{
 "meta": {
  "seeds": {
   "diag": 1,
   "x0": 2,
   "crosspartial_u": 3,
   "tridiag_rhs": 4,
   "adi_probe": 5,
   "rank1_v": 6,
   "rank1_b": 7,
   "cluster_q": 8,
   "cluster_widths": 9,
   "cluster_b": 10
  },
  "tol": 1e-10,
  "n": 32,
  "kappa_A": 440.6885603836583,
  "sqrt_kappa_A": 20.992583461395558,
  "runtime_seconds": 2.7,
  "n_checks": 44,
  "n_fail": 0
 },
 "checks": [
  {
   "name": "kappa(A) at n=32 equals the house number 440.69 (measured 440.69)",
   "pass": true
  },
  {
   "name": "diagonal A: Jacobi-GD converges in ONE step from random x0 (rel err after 1 step = 7.69e-17)",
   "pass": true
  },
  {
   "name": "d2J/du_i du_j == A_ij (finite differences on 5 pairs, max dev = 4.66e-10 on entries of size 4/h^2 = 4356)",
   "pass": true
  },
  {
   "name": "diag(A) constant => Jacobi-GD IS plain GD (iters 1998 vs 1998, max rel traj dev over first 200 its = 4.7e-15)",
   "pass": true
  },
  {
   "name": "rod rhs parity: lowest excited mode is (1,2) with lam = 49.221 = lam_1+lam_2 = 49.221; effective kappa = 176.00 (not 440.69)",
   "pass": true
  },
  {
   "name": "Jacobi-GD does NOT one-step: 1998 iterations to 1e-10 on the rod problem; asymptotic A-norm rate 0.988687 = (kappa_eff-1)/(kappa_eff+1) = 0.988700 at the parity-effective kappa",
   "pass": true
  },
  {
   "name": "on a generic (GRF) rhs Jacobi-GD attains the full-kappa rate: measured 0.995464 vs (kappa-1)/(kappa+1) = 0.995472 (4753 iterations to 1e-10)",
   "pass": true
  },
  {
   "name": "(kappa-1)/(kappa+1) == cos(pi h) == rho(Jacobi Richardson) of report 12 (0.995472 vs 0.995472)",
   "pass": true
  },
  {
   "name": "2-D DST-I basis is orthonormal (max |V'V - I| = 1.0e-14) and diagonalizes A (max offdiag of V'AV = 3.3e-15 rel to diag; diag == lam_i+lam_j to 5.3e-15)",
   "pass": true
  },
  {
   "name": "in the eigenbasis the problem is 1024 independent scalar parabolas: one-pass solve x = V(V'b/lam) matches spsolve to 1.3e-15",
   "pass": true
  },
  {
   "name": "A == H + V exactly (max |H+V-A| = 0.0e+00)",
   "pass": true
  },
  {
   "name": "H and V COMMUTE exactly: max |HV - VH| = 0.0e+00 (Kronecker algebra: HV = VH = kron(d1,d1)/h^4)",
   "pass": true
  },
  {
   "name": "Kronecker SUM spectrum: eig(A) == {lam_i + lam_j} (max rel dev 3.1e-15)",
   "pass": true
  },
  {
   "name": "H is block-diagonal: zero coupling between different grid rows (max = 0.0e+00); all 32 blocks == d1/h^2 (max dev 0.0e+00)",
   "pass": true
  },
  {
   "name": "V couples only within grid columns (max cross-column mass = 0.0e+00)",
   "pass": true
  },
  {
   "name": "(H + sigma I)x=y via 32 separate tridiagonal solves matches spsolve (rel dev 2.9e-16); same for V by columns (2.9e-16)",
   "pass": true
  },
  {
   "name": "ADI M is SPD: commuting SPD factors => M^{-1} symmetric (asym 4.9e-16), eigenvalues in [2.00e-05, 8.80e-03] > 0; tridiagonal apply == dense M^{-1}r (5.4e-16)",
   "pass": true
  },
  {
   "name": "spectrum of M^{-1}A == closed form f(lam_i,lam_j) = 2s(li+lj)/((li+s)(lj+s)) (max dev 3.4e-15)",
   "pass": true
  },
  {
   "name": "geometric-mean shift balances the corners: f(l1,l1) == f(ln,ln) = 0.173609 (rel dev 1.6e-16); min of f at pure corners (1, 1), max at mixed corner (1, 32) = 1.826391",
   "pass": true
  },
  {
   "name": "the square-root effect: kappa(M^-1 A) = 10.520 = 0.501 * sqrt(kappa(A)) = 0.501 * 20.993",
