunladder_16_firstrun.out ==================== pair births found: [np.float64(3.8717), np.float64(5.2802), np.float64(7.1576), np.float64(9.085), np.float64(11.0266), np.float64(12.9707)] FAIL K1 ladder reproduced height 0.05: loop around first 1 point(s) [3.472, 4.272]: cycles [[1, 2]] (return error 0.0e+00) height 0.05: loop around first 2 point(s) [3.472, 5.680]: cycles [[1, 2], [3, 4]] (return error 0.0e+00) ==================== bh_sunladder_16b.out ==================== sparse vs dense eigenvalues at alpha0 = 3+0.1i: 6.6e-12 real-axis branch points: [(np.float64(3.8717), 'birth'), (np.float64(5.2802), 'birth'), (np.float64(7.1576), 'birth'), (np.float64(9.085), 'birth'), (np.float64(11.0266), 'birth'), (np.float64(12.9707), 'birth')] PASS K1 the five published birth points are reproduced max offset 0.0054; extra points: [(np.float64(12.9707), 'birth')] lasso around point 1 (alpha0 = 3.8717, birth): cycles [[1, 2]] winding +1.000 return err 4.0e-13 min overlap 1.000 lasso around point 2 (alpha0 = 5.2802, birth): cycles [[3, 4]] winding +1.000 return err 3.1e-13 min overlap 1.000 lasso around point 3 (alpha0 = 7.1576, birth): cycles [[5, 6]] winding +1.000 return err 3.1e-13 min overlap 1.000 lasso around point 4 (alpha0 = 9.0850, birth): cycles [[7, 8]] winding +1.000 return err 3.3e-13 min overlap 1.000 lasso around point 5 (alpha0 = 11.0266, birth): cycles [[9, 10]] winding +1.000 return err 3.6e-13 min overlap 1.000 PASS K2 each lasso is one transposition loop around first 1: h=0.05 cycles [[1, 2]] winding +1.000 | h=0.2 cycles [[1, 2]] winding +1.000 | reversed [[1, 2]] | product of lassos [[1, 2]] loop around first 2: h=0.05 cycles [[1, 2], [3, 4]] winding +2.000 | h=0.2 cycles [[1, 2], [3, 4]] winding +2.000 | reversed [[1, 2], [3, 4]] | product of lassos [[1, 2], [3, 4]] loop around first 3: h=0.05 cycles [[1, 2], [3, 4], [5, 6]] winding +3.000 | h=0.2 cycles [[1, 2], [3, 4], [5, 6]] winding +3.000 | reversed [[1, 2], [3, 4], [5, 6]] | product of lassos [[1, 2], [3, 4], [5, 6]] loop around first 4: h=0.05 cycles [[1, 2], [3, 4], [5, 6], [7, 8]] winding +4.000 | h=0.2 cycles [[1, 2], [3, 4], [5, 6], [7, 8]] winding +4.000 | reversed [[1, 2], [3, 4], [5, 6], [7, 8]] | product of lassos [[1, 2], [3, 4], [5, 6], [7, 8]] loop around first 5: h=0.05 cycles [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10]] winding +5.000 | h=0.2 cycles [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10]] winding +5.000 | reversed [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10]] | product of lassos [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10]] PASS K3+K4 windings, two heights, lasso product and reversal all consistent verdict n=1: SWAPS [[1, 2]] verdict n=2: SWAPS [[1, 2], [3, 4]] verdict n=3: SWAPS [[1, 2], [3, 4], [5, 6]] verdict n=4: SWAPS [[1, 2], [3, 4], [5, 6], [7, 8]] verdict n=5: SWAPS [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10]] births in [12, 13.5] at N = 120: [np.float64(12.9707)] births in [12, 13.5] at N = 240: [np.float64(13.0171)] births in [12, 13.5] at N = 480: [np.float64(13.0289)] ALL PASS ==================== bh_sunladder_16b_run1.out ==================== sparse vs dense eigenvalues at alpha0 = 3+0.1i: 6.6e-12 real-axis branch points: [(np.float64(3.8717), 'birth'), (np.float64(5.2802), 'birth'), (np.float64(7.1576), 'birth'), (np.float64(9.085), 'birth'), (np.float64(11.0266), 'birth'), (np.float64(12.9707), 