{
  "schema_version": 1,
  "inputs": {
    "schema_version": 1,
    "J": 1.0,
    "chain_lengths": [
      4,
      5,
      6,
      7,
      8
    ],
    "selected_modes": {
      "4": [
        1
      ],
      "5": [
        1
      ],
      "6": [
        1,
        2
      ],
      "7": [
        1,
        2
      ],
      "8": [
        1,
        2,
        3
      ]
    },
    "tolerances": {
      "equation_atol": 1e-11,
      "minimum_control_residual": 0.01,
      "minimum_raw_norm": 1e-10,
      "saved_data_atol": 5e-12,
      "saved_data_rtol": 1e-10
    }
  },
  "environment": {
    "python": "3.9.6",
    "numpy": "2.0.2",
    "arithmetic": "IEEE 754 binary64; complex128 states"
  },
  "convention": "periodic spin-1/2 XXX, H=J sum(1/4-S.S), S=sigma/2, sites 0,...,N-1",
  "chains": [
    {
      "N": 4,
      "full_dimension": 16,
      "hermiticity_max_abs_over_J": 0.0,
      "Sz_commutator_max_abs_over_J": 0.0,
      "vacuum_residual_l2_over_J": 0.0,
      "M1_dimension": 4,
      "M1_block_difference_max_abs_over_J": 0.0,
      "M1_spectrum_over_J": [
        -2.1116044736089807e-16,
        1.0,
        1.0,
        1.9999999999999998
      ],
      "M2_dimension": 6,
      "M2_block_difference_max_abs_over_J": 0.0,
      "M2_spectrum_over_J": [
        2.5494010195096103e-16,
        0.9999999999999999,
        1.0000000000000002,
        1.0000000000000007,
        1.9999999999999996,
        2.9999999999999987
      ],
      "one_magnon_spectrum_error_max_abs_over_J": 2.220446049250313e-16,
      "one_magnon_state_residual_max_l2_over_J": 3.1507115938535577e-16,
      "descendant_residual_l2_over_J": 6.206335383118183e-17,
      "descendant_uniform_state_error_l2": 0.0
    },
    {
      "N": 5,
      "full_dimension": 32,
      "hermiticity_max_abs_over_J": 0.0,
      "Sz_commutator_max_abs_over_J": 0.0,
      "vacuum_residual_l2_over_J": 0.0,
      "M1_dimension": 5,
      "M1_block_difference_max_abs_over_J": 0.0,
      "M1_spectrum_over_J": [
        -8.326672684688685e-17,
        0.6909830056250524,
        0.6909830056250524,
        1.8090169943749468,
        1.8090169943749468
      ],
      "M2_dimension": 10,
      "M2_block_difference_max_abs_over_J": 0.0,
      "M2_spectrum_over_J": [
        -5.233033649136332e-16,
        0.6909830056250524,
        0.6909830056250524,
        0.8819660112501053,
        0.8819660112501053,
        1.809016994374947,
        1.8090169943749477,
        1.999999999999999,
        3.1180339887498936,
        3.1180339887498945
      ],
      "one_magnon_spectrum_error_max_abs_over_J": 6.661338147750939e-16,
      "one_magnon_state_residual_max_l2_over_J": 3.263493924073004e-16,
      "descendant_residual_l2_over_J": 8.777083671441753e-17,
      "descendant_uniform_state_error_l2": 0.0
    },
    {
      "N": 6,
      "full_dimension": 64,
      "hermiticity_max_abs_over_J": 0.0,
      "Sz_commutator_max_abs_over_J": 0.0,
      "vacuum_residual_l2_over_J": 0.0,
      "M1_dimension": 6,
      "M1_block_difference_max_abs_over_J": 0.0,
      "M1_spectrum_over_J": [
        -3.661434948392178e-16,
        0.4999999999999999,
        0.5,
        1.4999999999999998,
        1.5000000000000002,
        1.9999999999999996
      ],
      "M2_dimension": 15,
      "M2_block_difference_max_abs_over_J": 0.0,
      "M2_spectrum_over_J": [
        -2.36035029973896e-16,
        0.4999999999999998,
        0.4999999999999998,
        0.7192235935955846,
        0.7192235935955859,
        1.0000000000000002,
        1.3819660112501058,
        1.5000000000000002,
        1.5000000000000002,
        2.0,
        2.4999999999999996,
        2.5000000000000004,
        2.7807764064044145,
        2.780776406404415,
        3.618033988749895
      ],
