{
  "program": "klein-gordon-wave-packets.py",
  "script_sha256": "9cadce66001c85e85af9cc4a7ceba1223a77280149c2ae9470f5dbf86c43d895",
  "environment": {
    "python": "3.12.14",
    "numpy": "2.3.5",
    "platform": "Windows-11-10.0.26300-SP0",
    "tested_dependencies": "Python 3.12.14; NumPy 2.3.5"
  },
  "units": "c = hbar = 1; 1 spatial dimension; massive pairing i/(2m)",
  "method": "unitary spatial FFT; exact cosine/sine evolution of both Cauchy data",
  "randomness": "none",
  "parameters": {
    "mass": 1.0,
    "length": 128.0,
    "points": 1024,
    "dx": 0.125,
    "nyquist_abs_k": 25.132741228718345,
    "width": 2.0,
    "center": -6.0,
    "momentum": 0.9,
    "final_time": 24.0,
    "time_samples": 5
  },
  "interpretation": {
    "charge": "signed KG pairing, not a pointwise positive probability",
    "shape_moments": "computed with abs(phi)**2; reliable as line moments only before edge contamination",
    "box": "periodic numerical problem; field, derivative and energy edge fractions and sector/energy cutoff fractions are reported",
    "time_sampling": "no time integration; refinement dt only estimates derivatives for a residual diagnostic",
    "validation_scope": "fixed documented self-check fixtures; custom profile parameters require inspecting their edge/cutoff diagnostics",
    "massless": "excluded: the adopted massive pairing divides by m"
  },
  "sector_fixtures": {
    "positive_frequency": {
      "positive_sector_weight": 1.3585731262198564,
      "negative_sector_weight": 1.4575911579057712e-32,
      "signed_charge": 1.3585731262198564
    },
    "independent_cauchy_data": {
      "positive_sector_weight": 0.44236048296681535,
      "negative_sector_weight": 0.4320450558738188,
      "signed_charge": 0.010315427092996543
    },
    "real_zero_derivative": {
      "positive_sector_weight": 0.2574920317278387,
      "negative_sector_weight": 0.2574920317278387,
      "signed_charge": 0.0
    }
  },
  "refinement": {
    "cutoff_errors": [
      0.1022303369846069,
      8.35238967241074e-10,
      4.468624627625218e-15,
      4.290065310564051e-15,
      4.36616141602455e-15
    ],
    "box_errors": [
      0.018164272507370655,
      2.3378574697922075e-15,
      1.412040182944926e-15,
      1.0188999408928376e-15
    ],
    "time_difference_kg_errors": [
      0.001156971007497483,
      0.00028936441770227643,
      7.234871096025067e-05,
      1.8087652353252317e-05
    ],
    "time_difference_continuity_errors": [
      0.010869250659920752,
      0.0027224606189950856,
      0.0006809372667461841,
      0.0001702544543973915
    ],
    "time_integration_error": "not applicable: exact finite-mode evolution"
  },
  "group_velocity_fixture": {
    "weight": "normalized abs(fhat)**2, a field-shape diagnostic",
    "spectral_mean_group_velocity": 0.6498814800334042,
    "group_velocity_at_mean_momentum": 0.6689647316224496,
    "spectral_group_velocity_variance": 0.012483920694784777,
    "measured_centroid": 9.597155520801701,
    "predicted_centroid": 9.5971555208017,
    "measured_variance": 11.190738320196038,
    "predicted_variance": 11.190738320196033
  },
  "nonrelativistic_fixture": {
    "parameters": "m=2, mean k=0.2, sigma_k=0.1; rest phase removed",
    "norm": "flat field L2 norm",
    "rows": [
      {
        "time": 0,
        "mass": 2,
        "mean_momentum": 0.2,
        "momentum_standard_deviation": 0.1,
        "schrodinger_relative_field_error": 0.0,
        "phase_energy_rms_bound": 0.0,
        "quartic_momentum_moment_bound": 0.0,
        "fourth_order_relative_field_error": 0.0,
        "sixth_momentum_moment_bound": 0.0
      },
      {
        "time": 5,
        "mass": 2,
        "mean_momentum": 0.2,
        "momentum_standard_deviation": 0.1,
        "schrodinger_relative_field_error": 0.0006488566523145663,
        "phase_energy_rms_bound": 0.0006488569346724763,
        "quartic_momentum_moment_bound": 0.0006625902798157017,
        "fourth_order_relative_field_error": 1.4716300417820875e-05,
        "sixth_momentum_moment_bound": 1.5224214737288025e-05
      },
      {
        "time": 10,
        "mass": 2,
        "mean_momentum": 0.2,
        "momentum_standard_deviation": 0.1,
        "schrodinger_relative_field_error": 0.001297711610491175,
        "phase_energy_rms_bound": 0.0012977138693449526,