   "pass": true
  },
  {
   "name": "1-D Green's function: (A^-1)_ij == h x_i (1-x_j) for i<=j (max rel dev 1.4e-15) -- the upper triangle of A^-1 IS a rank-1 matrix (Brownian-bridge covariance), while A^-1 itself has full rank 32",
   "pass": true
  },
  {
   "name": "2-D H-matrix fact: far off-diagonal block of A^-1 (rows 0-7 x rows 24-31, 256x256) has numerical rank 11 at 1e-8 (sv1 = 1.76e-03), while A^-1 is dense (all entries > 0: min 2.91e-09) and full-rank (lam_min = 1.15e-04 > 0)",
   "pass": true
  },
  {
   "name": "no precision edges cross the separator: nnz(A_LR) = 0",
   "pass": true
  },
  {
   "name": "GMRF Markov property: Cov(u_L, u_R | u_I) == 0 exactly (max |cond cov| = 1.9e-19 vs marginal max |Sigma_LR| = 2.6e-04; ratio 7.2e-16)",
   "pass": true
  },
  {
   "name": "residual coupling is the interface: A_LR has nnz = 32 = one column of edges, rank = 32; M^-1 A has eigenvalue 1 with multiplicity 960 = N - 2*rank = 960, rest split 1 +/- mu (max pairing dev 4.4e-15)",
   "pass": true
  },
  {
   "name": "extreme eigenvectors LOCALIZE AT THE INTERFACE: column-energy peak of the lam_min = 0.0986 eigenvector at col 16, of lam_max = 1.9014 at col 15 (cols 14-17 carry 39% of the lam_min mode's energy)",
   "pass": true
  },
  {
   "name": "cg_err reproduces pcg.pcg's residual history (max dev = 0.0e+00 over 77 entries -- same recurrence, error tracking is passive)",
   "pass": true
  },
  {
   "name": "ladder ordering (iterations to rel l2 err 1e-10): GD >= CG >= ADI-CG >= blockJacobi2-CG-ish >= twolevel-CG >= DST(1): gd=1998, jacobi_gd=1998, cg=73, adi_cg=32, blockjacobi2_cg=12, twolevel_ic0_cg=31, adi_gd=88, blockjacobi2_gd=98, dst_direct=1",
   "pass": true
  },
  {
   "name": "rho=1e+02: CG converges in EXACTLY 2 iterations (rel err after 2 = 1.4e-15 random b, 1.3e-15 worst b)",
   "pass": true
  },
  {
   "name": "rho=1e+02: GD(worst b) rate 0.98039214 == (kappa-1)/(kappa+1) = 0.98039216",
   "pass": true
  },
  {
   "name": "rho=1e+02: GD(random b) rate 0.19548504 == closed form f(m0)^(1/2) = 0.19548494 (worst case NOT attained for generic b)",
   "pass": true
  },
  {
   "name": "rho=1e+04: CG converges in EXACTLY 2 iterations (rel err after 2 = 3.6e-14 random b, 3.7e-14 worst b)",
   "pass": true
  },
  {
   "name": "rho=1e+04: GD(worst b) rate 0.99980004 == (kappa-1)/(kappa+1) = 0.99980004",
   "pass": true
  },
  {
   "name": "rho=1e+04: GD(random b) rate 0.89470826 == closed form f(m0)^(1/2) = 0.89470826 (worst case NOT attained for generic b)",
   "pass": true
  },
  {
   "name": "rho=1e+06: CG converges in EXACTLY 2 iterations (rel err after 2 = 3.6e-12 random b, 3.6e-12 worst b)",
   "pass": true
  },
  {
   "name": "rho=1e+06: GD(worst b) rate 0.99999800 == (kappa-1)/(kappa+1) = 0.99999800",
   "pass": true
  },
  {
   "name": "rho=1e+06: GD(random b) rate 0.99875514 == closed form f(m0)^(1/2) = 0.99875637 (worst case NOT attained for generic b)",
   "pass": true
  },
  {
   "name": "GD iteration count to 1e-10 GROWS with rho: [1163, 115141, 11512937] for rho = 1e2/1e4/1e6 (worst-case rhs; CG stays at 2)",
   "pass": true
  },
  {
   "name": "clusters w=1e-3: GD hopeless -- after 30000 its rel err still 7.2e-01; tail rate 0.99998899 vs (kappa-1)/(kappa+1) = 0.99999800 (kappa = 1.001e+06), projected ~2.1e+06 its to 1e-10",
   "pass": true
  },
  {