      "one_magnon_spectrum_error_max_abs_over_J": 4.440892098500626e-16,
      "one_magnon_state_residual_max_l2_over_J": 1.1012857236204923e-15,
      "descendant_residual_l2_over_J": 0.0,
      "descendant_uniform_state_error_l2": 0.0
    },
    {
      "N": 7,
      "full_dimension": 128,
      "hermiticity_max_abs_over_J": 0.0,
      "Sz_commutator_max_abs_over_J": 0.0,
      "vacuum_residual_l2_over_J": 0.0,
      "M1_dimension": 7,
      "M1_block_difference_max_abs_over_J": 0.0,
      "M1_spectrum_over_J": [
        -4.508307119498701e-16,
        0.37651019814126624,
        0.3765101981412664,
        1.2225209339563143,
        1.2225209339563146,
        1.9009688679024188,
        1.90096886790242
      ],
      "M2_dimension": 21,
      "M2_block_difference_max_abs_over_J": 0.0,
      "M2_spectrum_over_J": [
        -1.1329508606390156e-16,
        0.37651019814126596,
        0.3765101981412673,
        0.5816663233380498,
        0.5816663233380505,
        0.9509785740354566,
        0.9509785740354578,
        1.0000000000000004,
        1.2225209339563143,
        1.2225209339563161,
        1.9009688679024197,
        1.9009688679024206,
        1.9681832877263348,
        1.9681832877263363,
        2.3705734920184383,
        2.3705734920184387,
        2.572322940846902,
        2.5723229408469046,
        3.000000000000001,
        3.5562753820348143,
        3.556275382034817
      ],
      "one_magnon_spectrum_error_max_abs_over_J": 8.881784197001252e-16,
      "one_magnon_state_residual_max_l2_over_J": 3.807550085865571e-16,
      "descendant_residual_l2_over_J": 9.205483015737869e-17,
      "descendant_uniform_state_error_l2": 0.0
    },
    {
      "N": 8,
      "full_dimension": 256,
      "hermiticity_max_abs_over_J": 0.0,
      "Sz_commutator_max_abs_over_J": 0.0,
      "vacuum_residual_l2_over_J": 0.0,
      "M1_dimension": 8,
      "M1_block_difference_max_abs_over_J": 0.0,
      "M1_spectrum_over_J": [
        4.297890376032674e-17,
        0.29289321881345237,
        0.2928932188134525,
        0.9999999999999998,
        0.9999999999999998,
        1.7071067811865475,
        1.7071067811865477,
        2.0
      ],
      "M2_dimension": 28,
      "M2_block_difference_max_abs_over_J": 0.0,
      "M2_spectrum_over_J": [
        5.093142494589197e-16,
        0.29289321881345215,
        0.29289321881345304,
        0.4743128791344807,
        0.4743128791344813,
        0.7530203962825329,
        0.8558771943646312,
        0.8558771943646313,
        0.9999999999999991,
        0.9999999999999997,
        0.9999999999999997,
        1.562983975551178,
        1.562983975551181,
        1.7071067811865475,
        1.707106781186548,
        1.9999999999999998,
        2.0,
        2.000000000000001,
        2.2586520225041524,
        2.258652022504153,
        2.4370160244488193,
        2.4370160244488215,
        2.4450418679126287,
        3.144122805635368,
        3.1441228056353694,
        3.2670350983613647,
        3.2670350983613656,
        3.8019377358048376
      ],
      "one_magnon_spectrum_error_max_abs_over_J": 4.440892098500626e-16,
      "one_magnon_state_residual_max_l2_over_J": 1.7289635876953225e-15,
      "descendant_residual_l2_over_J": 1.0385185452638061e-16,
      "descendant_uniform_state_error_l2": 0.0
    }
  ],
  "two_magnon_cases": [
    {
      "N": 4,
      "mode": 1,
      "k1": 2.0943951023931953,
      "k2": -2.0943951023931953,
      "S12_real": -0.49999999999999994,
      "S12_imag": -0.8660254037844389,
      "energy_over_J": 2.9999999999999996,
      "raw_norm_squared": 11.999999999999995,
      "raw_norm_squared_error": 5.329070518200751e-15,
      "normalization_error": 0.0,
      "Bethe_equation_residual_max_abs": 6.497413668604471e-16,
      "seam_residual_max_abs": 3.204937810639274e-16,