        "quartic_momentum_moment_bound": 0.0013251805596314033,
        "fourth_order_relative_field_error": 2.9432600683700553e-05,
        "sixth_momentum_moment_bound": 3.044842947457605e-05
      },
      {
        "time": 20,
        "mass": 2,
        "mean_momentum": 0.2,
        "momentum_standard_deviation": 0.1,
        "schrodinger_relative_field_error": 0.002595409668164163,
        "phase_energy_rms_bound": 0.0025954277386899052,
        "quartic_momentum_moment_bound": 0.0026503611192628066,
        "fourth_order_relative_field_error": 5.8865200151730394e-05,
        "sixth_momentum_moment_bound": 6.08968589491521e-05
      },
      {
        "time": 40,
        "mass": 2,
        "mean_momentum": 0.2,
        "momentum_standard_deviation": 0.1,
        "schrodinger_relative_field_error": 0.005190710922915588,
        "phase_energy_rms_bound": 0.0051908554773798105,
        "quartic_momentum_moment_bound": 0.005300722238525613,
        "fourth_order_relative_field_error": 0.00011773039057851627,
        "sixth_momentum_moment_bound": 0.0001217937178983042
      }
    ],
    "bounds": "weighted momentum moments include the Gaussian tail; no false hard cutoff"
  },
  "checks": [
    {
      "name": "positive_frequency: FFT Parseval",
      "measured": 1.1102230246251565e-16,
      "absolute_tolerance": 2e-14,
      "passed": true
    },
    {
      "name": "positive_frequency: reconstruct initial field",
      "measured": 3.171618976831175e-16,
      "absolute_tolerance": 2e-14,
      "passed": true
    },
    {
      "name": "positive_frequency: reconstruct initial derivative",
      "measured": 4.268233824645339e-16,
      "absolute_tolerance": 2e-14,
      "passed": true
    },
    {
      "name": "positive_frequency: cosine versus two-branch evolution",
      "measured": 2.536589899602778e-16,
      "absolute_tolerance": 2e-13,
      "passed": true
    },
    {
      "name": "positive_frequency: position charge versus sector difference",
      "measured": 3.268791361166643e-16,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "positive_frequency: positive energy conservation",
      "measured": 2.220446049250313e-16,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "positive_frequency: local continuity including product differentiation",
      "measured": 1.406204231588157e-14,
      "absolute_tolerance": 2e-11,
      "passed": true
    },
    {
      "name": "positive_frequency: Cauchy evolution composition",
      "measured": 3.1284412239191243e-15,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "positive preparation: negative sector weight",
      "measured": 1.0728838439204418e-32,
      "absolute_tolerance": 2e-27,
      "passed": true
    },
    {
      "name": "positive preparation: separately conserved flat field norm",
      "measured": 2.220446049250313e-16,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "independent_cauchy_data: FFT Parseval",
      "measured": 1.1102230246251565e-16,
      "absolute_tolerance": 2e-14,
      "passed": true
    },
    {
      "name": "independent_cauchy_data: reconstruct initial field",
      "measured": 3.171618976831175e-16,
      "absolute_tolerance": 2e-14,
      "passed": true
    },
    {
      "name": "independent_cauchy_data: reconstruct initial derivative",
      "measured": 2.401616614126497e-16,
      "absolute_tolerance": 2e-14,
      "passed": true
    },
    {
      "name": "independent_cauchy_data: cosine versus two-branch evolution",
      "measured": 3.146565978517455e-16,
      "absolute_tolerance": 2e-13,
      "passed": true
    },
    {
      "name": "independent_cauchy_data: position charge versus sector difference",
      "measured": 6.348444602130439e-17,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "independent_cauchy_data: positive energy conservation",
      "measured": 2.220446049250313e-16,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "independent_cauchy_data: local continuity including product differentiation",
      "measured": 7.461464318493794e-15,
      "absolute_tolerance": 2e-11,
      "passed": true
    },
    {
      "name": "independent_cauchy_data: Cauchy evolution composition",
      "measured": 3.1289898876842917e-15,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "real_zero_derivative: FFT Parseval",