   "name": "clusters: CG needs 18 its at width 1e-3 (l2 err after 3 its 1.0e+00, A-norm err 1.0e+00; largest single-sweep drop 2935x), and the count grows only gently with width: {'1e-3': 18, '1e-2': 31, '1e-1': 72} while kappa stays ~1e6 -- clustering, not range, is CG's currency",
   "pass": true
  },
  {
   "name": "anchor: unpreconditioned PCG on the report-03 GRF rhs takes 116 iterations (report 08: 116)",
   "pass": true
  },
  {
   "name": "sqrt(kappa) BOUNDS every ladder method (A-norm iters <= sqrt(kappa) ln(2/tol)/2): none: 75 <= 249, adi: 34 <= 38, blockjacobi2: 12 <= 52, twolevel_ic0: 32 <= 39",
   "pass": true
  },
  {
   "name": "...but sqrt(kappa) alone does NOT predict the fine ordering (ordering_match = False): c = iters/sqrt(kappa) spans [2.96, 11.10]; ADI (flat spectrum, 0 eigenvalues at 1) uses 88% of its bound while blockJacobi2 (960/1024 eigenvalues exactly at 1) beats its larger kappa (19.3 vs 10.5) with 13 vs 36 its -- spectral shape (clustering, section 11) decides within the bound",
   "pass": true
  },
  {
   "name": "CG's search directions are A-orthogonal (sequential Gram-Schmidt in the A-inner product = 09 SS5's sequential regression): max |p_i'Ap_j| / (|p_i|_A |p_j|_A) = 7.3e-16 over the first 10, 1.8e-15 over the first 30 directions (finite-precision loss grows with depth)",
   "pass": true
  }
 ],
 "figures": [
  "decoupling_interface_mode.png",
  "decoupling_error_curves.png",
  "decoupling_gd_vs_cg.png",
  "decoupling_adi_spectrum.png",
  "decoupling_semiseparable.png"
 ],
 "deviations": [
  "GD runs that would exceed the runtime budget (rank-1 rho=1e6 and the cluster problem, both kappa ~ 1e6) were capped (120000 / 30000 iterations); their iterations-to-1e-10 are extrapolated from the measured tail rate (for the rank-1 case the per-step rate is exactly constant, so the extrapolation is exact up to the last digit of the rate).",
  "Rank-1 refinement: for a RANDOM rhs the (kappa-1)/(kappa+1) GD rate is provably not attained -- the per-step A-norm contraction is the exact closed form sqrt(f(m0)), f(m) = (k-1)^2 m/((1+k^2 m)(1+m)) (derivation in comments, machine-verified). The worst-case rhs b = v + w_perp, which attains (kappa-1)/(kappa+1) exactly, was added; both are reported.",
  "Ladder iteration counts are ERROR-based (||x_k - x*||/||x*|| <= 1e-10); section 12's sqrt(kappa) fit uses pcg's residual-based counts to stay comparable with report 08's 116-iteration anchor. Both are in the JSON.",
  "Error histories come from cg_err, a re-implementation of pcg.pcg's recurrence (verified to reproduce its residual history to machine precision in section 9) rather than a callback, so the reference iteration is untouched.",
  "Section 11: at width 1e-3 CG needs ~18 iterations, not literally 3: three clusters of nonzero width are not three points; the count and per-sweep drops are reported as measured.",
  "The hot/cold-rod rhs excites no even eigenmodes (odd parity), so GD rate checks on the rod problem use the parity-effective kappa (176.00); the full-kappa rate 0.995472 = cos(pi h) is verified on the GRF rhs.",
  "Section 12 reframed after measurement: sqrt(kappa) is verified as the Chebyshev BOUND on iterations, but it does not predict the fine ordering -- blockJacobi2 (960 eigenvalues exactly at 1) beats ADI despite a 1.8x larger kappa. A single-c fit is reported for completeness with its (poor) max prediction error."