      "contact_equation_residual_abs": 9.930136612989092e-16,
      "sector_eigenstate_residual_l2_over_J": 7.152111528976949e-16,
      "tensor_eigenstate_residual_l2_over_J": 6.591583993652515e-16,
      "nearest_eigenvalue_over_J": 2.9999999999999996,
      "nearest_energy_error_over_J": 0.0,
      "matched_eigenspace_dimension": 1,
      "eigenspace_projection_error_l2": 1.1544451141201852e-15,
      "wrong_ratio_residual_l2_over_J": 1.6431676725154984,
      "wrong_ratio_Bethe_residual_max_abs": 1.7320508075688776,
      "missing_closing_bond_residual_l2_over_J": 0.8660254037844384
    },
    {
      "N": 5,
      "mode": 1,
      "k1": 1.5707963267948966,
      "k2": -1.5707963267948966,
      "S12_real": 1.1102230246251568e-16,
      "S12_imag": -1.0,
      "energy_over_J": 1.9999999999999998,
      "raw_norm_squared": 20.000000000000004,
      "raw_norm_squared_error": 3.552713678800501e-15,
      "normalization_error": 1.1102230246251565e-16,
      "Bethe_equation_residual_max_abs": 1.9513939732432264e-16,
      "seam_residual_max_abs": 5.551115123125783e-17,
      "contact_equation_residual_abs": 0.0,
      "sector_eigenstate_residual_l2_over_J": 3.3766115072321297e-16,
      "tensor_eigenstate_residual_l2_over_J": 3.3766115072321297e-16,
      "nearest_eigenvalue_over_J": 1.9999999999999996,
      "nearest_energy_error_over_J": 2.220446049250313e-16,
      "matched_eigenspace_dimension": 1,
      "eigenspace_projection_error_l2": 2.310891289047784e-15,
      "wrong_ratio_residual_l2_over_J": 1.3416407864998736,
      "wrong_ratio_Bethe_residual_max_abs": 2.0,
      "missing_closing_bond_residual_l2_over_J": 0.6324555320336757
    },
    {
      "N": 6,
      "mode": 1,
      "k1": 1.2566370614359172,
      "k2": -1.2566370614359172,
      "S12_real": 0.3090169943749474,
      "S12_imag": -0.9510565162951535,
      "energy_over_J": 1.381966011250105,
      "raw_norm_squared": 29.99999999999999,
      "raw_norm_squared_error": 1.0658141036401503e-14,
      "normalization_error": 0.0,
      "Bethe_equation_residual_max_abs": 2.7755575615628914e-16,
      "seam_residual_max_abs": 1.0134903240813531e-16,
      "contact_equation_residual_abs": 0.0,
      "sector_eigenstate_residual_l2_over_J": 1.9129210371407775e-16,
      "tensor_eigenstate_residual_l2_over_J": 1.8567214655012615e-16,
      "nearest_eigenvalue_over_J": 1.3819660112501055,
      "nearest_energy_error_over_J": 4.440892098500626e-16,
      "matched_eigenspace_dimension": 1,
      "eigenspace_projection_error_l2": 2.3225712435217586e-15,
      "wrong_ratio_residual_l2_over_J": 1.0454199412300018,
      "wrong_ratio_Bethe_residual_max_abs": 1.902113032590307,
      "missing_closing_bond_residual_l2_over_J": 0.4799246488165452,
      "normalized_coefficients": [
        {
          "positions": [
            0,
            1
          ],
          "real": 0.2389927119922573,
          "imag": -0.17363836914571174
        },
        {
          "positions": [
            0,
            2
          ],
          "real": -0.09128709291752767,
          "imag": 0.06632395526256084
        },
        {
          "positions": [
            0,
            3
          ],
          "real": -0.29541123814945924,
          "imag": 0.21462882776630182
        },
        {
          "positions": [
            0,
            4
          ],
          "real": -0.09128709291752769,
          "imag": 0.06632395526256088
        },
        {
          "positions": [
            0,
            5
          ],
          "real": 0.23899271199225722,
          "imag": -0.17363836914571168
        },
        {
          "positions": [
            1,
            2
          ],
          "real": 0.2389927119922573,
          "imag": -0.17363836914571174
        },
        {
