{
  "schema": "sandboxphysics-ac09-period-analysis-v1",
  "units": "G=M_primary=M_companion=a_inner=1;inner period 2pi; time normalized by tK=hierarchyRatio**3",
  "sampling": "Frozen IAS15 output cadence 0.005 tK; phase-sensitive short-period changes remain in raw Cartesian data",
  "smoothing": "Centered moving average of eccentricity over one outer period;peak groups separated by at least 1 tK;one-half/double window comparison. The large-excursion estimator requires eccentricity range at least 0.2.",
  "scope": "Synthetic dynamical benchmark, not observed exoplanet measurements. Direct references are Newtonian. Conservative GR precession is only added to the averaged model in the separate GR comparison.",
  "cases": [
    {
      "file": "ias15-default.csv",
      "forceModel": "full",
      "provenance": "Derived from saved IAS15 Cartesian positions/velocities and separately integrated quadrupole equations",
      "cadence_tau": 0.005,
      "outer_period_tau": 0.10687915251348867,
      "direct_max_e": 0.8434866491310399,
      "averaged_max_e": 0.8389091210671382,
      "averaged_periapsis_angle_advance_degrees": 407.8257661179557,
      "direct_peak_times_tau": [
        2.51,
        7.5649999999999995
      ],
      "averaged_peak_times_tau": [
        2.514999999999833,
        7.54
      ],
      "direct_period_tau": 5.055,
      "averaged_period_tau": 5.025000000000167,
      "period_range_for_half_to_double_smoothing_window": [
        5.039999999999999,
        5.06
      ],
      "raw_e_residual_rms": 0.012870861502463728,
      "max_axial_momentum_change": 0.01772280633760237,
      "direct_stopped": false,
      "direct_completed_tau": 12.0,
      "averaged_completed_tau": 11.99999999999979,
      "interpretation": "Period is measured from grouped peaks after one-outer-orbit smoothing;window variation is a method sensitivity, not a statistical confidence interval."
    },
    {
      "file": "ias15-stable.csv",
      "forceModel": "full",
      "provenance": "Derived from saved IAS15 Cartesian positions/velocities and separately integrated quadrupole equations",
      "cadence_tau": 0.005,
      "outer_period_tau": 0.10687915251348867,
      "direct_max_e": 0.06669049694520533,
      "averaged_max_e": 0.06696036152129689,
      "averaged_periapsis_angle_advance_degrees": 781.0610284243929,
      "direct_peak_times_tau": [],
      "averaged_peak_times_tau": [],
      "direct_period_tau": null,
      "averaged_period_tau": null,
      "period_range_for_half_to_double_smoothing_window": null,
      "raw_e_residual_rms": 0.008333649134661971,
      "max_axial_momentum_change": 0.0024393062589931347,
      "direct_stopped": false,
      "direct_completed_tau": 12.0,
      "averaged_completed_tau": 11.99999999999979,
      "interpretation": "Small-amplitude stable branch; the large-excursion cycle estimator is not applied."
    },
    {
      "file": "ias15-strong-hierarchy.csv",
      "forceModel": "full",
      "provenance": "Derived from saved IAS15 Cartesian positions/velocities and separately integrated quadrupole equations",
      "cadence_tau": 0.005,
      "outer_period_tau": 0.0496729413289805,
      "direct_max_e": 0.8410404916269149,
      "averaged_max_e": 0.8389091210671399,
      "averaged_periapsis_angle_advance_degrees": 407.8257661179466,
      "direct_peak_times_tau": [
        2.505,
        7.54
      ],
      "averaged_peak_times_tau": [
        2.514999999999833,
        7.54
      ],
      "direct_period_tau": 5.035,
      "averaged_period_tau": 5.025000000000167,
      "period_range_for_half_to_double_smoothing_window": [
        5.025,
        5.035
      ],
      "raw_e_residual_rms": 0.0032401532843077457,
      "max_axial_momentum_change": 0.008046630944436639,
      "direct_stopped": false,
      "direct_completed_tau": 12.0,
      "averaged_completed_tau": 11.999999999999789,
      "interpretation": "Period is measured from grouped peaks after one-outer-orbit smoothing;window variation is a method sensitivity, not a statistical confidence interval."
    },
    {
      "file": "ias15-wide-inclined.csv",
      "forceModel": "full",
      "provenance": "Derived from saved IAS15 Cartesian positions/velocities and separately integrated quadrupole equations",
      "cadence_tau": 0.005,
      "outer_period_tau": 0.0496729413289805,
      "direct_max_e": 0.9445867227019559,
      "averaged_max_e": 0.9427841238554849,
      "averaged_periapsis_angle_advance_degrees": 495.3063738557529,
      "direct_peak_times_tau": [
        2.335,
        7.0,
        11.685
      ],
      "averaged_peak_times_tau": [
        2.334999999999853,
        7.005,
        11.674999999999969
      ],
      "direct_period_tau": 4.675000000000001,
      "averaged_period_tau": 4.6700000000000585,
      "period_range_for_half_to_double_smoothing_window": [
        4.67,
        4.675000000000001
      ],
      "raw_e_residual_rms": 0.003083537992020861,
      "max_axial_momentum_change": 0.01053093410689726,
      "direct_stopped": false,
      "direct_completed_tau": 12.0,
      "averaged_completed_tau": 11.999999999999789,
      "interpretation": "Period is measured from grouped peaks after one-outer-orbit smoothing;window variation is a method sensitivity, not a statistical confidence interval."
