{
  "sweep": {
    "n": [
      16,
      32,
      64,
      128,
      256
    ],
    "theta": [
      0.0,
      0.1,
      1.0,
      10.0
    ],
    "tau": [
      0.01,
      0.1,
      1.0
    ]
  },
  "predictions": [
    {
      "prediction": "commutator remains at roundoff for every theta,tau",
      "predicted_bound": 1e-12,
      "observed_max": 0.0,
      "confirmed": true
    },
    {
      "prediction": "symmetry basis removes off-block entries for every theta,tau",
      "predicted_bound": 1e-12,
      "observed_max": 4.027672990746775e-14,
      "confirmed": true
    },
    {
      "prediction": "transformed solve equals original solve for every theta,tau",
      "predicted_bound": 1e-12,
      "observed_max": 3.674032258059059e-14,
      "confirmed": true
    }
  ],
  "timing_by_n": [
    {
      "n": 16,
      "median_sparse_ms": 0.35162249969289405,
      "median_fourier_ms": 0.06439449953177245,
      "fourier_over_sparse": 0.1831353215110364
    },
    {
      "n": 32,
      "median_sparse_ms": 0.38427349954872625,
      "median_fourier_ms": 0.06886449955345597,
      "fourier_over_sparse": 0.17920699614812727
    },
    {
      "n": 64,
      "median_sparse_ms": 0.694793498951185,
      "median_fourier_ms": 0.5200475006859051,
      "fourier_over_sparse": 0.7484921800087867
    },
    {
      "n": 128,
      "median_sparse_ms": 1.0094344988829107,
      "median_fourier_ms": 1.6049465002652141,
      "fourier_over_sparse": 1.589946155041586
    },
    {
      "n": 256,
      "median_sparse_ms": 2.0164585002930835,
      "median_fourier_ms": 0.3904195000359323,
      "fourier_over_sparse": 0.1936164319668302
    }
  ],
  "projectors": {
    "resolution_error": 1.0844765365859752e-14,
    "idempotence_error": 6.799953763053708e-15,
    "orthogonality_error": 1.8994954887520917e-15
  },
  "max_commutator": 0.0,
  "max_block_error": 4.027672990746775e-14,
  "max_solution_relative_error": 3.674032258059059e-14,
  "max_residual": 4.3829057262518325e-14,
  "median_sparse_ms": 0.6974825000725104,
  "median_fourier_ms": 0.22325150075630518,
  "rows": [
    {
      "n": 16,
      "theta": 0.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 2.6760500847022975e-15,
      "equivalence": 2.709678047140064e-15,
      "residual": 2.709678047140064e-15,
      "sparse_ms": 0.7409589998133015,
      "fourier_ms": 0.13744999887421727
    },
    {
      "n": 16,
      "theta": 0.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 2.6760500847022975e-15,
      "equivalence": 2.3326339669431623e-15,
      "residual": 2.3326339669431623e-15,
      "sparse_ms": 0.35347299854038283,
      "fourier_ms": 0.06900499829498585
    },
    {
      "n": 16,
      "theta": 0.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 2.6760500847022975e-15,
      "equivalence": 2.408000762442528e-15,
      "residual": 2.408000762442528e-15,
      "sparse_ms": 0.3223929998057429,
      "fourier_ms": 0.06411499998648651
    },
    {
      "n": 16,
      "theta": 0.1,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 2.6750981847330076e-15,
      "equivalence": 2.1278473091474048e-15,
      "residual": 2.1196984471608624e-15,
      "sparse_ms": 0.46207800005504396,
      "fourier_ms": 0.0667000003886642
    },
    {
      "n": 16,
      "theta": 0.1,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 2.6524444509461343e-15,
      "equivalence": 3.244217391741111e-15,
      "residual": 3.207301022081158e-15,
      "sparse_ms": 0.34516200139478315,
      "fourier_ms": 0.07787400136294309
    },
    {
      "n": 16,
      "theta": 0.1,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 2.646108537908067e-15,
      "equivalence": 1.4795737721952942e-15,
      "residual": 1.426518521052921e-15,
      "sparse_ms": 0.3497720008454053,
      "fourier_ms": 0.06376600140356459
    },
    {
      "n": 16,
