{
  "units": "dimensionless; illustrative, not empirical",
  "software": {
    "python": "3.12.14",
    "numpy": "2.5.4",
    "scipy": "1.18.1",
    "sympy": "1.14.0",
    "matplotlib": "3.11.2"
  },
  "symbolic": {
    "potential_first": true,
    "potential_second": true,
    "hamilton_velocity": true,
    "hamilton_chi_source": true,
    "mass_squared_derivative": true,
    "wall_equation": true,
    "wall_first_integral": true
  },
  "symbolic_scope": "Scalar derivatives, Hamiltonian derivatives and wall identities only; tensor variations are derived in the report.",
  "wall": {
    "max_profile_error": 4.6898989691790405e-08,
    "tension": 0.23570226039548708,
    "tension_exact": 0.23570226039551587,
    "tension_abs_error": 2.8782531913407183e-14,
    "spectrum": [
      {
        "N": 600,
        "eigenvalues": [
          -7.595080464765787e-05,
          1.4998533293231047,
          2.0251165921179535,
          2.099198568346904
        ],
        "zero_error": 7.595080464765787e-05,
        "shape_error": 0.00014667067689533297
      },
      {
        "N": 1200,
        "eigenvalues": [
          -1.9016769256164995e-05,
          1.4999636647376864,
          2.0251426843149183,
          2.0992109453349386
        ],
        "zero_error": 1.9016769256164995e-05,
        "shape_error": 3.6335262313613725e-05
      },
      {
        "N": 2400,
        "eigenvalues": [
          -4.757990149833354e-06,
          1.499991292865314,
          2.025149215739022,
          2.0992140447583783
        ],
        "zero_error": 4.757990149833354e-06,
        "shape_error": 8.707134685925766e-06
      }
    ],
    "boundary": "Dirichlet at exact tail values, z in [-12,12]"
  },
  "weak_response": {
    "gaussian_source": {
      "s0": -0.02,
      "width": 1.0
    },
    "refinement": [
      {
        "N": 400,
        "max_sample_error": 4.3631456151001224e-07,
        "peak": 0.007577652281369518
      },
      {
        "N": 800,
        "max_sample_error": 1.094133680662121e-07,
        "peak": 0.0075784536045069364
      },
      {
        "N": 1600,
        "max_sample_error": 2.739142618973789e-08,
        "peak": 0.007578654490420207
      }
    ],
    "mu": 1.4142135623730951,
    "sign": "negative electric-dominated source raises chi for beta>0"
  },
  "cycle": {
    "p0": 0.3,
    "initial_energy": 1.044030650891055,
    "plateau_energy": 0.3,
    "initial_speed": 0.2873478855663454,
    "plateau_speed": 1.0,
    "energy_to_controller_on_entry": 0.744030650891055,
    "convergence": [
      {
        "rtol": 1e-08,
        "max_work_balance_error": 6.656567658192003e-15,
        "x_final": 18.80617912122727,
        "p_final": 0.3
      },
      {
        "rtol": 1e-10,
        "max_work_balance_error": 4.9960036108132044e-15,
        "x_final": 18.80617912122729,
        "p_final": 0.3
      },
      {
        "rtol": 1e-12,
        "max_work_balance_error": 5.5528498466017595e-15,
        "x_final": 18.80617912122727,
        "p_final": 0.3
      }
    ],
    "max_endpoint_error": 0.0
  },
  "mass_well": {
    "initial_energy": 1.1659890372674246,
    "max_energy_error": 6.661338147750939e-16,
    "max_speed": 0.9320403988375316,
    "final_position": 9.31328541032594,
    "final_momentum": 0.5996085731426125,
    "note": "finite start/end positions, not identical asymptotic endpoints"
  },
  "closed_system": {
    "parameters": {
      "box_length": 6.283185307179586,
      "beta": 0.4,
      "sigma": 0.35,
      "probe_m0": [
        0.03,
        0.05
      ],
      "duration": 10.0,
      "boundary": "periodic",
      "method": "Fourier collocation + DOP853"
    },
    "runs": [
      {
        "N": 64,
        "rtol": 1e-08,
        "max_step": 0.03,
        "initial_energy": 0.18397528924717535,
        "initial_total_momentum": 0.10233626104264473,
        "max_relative_energy_error": 6.034631222313473e-16,
        "max_absolute_total_momentum_error": 1.1102230246251565e-16,
        "final_probe_momenta": [
          0.0003900184226365624,
          -0.0005441015636098244
        ],
        "final_em_momentum": 0.10231711880266671,
        "final_scalar_momentum": 0.0001732253809511752,
        "max_abs_chi": 0.05698170758898026,
        "max_particle_vs_field_impulse_residual": 1.1281123604711674e-16,