 ],
 "part_a": {
  "diag_onestep_relerr": 7.691833900781323e-17,
  "cross_partial_max_dev": 4.656612873077393e-10,
  "jacobi_gd": {
   "iters_to_1e-10_rod": 1998,
   "rate_measured_rod": 0.9886865444590643,
   "kappa_eff_rod": 175.99600479905723,
   "rate_formula_eff": 0.9887003099178957,
   "lam_min_excited": 49.22144999764652,
   "iters_to_1e-10_grf": 4753,
   "rate_measured_grf": 0.9954638023993944,
   "rate_formula": 0.9954719225730846,
   "cos_pi_h": 0.9954719225730846
  },
  "dst": {
   "orth_dev": 1.021405182655144e-14,
   "offdiag_rel": 3.2699985843403318e-15,
   "diag_dev_rel": 5.336258468947343e-15,
   "onepass_relerr": 1.2732058018548402e-15
  }
 },
 "part_b": {
  "split": {
   "split_dev": 0.0,
   "commutator_fro_max": 0.0,
   "eigsum_dev_rel": 3.1389755699690254e-15
  },
  "tridiag_solve_dev": {
   "H": 2.8718298075985707e-16,
   "V": 2.916114586888915e-16
  },
  "adi": {
   "sigma": 207.0320623165098,
   "lam1d_min": 9.862152635821728,
   "lam1d_max": 4346.137847364178,
   "kappa_MinvA": 10.52010966625205,
   "kappa_A": 440.6885603836583,
   "sqrt_kappa_A": 20.992583461395558,
   "ratio_to_sqrt": 0.5011345881081322,
   "f_min": 0.17360945841157857,
   "f_max": 1.8263905415884216,
   "f_argmin_1based": [
    1,
    1
   ],
   "f_argmax_1based": [
    1,
    32
   ],
   "spec_dev": 3.3585271448675115e-15,
   "corner_balance_dev": 1.5987363747099741e-16,
   "cg_iters_err": 32,
   "gd_iters_err": 88
  },
  "semiseparable": {
   "tri_dev_rel": 1.3751637696215856e-15,
   "rank_S1": 32,
   "far_block_rows": [
    0,
    7
   ],
   "far_block_cols_rows": [
    24,
    31
   ],
   "far_block_sv": [
    0.0017574625160307368,
    0.00022369193607741562,
    2.734277203389175e-05,
    3.674947742056541e-06,
    5.508736128920439e-07,
    9.120153275474571e-08,
    1.6490170970436607e-08,
    3.2302664651717472e-09,
    6.821421159070756e-10,
    1.5482773036121816e-10,
    3.770477325368435e-11,
    9.841059776175763e-12,
    2.7507812638667116e-12,
    8.229515514245041e-13,
    2.63360961140096e-13,
    9.010291059024052e-14,
    3.293628676472944e-14,
    1.2855281013289126e-14,
    5.3539017232680734e-15,
    2.3775913288417942e-15,
    1.1250828598015283e-15,
    5.669083904805239e-16,
    3.0397976464162276e-16,
    1.7331382099791425e-16,
    1.0502380171805412e-16,
    6.759805791800566e-17,
    4.6206571497151414e-17,
    3.350187674055267e-17,
    2.5774826785664686e-17,
    2.103426879483417e-17
   ],
   "far_block_numrank_1e-8": 11,
   "S2_min_entry": 2.9059245266818398e-09,
   "S2_lam_min": 0.0001150446712828588
  }
 },
 "part_c": {
  "markov": {
   "A_LR_nnz": 0,
   "marginal_max": 0.00026283982659332586,
   "conditional_max": 1.8973538018496328e-19,
   "conditional_rel": 7.218669356319729e-16
  },
  "block_jacobi": {
   "kappa_MinvA": 19.2930724493778,
   "lam_min": 0.09855580050724765,
   "lam_max": 1.901444199492754,
   "n_unit_eigs": 960,
   "A_LR_nnz": 32,
   "A_LR_rank": 32,
   "eig_pairing_dev": 4.440892098500626e-15,
   "spectrum_low10": [
    0.09855580050724765,
    0.17349987541094183,
    0.24702780814735376,