      "theta": 1.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 2.6521141492259808e-15,
      "equivalence": 1.784665130558992e-15,
      "residual": 1.7208268206285012e-15,
      "sparse_ms": 0.33880600130942184,
      "fourier_ms": 0.06383599975379184
    },
    {
      "n": 16,
      "theta": 1.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 2.646108537908067e-15,
      "equivalence": 2.5576181991506586e-15,
      "residual": 2.5749094526054446e-15,
      "sparse_ms": 0.3614350007410394,
      "fourier_ms": 0.06341700100165326
    },
    {
      "n": 16,
      "theta": 1.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 2.665416242925186e-15,
      "equivalence": 2.1357320937485247e-15,
      "residual": 2.2495876340259583e-15,
      "sparse_ms": 0.3372690007381607,
      "fourier_ms": 0.06369600123434793
    },
    {
      "n": 16,
      "theta": 10.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 2.646108537908067e-15,
      "equivalence": 1.6730845381554915e-15,
      "residual": 1.6566750932141229e-15,
      "sparse_ms": 0.44797000009566545,
      "fourier_ms": 0.0646739990770584
    },
    {
      "n": 16,
      "theta": 10.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 2.665416242925186e-15,
      "equivalence": 2.686072179150826e-15,
      "residual": 2.378444477645389e-15,
      "sparse_ms": 0.33803800033638254,
      "fourier_ms": 0.06369599941535853
    },
    {
      "n": 16,
      "theta": 10.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 2.7396206682007676e-15,
      "equivalence": 1.8535991852818333e-15,
      "residual": 2.053536684353859e-15,
      "sparse_ms": 0.3564060007192893,
      "fourier_ms": 0.06593100079044234
    },
    {
      "n": 32,
      "theta": 0.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 5.055792094624233e-15,
      "equivalence": 4.450549524910488e-15,
      "residual": 4.450549524910488e-15,
      "sparse_ms": 0.38043299900891725,
      "fourier_ms": 0.10350699994887691
    },
    {
      "n": 32,
      "theta": 0.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 5.055792094624233e-15,
      "equivalence": 3.168883468360465e-15,
      "residual": 3.168883468360465e-15,
      "sparse_ms": 0.5082440002297517,
      "fourier_ms": 0.06984299943724182
    },
    {
      "n": 32,
      "theta": 0.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 5.055792094624233e-15,
      "equivalence": 2.8861366890099088e-15,
      "residual": 2.8861366890099088e-15,
      "sparse_ms": 0.3448129991738824,
      "fourier_ms": 0.06753799971193075
    },
    {
      "n": 32,
      "theta": 0.1,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 5.053653027103107e-15,
      "equivalence": 5.196234914097531e-15,
      "residual": 5.20150249093692e-15,
      "sparse_ms": 0.492389999635634,
      "fourier_ms": 0.09198299994750414
    },
    {
      "n": 32,
      "theta": 0.1,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 5.057090644071131e-15,
      "equivalence": 4.573844830822943e-15,
      "residual": 4.531974267619376e-15,
      "sparse_ms": 0.3800129998126067,
      "fourier_ms": 0.06830599886598065
    },
    {
      "n": 32,
      "theta": 0.1,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 5.054883926949303e-15,
      "equivalence": 4.6976833796298724e-15,
      "residual": 4.841153009992154e-15,
      "sparse_ms": 0.385740999263362,
      "fourier_ms": 0.06879399916215334
    },
    {
      "n": 32,
      "theta": 1.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 5.05627715440115e-15,
      "equivalence": 3.81181536066034e-15,
      "residual": 3.797032208580567e-15,
      "sparse_ms": 0.38280599983409047,
      "fourier_ms": 0.06781699994462542
    },
    {
      "n": 32,
      "theta": 1.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 5.054883926949303e-15,
      "equivalence": 5.100832374029288e-15,