          "positions": [
            1,
            3
          ],
          "real": -0.09128709291752768,
          "imag": 0.06632395526256081
        },
        {
          "positions": [
            1,
            4
          ],
          "real": -0.2954112381494592,
          "imag": 0.21462882776630182
        },
        {
          "positions": [
            1,
            5
          ],
          "real": -0.0912870929175277,
          "imag": 0.06632395526256085
        },
        {
          "positions": [
            2,
            3
          ],
          "real": 0.2389927119922573,
          "imag": -0.17363836914571174
        },
        {
          "positions": [
            2,
            4
          ],
          "real": -0.09128709291752768,
          "imag": 0.06632395526256081
        },
        {
          "positions": [
            2,
            5
          ],
          "real": -0.29541123814945913,
          "imag": 0.21462882776630174
        },
        {
          "positions": [
            3,
            4
          ],
          "real": 0.2389927119922573,
          "imag": -0.17363836914571174
        },
        {
          "positions": [
            3,
            5
          ],
          "real": -0.09128709291752765,
          "imag": 0.06632395526256084
        },
        {
          "positions": [
            4,
            5
          ],
          "real": 0.23899271199225727,
          "imag": -0.17363836914571174
        }
      ]
    },
    {
      "N": 6,
      "mode": 2,
      "k1": 2.5132741228718345,
      "k2": -2.5132741228718345,
      "S12_real": -0.8090169943749475,
      "S12_imag": -0.5877852522924732,
      "energy_over_J": 3.618033988749895,
      "raw_norm_squared": 30.00000000000001,
      "raw_norm_squared_error": 1.0658141036401503e-14,
      "normalization_error": 5.551115123125783e-16,
      "Bethe_equation_residual_max_abs": 8.671119018262734e-16,
      "seam_residual_max_abs": 3.3978387307159452e-16,
      "contact_equation_residual_abs": 4.440892098500626e-16,
      "sector_eigenstate_residual_l2_over_J": 7.338187872020388e-16,
      "tensor_eigenstate_residual_l2_over_J": 6.899926682599784e-16,
      "nearest_eigenvalue_over_J": 3.6180339887498962,
      "nearest_energy_error_over_J": 1.3322676295501878e-15,
      "matched_eigenspace_dimension": 1,
      "eigenspace_projection_error_l2": 1.3608018377149287e-15,
      "wrong_ratio_residual_l2_over_J": 1.032295370446131,
      "wrong_ratio_Bethe_residual_max_abs": 1.1755705045849472,
      "missing_closing_bond_residual_l2_over_J": 0.776534393824027,
      "normalized_coefficients": [
        {
          "positions": [
            0,
            1
          ],
          "real": 0.034868566760325785,
          "imag": -0.10731441388315091
        },
        {
          "positions": [
            0,
            2
          ],
          "real": -0.0912870929175277,
          "imag": 0.2809527830288626
        },
        {
          "positions": [
            0,
            3
          ],
          "real": 0.1128370523144038,
          "imag": -0.3472767382914234
        },
        {
          "positions": [
            0,
            4
          ],
          "real": -0.09128709291752764,
          "imag": 0.28095278302886245
        },
        {
          "positions": [
            0,
            5
          ],
          "real": 0.03486856676032568,
          "imag": -0.10731441388315063
        },
        {
          "positions": [
            1,
            2
          ],
          "real": 0.034868566760325785,
          "imag": -0.1073144138831509
        },
        {
          "positions": [
            1,
            3
          ],
          "real": -0.0912870929175277,
          "imag": 0.2809527830288626
        },
        {