'birth')] PASS K1 the five published birth points are reproduced max offset 0.0054; extra points: [(np.float64(12.9707), 'birth')] lasso around point 1 (alpha0 = 3.8717, birth): cycles [[1, 2]] winding +1.029 return err 2.3e-02 min overlap 1.000 lasso around point 2 (alpha0 = 5.2802, birth): cycles [[3, 4]] winding +1.029 return err 2.3e-02 min overlap 1.000 lasso around point 3 (alpha0 = 7.1576, birth): cycles [[5, 6]] winding +1.029 return err 2.3e-02 min overlap 1.000 lasso around point 4 (alpha0 = 9.0850, birth): cycles [[7, 8]] winding +1.029 return err 2.3e-02 min overlap 1.000 lasso around point 5 (alpha0 = 11.0266, birth): cycles [[9, 10]] winding +1.029 return err 2.3e-02 min overlap 1.000 PASS K2 each lasso is one transposition loop around first 1: h=0.05 cycles [[1, 2]] winding +1.000 | h=0.2 cycles [[1, 2]] winding +1.000 | reversed [[1, 2]] | product of lassos [[1, 2]] loop around first 2: h=0.05 cycles [[1, 2], [3, 4]] winding +2.000 | h=0.2 cycles [[1, 2], [3, 4]] winding +2.000 | reversed [[1, 2], [3, 4]] | product of lassos [[1, 2], [3, 4]] loop around first 3: h=0.05 cycles [[1, 2], [3, 4], [5, 6]] winding +3.000 | h=0.2 cycles [[1, 2], [3, 4], [5, 6]] winding +3.000 | reversed [[1, 2], [3, 4], [5, 6]] | product of lassos [[1, 2], [3, 4], [5, 6]] loop around first 4: h=0.05 cycles [[1, 2], [3, 4], [5, 6], [7, 8]] winding +4.000 | h=0.2 cycles [[1, 2], [3, 4], [5, 6], [7, 8]] winding +4.000 | reversed [[1, 2], [3, 4], [5, 6], [7, 8]] | product of lassos [[1, 2], [3, 4], [5, 6], [7, 8]] loop around first 5: h=0.05 cycles [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10]] winding +5.000 | h=0.2 cycles [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10]] winding +5.000 | reversed [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10]] | product of lassos [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10]] PASS K3+K4 windings, two heights, lasso product and reversal all consistent verdict n=1: SWAPS [[1, 2]] verdict n=2: SWAPS [[1, 2], [3, 4]] verdict n=3: SWAPS [[1, 2], [3, 4], [5, 6]] verdict n=4: SWAPS [[1, 2], [3, 4], [5, 6], [7, 8]] verdict n=5: SWAPS [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10]] ALL PASS ==================== bh_wall_03.out ==================== C9 D(w3) coefficients [w3^2, w3^1, w3^0]: (t1 + t2 - 2*t3)**2 -2*(dl*t1**2 - 2*dl*t1*t3 - dl*t2**2 + 2*dl*t2*t3 - t1**2 + 6*t1*t2 - 4*t1*t3 - t2**2 - 4*t2*t3 + 4*t3**2) (dl*t1 - dl*t2 - t1 - t2 + 2*t3)**2 sampled 50000 (t, delta): roots inside the physical segment: 0; largest real root seen: 1.998e+00 D at the wall (w3=0) = (dl*t1 - dl*t2 - t1 - t2 + 2*t3)**2 -> zero exactly on the joint condition slope dD/dw3 at w3 = 0 on the joint condition: -4*(dl - 1)*(dl + 1)*(t1 - t2)**2 C10 Schur-Cayley: max identity error 5.86e-15; max ||S|| over 200 passive samples 0.945022 matched load Z = I gives S = [0. 0. 0. 0.] C11 k=0.0: projector metric g_kk at N=1: 0.249999999975; at N=1e-8: 0.249999999975; exact m^2/(4(k^2+m^2)^2) = 0.250000000000 C11 k=0.7: projector metric g_kk at N=1: 0.112607540199; at N=1e-8: 0.112607540199; exact m^2/(4(k^2+m^2)^2) = 0.112607540201 local proper-time gap 2 sqrt(k^2+m^2) stays open; coordinate gap 2 N sqrt(k^2+m^2) -> 0 is the lapse, not the projector RESULT PASS ==================== bh_wall_check.out ==================== K1 Kendall w=(0.7,0.3) sectional 4.0000000 K1 Kendall w=(0.5,0.5) sectional 4.0000000 K1 Kendall w=(0.9,0.1) sectional 4.0000000 K2 shape metric in sigma coords: [[1. 