      "residual": 5.012790696845417e-15,
      "sparse_ms": 0.38608999966527335,
      "fourier_ms": 0.06760699943697546
    },
    {
      "n": 32,
      "theta": 1.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 5.210663106400905e-15,
      "equivalence": 4.967460597886471e-15,
      "residual": 4.806790527548728e-15,
      "sparse_ms": 0.38818499888293445,
      "fourier_ms": 0.06788700011384208
    },
    {
      "n": 32,
      "theta": 10.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 5.054883926949303e-15,
      "equivalence": 4.259756060155669e-15,
      "residual": 4.246409140082132e-15,
      "sparse_ms": 0.37994299964339007,
      "fourier_ms": 0.06914400000823662
    },
    {
      "n": 32,
      "theta": 10.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 5.210663106400905e-15,
      "equivalence": 3.895782419055195e-15,
      "residual": 4.077141094244391e-15,
      "sparse_ms": 0.3655559994513169,
      "fourier_ms": 0.10134199874300975
    },
    {
      "n": 32,
      "theta": 10.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 5.39018995035214e-15,
      "equivalence": 3.472639555657943e-15,
      "residual": 4.281824868230312e-15,
      "sparse_ms": 0.3941209997719852,
      "fourier_ms": 0.0689349999447586
    },
    {
      "n": 64,
      "theta": 0.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 1.0095345478298608e-14,
      "equivalence": 9.440884524951155e-15,
      "residual": 9.440884524951155e-15,
      "sparse_ms": 0.7551369999418966,
      "fourier_ms": 0.15777399858052377
    },
    {
      "n": 64,
      "theta": 0.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 1.0095345478298608e-14,
      "equivalence": 8.353669147409391e-15,
      "residual": 8.353669147409391e-15,
      "sparse_ms": 0.5711719986720709,
      "fourier_ms": 0.12543699995148927
    },
    {
      "n": 64,
      "theta": 0.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 1.0095345478298608e-14,
      "equivalence": 8.013303761052601e-15,
      "residual": 8.013303761052601e-15,
      "sparse_ms": 0.6577069998456864,
      "fourier_ms": 0.35277400093036704
    },
    {
      "n": 64,
      "theta": 0.1,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 1.0100504236868143e-14,
      "equivalence": 7.049842769810993e-15,
      "residual": 7.055035787972347e-15,
      "sparse_ms": 0.7547879995399853,
      "fourier_ms": 0.3600390009523835
    },
    {
      "n": 64,
      "theta": 0.1,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 1.0103624221157238e-14,
      "equivalence": 7.954041420318364e-15,
      "residual": 8.014072643749804e-15,
      "sparse_ms": 0.7232190000650007,
      "fourier_ms": 0.6800560004194267
    },
    {
      "n": 64,
      "theta": 0.1,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 1.0150615902717781e-14,
      "equivalence": 8.031301908605485e-15,
      "residual": 7.98802430032254e-15,
      "sparse_ms": 0.7390039991150843,
      "fourier_ms": 0.2641440005390905
    },
    {
      "n": 64,
      "theta": 1.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 1.0103253944294337e-14,
      "equivalence": 6.740260629763592e-15,
      "residual": 6.681883496302359e-15,
      "sparse_ms": 0.6938159986020764,
      "fourier_ms": 0.1257859985344112
    },
    {
      "n": 64,
      "theta": 1.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 1.0150615902717781e-14,
      "equivalence": 8.956480256372376e-15,
      "residual": 9.090074749508547e-15,
      "sparse_ms": 0.6871810001030099,
      "fourier_ms": 5.239168000116479
    },
    {
      "n": 64,
      "theta": 1.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 1.0609644113595448e-14,
      "equivalence": 9.214085603252036e-15,
      "residual": 8.543392127964536e-15,
      "sparse_ms": 0.6957709993002936,
      "fourier_ms": 3.0419209997489816
    },
    {
      "n": 64,