      "measured": 0.0,
      "absolute_tolerance": 2e-14,
      "passed": true
    },
    {
      "name": "real_zero_derivative: reconstruct initial field",
      "measured": 3.510257927057845e-16,
      "absolute_tolerance": 2e-14,
      "passed": true
    },
    {
      "name": "real_zero_derivative: reconstruct initial derivative",
      "measured": 0.0,
      "absolute_tolerance": 2e-14,
      "passed": true
    },
    {
      "name": "real_zero_derivative: cosine versus two-branch evolution",
      "measured": 2.1922892352657002e-16,
      "absolute_tolerance": 2e-13,
      "passed": true
    },
    {
      "name": "real_zero_derivative: position charge versus sector difference",
      "measured": 0.0,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "real_zero_derivative: positive energy conservation",
      "measured": 6.661338147750939e-16,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "real_zero_derivative: local continuity including product differentiation",
      "measured": 1.3737112912090724e-14,
      "absolute_tolerance": 2e-11,
      "passed": true
    },
    {
      "name": "real_zero_derivative: Cauchy evolution composition",
      "measured": 2.439194048504503e-15,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "real data: b(k)=conj(a(-k))",
      "measured": 3.035302223861324e-16,
      "absolute_tolerance": 2e-14,
      "passed": true
    },
    {
      "name": "real nonzero solution: charge is zero",
      "measured": 0.0,
      "absolute_tolerance": 2e-14,
      "passed": true
    },
    {
      "name": "real h=0 data contain both sectors",
      "passed": true,
      "evidence": {
        "positive_weight": 0.2574920317278387,
        "negative_weight": 0.2574920317278387
      }
    },
    {
      "name": "real mixed-sector flat field norm need not be conserved",
      "passed": true,
      "evidence": 0.908173478684491
    },
    {
      "name": "plane wave mode 0, frequency sign -1",
      "measured": 0.0,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "plane wave mode 0, frequency sign 1",
      "measured": 0.0,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "plane wave mode 7, frequency sign -1",
      "measured": 7.586921951253025e-15,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "plane wave mode 7, frequency sign 1",
      "measured": 7.58002964000085e-15,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "plane wave mode -11, frequency sign -1",
      "measured": 7.309387269557312e-15,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "plane wave mode -11, frequency sign 1",
      "measured": 7.298562444496008e-15,
      "absolute_tolerance": 3e-13,
      "passed": true
    },
    {
      "name": "negative control: wrong phase sign is detected",
      "passed": true,
      "evidence": 1.6222839596642624
    },
    {
      "name": "cutoff refinement: finest versus 2048-point reference",
      "measured": 4.36616141602455e-15,
      "absolute_tolerance": 2e-11,
      "passed": true
    },
    {
      "name": "cutoff refinement resolves a deliberately coarse grid",
      "passed": true,
      "evidence": [
        0.1022303369846069,
        8.35238967241074e-10,
        4.468624627625218e-15,
        4.290065310564051e-15,
        4.36616141602455e-15
      ]
    },
    {
      "name": "box refinement: largest versus length-512 reference",
      "measured": 1.0188999408928376e-15,
      "absolute_tolerance": 2e-11,
      "passed": true
    },
    {
      "name": "box refinement detects finite-box contamination",
      "passed": true,
      "evidence": [
        0.018164272507370655,
        2.3378574697922075e-15,
        1.412040182944926e-15,
        1.0188999408928376e-15
      ]
    },
    {
      "name": "time-difference KG residual converges quadratically",
      "passed": true,
      "evidence": [
        3.99831816463307,
        3.9995794515434704,
        3.9998950415055794
      ]
    },
    {
      "name": "time-difference finest KG residual",
      "measured": 1.8087652353252317e-05,
      "absolute_tolerance": 3e-05,
      "passed": true
    },
    {
      "name": "time-difference finest continuity residual",
      "measured": 0.0001702544543973915,
      "absolute_tolerance": 0.0003,
      "passed": true
    },
    {