0. 0.] [0. 1. 0.] [0. 0. 1.]] K3 eps=0.1 length 0.32175055 arcsin sqrt 0.32175055 K3 eps=0.01 length 0.10016742 arcsin sqrt 0.10016742 K3 eps=0.0001 length 0.01000017 arcsin sqrt 0.01000017 K4 w3=0.01: sectional K(E2,V)=5.48044 K(E4,V)=4.93415 K(E5,V)=6.75797 K4 w3=0.001: sectional K(E2,V)=5.59820 K(E4,V)=6.22501 K(E5,V)=7.55863 K4 w3=0.0001: sectional K(E2,V)=5.61019 K(E4,V)=6.43034 K(E5,V)=7.65566 K4 growth ratio 1e-2 -> 1e-4: 1.3032312615736628 RESULT PASS ==================== photon_ring_wall.out ==================== photon orbit r = 3*M ; generator eigenvalues +- L/(9*M**2) (affine), real: unstable wall Lyapunov rate (coordinate time) = sqrt(3)/(9*M) ; orbital rate = sqrt(3)/(9*M) echo demagnification per half orbit gamma = pi -> factor 0.0432139 ; delay = 3*sqrt(3)*pi*M PASS: reproduces gamma = pi and delay 3 sqrt(3) pi M ==================== v10_corrections_check.out ==================== C1 non-commuting T: g(AH,K) - g(H,AK) = +0.133333 (magnitude 2/15 = 0.133333); diagonal T, 200 random pairs: 1.4e-14 APPLY C1b Jacobian of the normalised flow = A(H) - 2Tr(TG)H symbolically: YES C2 Gaussian curvature of the eigenvalue block at three points: [1.0000000000000007, 0.9999999999999999, 0.9999999999999996] APPLY (round, not flat) C3 real-qubit Bures = (1/4)(dchi^2 + sin^2 chi dphi^2), chi in [0, pi/2): max dev 2.0e-15; boundary circle length pi APPLY C4 rank of P_W D_T Q_W D_T P_W: anisotropic 1, isotropic 0 APPLY (rank at most one) C5 rotation frame change adds oint = -6.283185 (exp = 535.5, not 1) APPLY (claim false in general) C5b diagonal F: extra term = share-weighted dlog F_i: YES C6 ||P+||^2 = 1 + (b-c)^2/g^2 on 500 random real matrices: max dev 4.3e-14 APPLY C7 s, F_Q (SLD), 8K: 0.1: 4.040404, 4.040404; 0.01: 4.000400, 4.000400; 0.001: 4.000004, 4.000004; 0: 0.000000, 4.000000 APPLY (full rank only) C8 quartic coefficient -3*sqrt(2)/2 - 3/4 = -2.8713; at sigma3 = 0.3 exact rise 0.1420008842 vs harmonic 0.1645584412; energy 0.1420 reaches sigma3 = 0.3000, harmonic bound says 0.2787 APPLY (at most is false) C9 at N = 0 the symbol is zero (every vector an eigenvector); the N > 0 projector is constant, so its limit exists APPLY SUMMARY: C1 apply, C1b apply, C2 apply, C3 apply, C4 apply, C5 apply, C5b apply, C6 apply, C7 apply, C8 apply, C9 apply ==================== v10_push_others.out ==================== P1 photon ring, nonlinear orbit: growth per radian 0.999983 (exact 1) -> per half orbit e^pi, echo factor 0.043216 opposite, charged hole Q = 0.5 M: growth per radian 0.968095 (formula 0.968119), echo factor 0.047770 PASS half-orbit delay pi/Omega = 16.324194 M against 3 sqrt3 pi M = 16.324194 M P2 sine guard: Gram identity max dev 7.1e-15; two-level max(|Omega| - bound) = 1.3e-14 (median slack 0.0e+00: equality), three-level -7.8e-05 (median slack 0.37: strict) PASS P3 fermion offset is a Smith coordinate: symbolic yes; partial trace over 6 frequencies x 7 phases max dev 2.8e-16; Gamma = 0 only at omega = 0 PASS P4 compression D(X) = PXP: Hermitian P idempotent 7.8e-16, self-adjoint 0.0e+00; oblique Riesz P (norm 1.44) idempotent 0.0e+00, NOT self-adjoint (0.94) PASS P5 tip and tail, 2000 trials each: max DK ratio 0.904 (must be <= 1); tight cluster: median vector turn 73.72 deg vs subspace 0.272 deg; wide split (opposite): vector turn 2.653 deg PASS SUMMARY: P1 pass, P2 pass, P3 pass, P4 pass, P5 pass