      "theta": 10.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 1.0150615902717781e-14,
      "equivalence": 7.730309923542312e-15,
      "residual": 7.622958499560755e-15,
      "sparse_ms": 0.6991940008447273,
      "fourier_ms": 3.321432001030189
    },
    {
      "n": 64,
      "theta": 10.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 1.0609644113595448e-14,
      "equivalence": 8.757948630621708e-15,
      "residual": 1.0294888641167345e-14,
      "sparse_ms": 0.6783099997846875,
      "fourier_ms": 5.803914000352961
    },
    {
      "n": 64,
      "theta": 10.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 1.098014631399633e-14,
      "equivalence": 8.384854784524718e-15,
      "residual": 1.1245323077010524e-14,
      "sparse_ms": 0.6822209998063045,
      "fourier_ms": 5.191325000851066
    },
    {
      "n": 128,
      "theta": 0.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 2.0128694050867613e-14,
      "equivalence": 1.5576074546727524e-14,
      "residual": 1.5576074546727524e-14,
      "sparse_ms": 0.8657679991301848,
      "fourier_ms": 3.1501770008617314
    },
    {
      "n": 128,
      "theta": 0.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 2.0128694050867613e-14,
      "equivalence": 1.5311253819134602e-14,
      "residual": 1.5311253819134602e-14,
      "sparse_ms": 0.8145030005834997,
      "fourier_ms": 1.1954250003327616
    },
    {
      "n": 128,
      "theta": 0.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 2.0128694050867613e-14,
      "equivalence": 1.66674077995346e-14,
      "residual": 1.66674077995346e-14,
      "sparse_ms": 0.8459330001642229,
      "fourier_ms": 1.1716780009010108
    },
    {
      "n": 128,
      "theta": 0.1,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 2.013560509853296e-14,
      "equivalence": 1.3775690961688278e-14,
      "residual": 1.3779134358710435e-14,
      "sparse_ms": 1.0952699994959403,
      "fourier_ms": 2.613367998492322
    },
    {
      "n": 128,
      "theta": 0.1,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 2.013053580919517e-14,
      "equivalence": 1.8470832257551765e-14,
      "residual": 1.84271254618493e-14,
      "sparse_ms": 1.072572000339278,
      "fourier_ms": 2.0144680001976667
    },
    {
      "n": 128,
      "theta": 0.1,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 2.0098106164090788e-14,
      "equivalence": 1.7464217508517732e-14,
      "residual": 1.734377622203098e-14,
      "sparse_ms": 1.0203990004811203,
      "fourier_ms": 2.5905289985530544
    },
    {
      "n": 128,
      "theta": 1.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 2.013038443914069e-14,
      "equivalence": 1.7817758860396598e-14,
      "residual": 1.7804814499093742e-14,
      "sparse_ms": 1.0074089987028856,
      "fourier_ms": 0.9184989994537318
    },
    {
      "n": 128,
      "theta": 1.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 2.0098106164090788e-14,
      "equivalence": 1.8827666233036678e-14,
      "residual": 1.8133461417053446e-14,
      "sparse_ms": 1.0034280003310414,
      "fourier_ms": 2.193055001043831
    },
    {
      "n": 128,
      "theta": 1.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 2.048290319002887e-14,
      "equivalence": 2.1596039490983333e-14,
      "residual": 2.422966997532305e-14,
      "sparse_ms": 1.0919179985648952,
      "fourier_ms": 3.3110240001406055
    },
    {
      "n": 128,
      "theta": 10.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 2.0098106164090788e-14,
      "equivalence": 1.6289257608556863e-14,
      "residual": 1.5860488242203388e-14,
      "sparse_ms": 1.0114599990629358,
      "fourier_ms": 0.17369899978803005
    },
    {
      "n": 128,
      "theta": 10.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 2.048290319002887e-14,
      "equivalence": 1.8061229345701073e-14,