          "positions": [
            1,
            4
          ],
          "real": 0.1128370523144038,
          "imag": -0.3472767382914234
        },
        {
          "positions": [
            1,
            5
          ],
          "real": -0.09128709291752762,
          "imag": 0.28095278302886245
        },
        {
          "positions": [
            2,
            3
          ],
          "real": 0.034868566760325785,
          "imag": -0.1073144138831509
        },
        {
          "positions": [
            2,
            4
          ],
          "real": -0.09128709291752772,
          "imag": 0.2809527830288626
        },
        {
          "positions": [
            2,
            5
          ],
          "real": 0.1128370523144038,
          "imag": -0.3472767382914234
        },
        {
          "positions": [
            3,
            4
          ],
          "real": 0.0348685667603258,
          "imag": -0.10731441388315094
        },
        {
          "positions": [
            3,
            5
          ],
          "real": -0.0912870929175277,
          "imag": 0.2809527830288626
        },
        {
          "positions": [
            4,
            5
          ],
          "real": 0.034868566760325785,
          "imag": -0.10731441388315091
        }
      ]
    },
    {
      "N": 7,
      "mode": 1,
      "k1": 1.0471975511965976,
      "k2": -1.0471975511965976,
      "S12_real": 0.5000000000000001,
      "S12_imag": -0.8660254037844386,
      "energy_over_J": 0.9999999999999998,
      "raw_norm_squared": 42.000000000000014,
      "raw_norm_squared_error": 1.4210854715202004e-14,
      "normalization_error": 0.0,
      "Bethe_equation_residual_max_abs": 8.455206652451151e-16,
      "seam_residual_max_abs": 3.1448543398982914e-16,
      "contact_equation_residual_abs": 0.0,
      "sector_eigenstate_residual_l2_over_J": 4.304327236548564e-16,
      "tensor_eigenstate_residual_l2_over_J": 4.4016958211282834e-16,
      "nearest_eigenvalue_over_J": 1.0000000000000007,
      "nearest_energy_error_over_J": 8.881784197001252e-16,
      "matched_eigenspace_dimension": 1,
      "eigenspace_projection_error_l2": 8.306612014974488e-15,
      "wrong_ratio_residual_l2_over_J": 0.8416254115301729,
      "wrong_ratio_Bethe_residual_max_abs": 1.732050807568877,
      "missing_closing_bond_residual_l2_over_J": 0.3779644730092272
    },
    {
      "N": 7,
      "mode": 2,
      "k1": 2.0943951023931953,
      "k2": -2.0943951023931953,
      "S12_real": -0.49999999999999994,
      "S12_imag": -0.8660254037844389,
      "energy_over_J": 2.9999999999999996,
      "raw_norm_squared": 41.99999999999998,
      "raw_norm_squared_error": 2.1316282072803006e-14,
      "normalization_error": 2.220446049250313e-16,
      "Bethe_equation_residual_max_abs": 1.1322097734007351e-15,
      "seam_residual_max_abs": 3.2152228933423613e-16,
      "contact_equation_residual_abs": 9.930136612989092e-16,
      "sector_eigenstate_residual_l2_over_J": 7.751710408992441e-16,
      "tensor_eigenstate_residual_l2_over_J": 8.233634315563857e-16,
      "nearest_eigenvalue_over_J": 3.000000000000001,
      "nearest_energy_error_over_J": 1.3322676295501878e-15,
      "matched_eigenspace_dimension": 1,
      "eigenspace_projection_error_l2": 1.842612791380831e-15,
      "wrong_ratio_residual_l2_over_J": 1.26243811729526,
      "wrong_ratio_Bethe_residual_max_abs": 1.7320508075688776,
      "missing_closing_bond_residual_l2_over_J": 0.6546536707079769
    },
    {
      "N": 8,
      "mode": 1,
      "k1": 0.8975979010256552,
      "k2": -0.8975979010256552,
      "S12_real": 0.6234898018587337,
      "S12_imag": -0.7818314824680298,
      "energy_over_J": 0.7530203962825328,
      "raw_norm_squared": 56.00000000000005,
      "raw_norm_squared_error": 4.973799150320701e-14,