    0.31359116240956203,
    0.37328026334757286,
    0.4266537237726307,
    0.4742878580250931,
    0.5167376985869111,
    0.5545271416131125,
    0.5881434039043091
   ],
   "spectrum_high10": [
    1.4118565960956921,
    1.4454728583868874,
    1.483262301413089,
    1.5257121419749087,
    1.5733462762273736,
    1.6267197366524278,
    1.6864088375904378,
    1.7529721918526455,
    1.8265001245890593,
    1.901444199492754
   ],
   "interface_mode": {
    "lam_min": 0.09855580050724672,
    "lam_max": 1.9014441994927527,
    "col_energy_peak_min": 16,
    "col_energy_peak_max": 15,
    "energy_frac_cols14_17": 0.39178978009976084,
    "col_energy_min": [
     0.00020644904076617073,
     0.0008332916326944909,
     0.0019032863199420647,
     0.00345528100731172,
     0.005545623396888338,
     0.00825020678339891,
     0.011667225484238352,
     0.015920739944548856,
     0.021165180954315615,
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     0.07060802805141406,
     0.08763608296229623,
     0.10825880708758413,
     0.10825880708758415,
     0.08763608296229634,
     0.07060802805141421,
     0.05655641100852153,
     0.04497106492558212,
     0.03543136489005971,
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     0.015920739944548908,
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     0.005545623396888343,
     0.003455281007311719,
     0.0019032863199420636,
     0.0008332916326944906,
     0.0002064490407661706
    ]
   },
   "cg_iters_err": 12,
   "gd_iters_err": 98
  }
 },
 "ladder": {
  "iters_to_1e-10_err": {
   "gd": 1998,
   "jacobi_gd": 1998,
   "cg": 73,
   "adi_cg": 32,
   "blockjacobi2_cg": 12,
   "twolevel_ic0_cg": 31,
   "adi_gd": 88,
   "blockjacobi2_gd": 98,
   "dst_direct": 1
  },
  "dst_direct_relerr": 1.2732058018548402e-15,
  "curves": {
   "gd": [
    1.0,
    0.7871803943148659,
    0.6638893384686108,
    0.5705149757463236,
    0.4962861255467868,
    0.4352659193299839,
    0.3838703824836409,
    0.33981851835366145,
    0.3015918002840911,
    0.2681323968202493,
    0.23866961281301463,
    0.212618064364245,
    0.18951652826419466,
    0.16899027036654143,
    0.15072722154662313,
    0.1344624900431908,
    0.11996798610058922,
    0.10704523781326108,
    0.0955202367184271,
    0.08523960235380658,
    0.07606762576356031,
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    0.060581475387821875,
    0.0540650865411539,
    0.048249941762088105,
    0.04306045798382128,
    0.038429244938617366,
    0.03429619946095323,
    0.03060770662646366,
    0.027315932617297544,
    0.02437819722,
    0.02175641601831672,
    0.019416603938894855,
    0.01732843302956434,
    0.015464838315682034,
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    0.0022351738449990417,
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    0.0017802615431817851,
    0.0015888030404512493,
    0.0014179349720821867,
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    5.375051864783317e-06,
    4.796991082541095e-06,
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