      "residual": 1.9534109453014828e-14,
      "sparse_ms": 0.9960940005839802,
      "fourier_ms": 0.17537500025355257
    },
    {
      "n": 128,
      "theta": 10.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 2.0931704105495374e-14,
      "equivalence": 2.0268346728771054e-14,
      "residual": 2.23833327892793e-14,
      "sparse_ms": 1.0201200002484256,
      "fourier_ms": 0.18235900097351987
    },
    {
      "n": 256,
      "theta": 0.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 3.959678754117359e-14,
      "equivalence": 2.997842573428557e-14,
      "residual": 2.997842573428557e-14,
      "sparse_ms": 1.6775479998614173,
      "fourier_ms": 0.36401900069904514
    },
    {
      "n": 256,
      "theta": 0.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 3.959678754117359e-14,
      "equivalence": 3.196636039681922e-14,
      "residual": 3.196636039681922e-14,
      "sparse_ms": 1.6662329999235226,
      "fourier_ms": 0.3386670014151605
    },
    {
      "n": 256,
      "theta": 0.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 3.959678754117359e-14,
      "equivalence": 2.882213433333891e-14,
      "residual": 2.882213433333891e-14,
      "sparse_ms": 1.6305430017382605,
      "fourier_ms": 0.4082989999005804
    },
    {
      "n": 256,
      "theta": 0.1,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 3.959361183281542e-14,
      "equivalence": 3.477174045503652e-14,
      "residual": 3.476368692597631e-14,
      "sparse_ms": 2.044779001153074,
      "fourier_ms": 0.4126999992877245
    },
    {
      "n": 256,
      "theta": 0.1,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 3.957509766111119e-14,
      "equivalence": 3.372422357202544e-14,
      "residual": 3.365228810359602e-14,
      "sparse_ms": 2.1624640012305463,
      "fourier_ms": 0.35186700006306637
    },
    {
      "n": 256,
      "theta": 0.1,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 3.945467996995351e-14,
      "equivalence": 3.487524145895868e-14,
      "residual": 3.4051098721310104e-14,
      "sparse_ms": 2.026272000875906,
      "fourier_ms": 0.33447600071667694
    },
    {
      "n": 256,
      "theta": 1.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 3.9574981687018446e-14,
      "equivalence": 2.744950057017979e-14,
      "residual": 2.747614746018048e-14,
      "sparse_ms": 2.029553999818745,
      "fourier_ms": 0.3725400001712842
    },
    {
      "n": 256,
      "theta": 1.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 3.945467996995351e-14,
      "equivalence": 3.167371693929385e-14,
      "residual": 3.105849832668305e-14,
      "sparse_ms": 1.999242000238155,
      "fourier_ms": 0.342786999681266
    },
    {
      "n": 256,
      "theta": 1.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 3.974724466797317e-14,
      "equivalence": 3.3234377112927264e-14,
      "residual": 3.5100731540901315e-14,
      "sparse_ms": 2.013490000535967,
      "fourier_ms": 0.4676650005421834
    },
    {
      "n": 256,
      "theta": 10.0,
      "tau": 0.01,
      "comm": 0.0,
      "block_error": 3.945467996995351e-14,
      "equivalence": 3.21524084829462e-14,
      "residual": 3.226431528013121e-14,
      "sparse_ms": 2.0210329985275166,
      "fourier_ms": 4.486403999180766
    },
    {
      "n": 256,
      "theta": 10.0,
      "tau": 0.1,
      "comm": 0.0,
      "block_error": 3.974724466797317e-14,
      "equivalence": 3.5219564516731916e-14,
      "residual": 3.470738922407767e-14,
      "sparse_ms": 2.0194270000502,
      "fourier_ms": 2.967399001136073
    },
    {
      "n": 256,
      "theta": 10.0,
      "tau": 1.0,
      "comm": 0.0,
      "block_error": 4.027672990746775e-14,
      "equivalence": 3.674032258059059e-14,
      "residual": 4.3829057262518325e-14,
      "sparse_ms": 1.9883460008713882,
      "fourier_ms": 5.037950999394525
    }
  ]
}