        "nfev": 5012
      },
      {
        "N": 128,
        "rtol": 1e-10,
        "max_step": 0.03,
        "initial_energy": 0.18397528924717538,
        "initial_total_momentum": 0.10233626104264476,
        "max_relative_energy_error": 6.034631222313472e-16,
        "max_absolute_total_momentum_error": 9.71445146547012e-17,
        "final_probe_momenta": [
          0.0003900184226365597,
          -0.0005441015636098342
        ],
        "final_em_momentum": 0.10231711880266678,
        "final_scalar_momentum": 0.00017322538095118598,
        "max_abs_chi": 0.05698170758898023,
        "max_particle_vs_field_impulse_residual": 1.0267394573437727e-16,
        "nfev": 5012
      },
      {
        "N": 256,
        "rtol": 1e-11,
        "max_step": 0.03,
        "initial_energy": 0.18397528924717535,
        "initial_total_momentum": 0.10233626104264473,
        "max_relative_energy_error": 4.525973416735105e-16,
        "max_absolute_total_momentum_error": 6.938893903907228e-17,
        "final_probe_momenta": [
          0.0003900184226365588,
          -0.000544101563609829
        ],
        "final_em_momentum": 0.10231711880266682,
        "final_scalar_momentum": 0.00017322538095118306,
        "max_abs_chi": 0.05698170758897987,
        "max_particle_vs_field_impulse_residual": 6.418476861114186e-17,
        "nfev": 5012
      }
    ],
    "endpoint_probe_resolution_difference": 5.204170427930421e-18,
    "independent_step_refinement": [
      {
        "N": 64,
        "rtol": 1e-05,
        "max_step": 0.12,
        "initial_energy": 0.18397528924717535,
        "initial_total_momentum": 0.10233626104264473,
        "max_relative_energy_error": 8.840885606469796e-12,
        "max_absolute_total_momentum_error": 6.729200530131152e-13,
        "final_probe_momenta": [
          0.00039001842263326926,
          -0.0005441015636253063
        ],
        "final_em_momentum": 0.10231711880259109,
        "final_scalar_momentum": 0.00017322538096963206,
        "max_abs_chi": 0.056981707589519176,
        "max_particle_vs_field_impulse_residual": 6.729138188506234e-13,
        "nfev": 1262,
        "probe_endpoint_difference_from_main": 1.547731127288232e-14
      },
      {
        "N": 64,
        "rtol": 1e-05,
        "max_step": 0.06,
        "initial_energy": 0.18397528924717535,
        "initial_total_momentum": 0.10233626104264473,
        "max_relative_energy_error": 9.353678394585883e-15,
        "max_absolute_total_momentum_error": 6.800116025829084e-16,
        "final_probe_momenta": [
          0.0003900184226362714,
          -0.0005441015636096116
        ],
        "final_em_momentum": 0.10231711880266656,
        "final_scalar_momentum": 0.00017322538095125348,
        "max_abs_chi": 0.05698170758898347,
        "max_particle_vs_field_impulse_residual": 6.855579743215295e-16,
        "nfev": 2507,
        "probe_endpoint_difference_from_main": 2.8736778581728295e-16
      },
      {
        "N": 64,
        "rtol": 1e-05,
        "max_step": 0.03,
        "initial_energy": 0.18397528924717535,
        "initial_total_momentum": 0.10233626104264473,
        "max_relative_energy_error": 6.034631222313473e-16,
        "max_absolute_total_momentum_error": 1.1102230246251565e-16,
        "final_probe_momenta": [
          0.0003900184226365624,
          -0.0005441015636098244
        ],
        "final_em_momentum": 0.10231711880266671,
        "final_scalar_momentum": 0.0001732253809511752,
        "max_abs_chi": 0.05698170758898026,
        "max_particle_vs_field_impulse_residual": 1.1281123604711674e-16,
        "nfev": 5012,
        "probe_endpoint_difference_from_main": 4.662069341687669e-18
      }
    ]
  },
  "endpoint": {
    "fixed_nonzero_p_limit": 1.0,
    "p_equals_a_delta_squared_limits": {
      "0.1": 0.09950371902099893,
      "1": 0.7071067811865475,
      "10": 0.9950371902099892
    },
    "zero_state": "p=0 and m=0 has zero energy and is not a regular propagating particle state",
    "recovery": "demonstrated for prescribed homogeneous control; not an autonomous device solution",
    "direction_locking": "absent: no direct matter-null-flow interaction in the action"
  },
  "script_sha256": "8a737011cbd43af6392417cee4e32de9d36b91f2a326c7932feb614010f1322e"
}