      "name": "group velocity predicts field-shape centroid",
      "measured": 1.7763568394002505e-15,
      "absolute_tolerance": 2e-10,
      "passed": true
    },
    {
      "name": "velocity variance predicts Gaussian field-shape spread",
      "measured": 5.329070518200751e-15,
      "absolute_tolerance": 2e-09,
      "passed": true
    },
    {
      "name": "moment comparison has negligible edge intensity",
      "measured": 2.019277344297852e-30,
      "absolute_tolerance": 1e-15,
      "passed": true
    },
    {
      "name": "all retained massive group velocities are subluminal",
      "passed": true,
      "evidence": 0.9992093668932869
    },
    {
      "name": "low-momentum fixture controls its tail",
      "passed": true,
      "evidence": {
        "mean_k_over_m": 0.1,
        "rms_width_k_over_m": 0.05,
        "weight_outside_abs_k_over_m_0.4": 1.0394817297733642e-09
      }
    },
    {
      "name": "Schrodinger error below phase-energy bound at t=0",
      "measured": 0.0,
      "absolute_tolerance": 3e-14,
      "passed": true
    },
    {
      "name": "phase-energy bound below quartic moment bound at t=0",
      "measured": 0.0,
      "absolute_tolerance": 3e-14,
      "passed": true
    },
    {
      "name": "fourth-order envelope error below sixth moment bound at t=0",
      "measured": 0.0,
      "absolute_tolerance": 3e-14,
      "passed": true
    },
    {
      "name": "Schrodinger error below phase-energy bound at t=5",
      "measured": 0.0,
      "absolute_tolerance": 3e-14,
      "passed": true
    },
    {
      "name": "phase-energy bound below quartic moment bound at t=5",
      "measured": 0.0,
      "absolute_tolerance": 3e-14,
      "passed": true
    },
    {
      "name": "fourth-order envelope error below sixth moment bound at t=5",
      "measured": 0.0,
      "absolute_tolerance": 3e-14,
      "passed": true
    },
    {
      "name": "Schrodinger error below phase-energy bound at t=10",
      "measured": 0.0,
      "absolute_tolerance": 3e-14,
      "passed": true
    },
    {
      "name": "phase-energy bound below quartic moment bound at t=10",
      "measured": 0.0,
      "absolute_tolerance": 3e-14,
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    },
    {
      "name": "fourth-order envelope error below sixth moment bound at t=10",
      "measured": 0.0,
      "absolute_tolerance": 3e-14,
      "passed": true
    },
    {
      "name": "Schrodinger error below phase-energy bound at t=20",
      "measured": 0.0,
      "absolute_tolerance": 3e-14,
      "passed": true
    },
    {
      "name": "phase-energy bound below quartic moment bound at t=20",
      "measured": 0.0,
      "absolute_tolerance": 3e-14,
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    },
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      "name": "fourth-order envelope error below sixth moment bound at t=20",
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    },
    {
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      "absolute_tolerance": 3e-14,
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    },
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      "name": "phase-energy bound below quartic moment bound at t=40",
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      "absolute_tolerance": 3e-14,
      "passed": true
    },
    {
      "name": "fourth-order envelope error below sixth moment bound at t=40",
      "measured": 0.0,
      "absolute_tolerance": 3e-14,
      "passed": true
    }
  ],
  "passed": true,
  "output_files": {
    "profiles": "/data/relativistic-qm/klein-gordon-wave-packets/profiles.csv",
    "diagnostics": "/data/relativistic-qm/klein-gordon-wave-packets/diagnostics.csv",
    "refinement": "/data/relativistic-qm/klein-gordon-wave-packets/refinement.csv",
    "nonrelativistic": "/data/relativistic-qm/klein-gordon-wave-packets/nonrelativistic.csv",
    "summary": "/data/relativistic-qm/klein-gordon-wave-packets/summary.json"
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  "publication": {
    "normalization": "Absolute output locations replaced by site download paths; numerical results and CSV bytes unchanged.",
    "source_run_report_sha256": "efff4f280fce3821c8667de8318bb9cfeeb923cdcce3a268155addf442e55a0e",
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}