      "normalization_error": 0.0,
      "Bethe_equation_residual_max_abs": 6.753223014464259e-16,
      "seam_residual_max_abs": 1.8766194259190266e-16,
      "contact_equation_residual_abs": 2.220446049250313e-16,
      "sector_eigenstate_residual_l2_over_J": 3.315092418796143e-16,
      "tensor_eigenstate_residual_l2_over_J": 3.384089379997609e-16,
      "nearest_eigenvalue_over_J": 0.7530203962825324,
      "nearest_energy_error_over_J": 4.440892098500626e-16,
      "matched_eigenspace_dimension": 1,
      "eigenspace_projection_error_l2": 3.5848773107188965e-15,
      "wrong_ratio_residual_l2_over_J": 0.6922057961174433,
      "wrong_ratio_Bethe_residual_max_abs": 1.5636629649360596,
      "missing_closing_bond_residual_l2_over_J": 0.30680213417660024
    },
    {
      "N": 8,
      "mode": 2,
      "k1": 1.7951958020513104,
      "k2": -1.7951958020513104,
      "S12_real": -0.22252093395631434,
      "S12_imag": -0.9749279121818236,
      "energy_over_J": 2.4450418679126287,
      "raw_norm_squared": 56.00000000000003,
      "raw_norm_squared_error": 2.842170943040401e-14,
      "normalization_error": 0.0,
      "Bethe_equation_residual_max_abs": 6.087272963323595e-16,
      "seam_residual_max_abs": 1.483597921805437e-16,
      "contact_equation_residual_abs": 2.220446049250313e-16,
      "sector_eigenstate_residual_l2_over_J": 3.885780586188048e-16,
      "tensor_eigenstate_residual_l2_over_J": 3.9448084808579573e-16,
      "nearest_eigenvalue_over_J": 2.445041867912629,
      "nearest_energy_error_over_J": 4.440892098500626e-16,
      "matched_eigenspace_dimension": 1,
      "eigenspace_projection_error_l2": 1.980605743418802e-14,
      "wrong_ratio_residual_l2_over_J": 1.206250444563433,
      "wrong_ratio_Bethe_residual_max_abs": 1.9498558243636477,
      "missing_closing_bond_residual_l2_over_J": 0.5528383429982752
    },
    {
      "N": 8,
      "mode": 3,
      "k1": 2.6927937030769655,
      "k2": -2.6927937030769655,
      "S12_real": -0.900968867902419,
      "S12_imag": -0.43388373911755823,
      "energy_over_J": 3.801937735804838,
      "raw_norm_squared": 55.99999999999996,
      "raw_norm_squared_error": 4.263256414560601e-14,
      "normalization_error": 2.220446049250313e-16,
      "Bethe_equation_residual_max_abs": 1.0990647210786425e-15,
      "seam_residual_max_abs": 3.025957901444652e-16,
      "contact_equation_residual_abs": 1.1102230246251565e-16,
      "sector_eigenstate_residual_l2_over_J": 6.413129810193258e-16,
      "tensor_eigenstate_residual_l2_over_J": 7.421048898892461e-16,
      "nearest_eigenvalue_over_J": 3.8019377358048385,
      "nearest_energy_error_over_J": 4.440892098500626e-16,
      "matched_eigenspace_dimension": 1,
      "eigenspace_projection_error_l2": 1.5081831956208128e-15,
      "wrong_ratio_residual_l2_over_J": 0.675786627647446,
      "wrong_ratio_Bethe_residual_max_abs": 0.8677674782351172,
      "missing_closing_bond_residual_l2_over_J": 0.6893781378718105
    }
  ],
  "regularity_examples": {
    "z1_z2_equal_one_singular_ratio_rejected": true,
    "coincident_minus_one_N5": {
      "S12_real": -1.0,
      "Bethe_equation_residual_max_abs": 0.0,
      "raw_norm_squared": 0.0,
      "normalization_rejected": true
    }
  },
  "summary": {
    "chain_count": 5,
    "selected_state_count": 9,
    "maximum_tensor_sector_block_difference_over_J": 0.0,
    "maximum_one_magnon_spectrum_error_over_J": 8.881784197001252e-16,
    "maximum_Bethe_equation_residual": 1.1322097734007351e-15,
    "maximum_two_magnon_eigenstate_residual_over_J": 8.233634315563857e-16,
    "maximum_nearest_energy_error_over_J": 1.3322676295501878e-15,
    "minimum_wrong_ratio_residual_over_J": 0.675786627647446,
    "minimum_missing_bond_residual_over_J": 0.30680213417660024
  }
}