    },
    {
      "file": "ias15-compact.csv",
      "forceModel": "full",
      "provenance": "Derived from saved IAS15 Cartesian positions/velocities and separately integrated quadrupole equations",
      "cadence_tau": 0.005,
      "outer_period_tau": 0.3022998940390363,
      "direct_max_e": 0.9530351723814942,
      "averaged_max_e": 0.9427841238554847,
      "averaged_periapsis_angle_advance_degrees": 495.30637385572606,
      "direct_peak_times_tau": [
        2.525,
        7.53
      ],
      "averaged_peak_times_tau": [
        2.334999999999853,
        7.005,
        11.67499999999997
      ],
      "direct_period_tau": 5.005000000000001,
      "averaged_period_tau": 4.6700000000000585,
      "period_range_for_half_to_double_smoothing_window": [
        5.0,
        5.074999999999999
      ],
      "raw_e_residual_rms": 0.23208016113403412,
      "max_axial_momentum_change": 0.06131331335069179,
      "direct_stopped": false,
      "direct_completed_tau": 12.0,
      "averaged_completed_tau": 11.99999999999979,
      "interpretation": "Period is measured from grouped peaks after one-outer-orbit smoothing;window variation is a method sensitivity, not a statistical confidence interval."
    },
    {
      "file": "ias15-near-polar.csv",
      "forceModel": "full",
      "provenance": "Derived from saved IAS15 Cartesian positions/velocities and separately integrated quadrupole equations",
      "cadence_tau": 0.005,
      "outer_period_tau": 0.10687915251348867,
      "direct_max_e": 0.9673640744117187,
      "averaged_max_e": 0.9699089237941216,
      "averaged_periapsis_angle_advance_degrees": 46.82703566288035,
      "direct_peak_times_tau": [],
      "averaged_peak_times_tau": [],
      "direct_period_tau": null,
      "averaged_period_tau": null,
      "period_range_for_half_to_double_smoothing_window": null,
      "raw_e_residual_rms": 0.010539693131059109,
      "max_axial_momentum_change": 0.023720766258074677,
      "direct_stopped": true,
      "direct_completed_tau": 2.18,
      "averaged_completed_tau": 2.1599999999998722,
      "interpretation": "No full secular period is measured from a stopped partial cycle."
    },
    {
      "file": "ias15-quadrupole-compact.csv",
      "forceModel": "instantaneous-quadrupole",
      "provenance": "Derived from saved IAS15 Cartesian positions/velocities and separately integrated quadrupole equations",
      "cadence_tau": 0.005,
      "outer_period_tau": 0.3022998940390363,
      "direct_max_e": 0.9524067828716327,
      "averaged_max_e": 0.9427841238554847,
      "averaged_periapsis_angle_advance_degrees": 495.30637385572606,
      "direct_peak_times_tau": [
        2.4050000000000002,
        7.215
      ],
      "averaged_peak_times_tau": [
        2.334999999999853,
        7.005,
        11.67499999999997
      ],
      "direct_period_tau": 4.81,
      "averaged_period_tau": 4.6700000000000585,
      "period_range_for_half_to_double_smoothing_window": [
        4.744999999999999,
        4.81
      ],
      "raw_e_residual_rms": 0.1037702164603973,
      "max_axial_momentum_change": 0.06348468309929423,
      "direct_stopped": false,
      "direct_completed_tau": 12.0,
      "averaged_completed_tau": 11.99999999999979,
      "interpretation": "Period is measured from grouped peaks after one-outer-orbit smoothing;window variation is a method sensitivity, not a statistical confidence interval."
    },
    {
      "file": "ias15-quadrupole-wide.csv",
      "forceModel": "instantaneous-quadrupole",
      "provenance": "Derived from saved IAS15 Cartesian positions/velocities and separately integrated quadrupole equations",
      "cadence_tau": 0.005,
      "outer_period_tau": 0.0496729413289805,
      "direct_max_e": 0.9445580816659025,
      "averaged_max_e": 0.9427841238554849,
      "averaged_periapsis_angle_advance_degrees": 495.3063738557529,
      "direct_peak_times_tau": [
        2.335,
        7.0,
        11.665
      ],
      "averaged_peak_times_tau": [
        2.334999999999853,
        7.005,
        11.674999999999969
      ],
      "direct_period_tau": 4.664999999999999,
      "averaged_period_tau": 4.6700000000000585,
      "period_range_for_half_to_double_smoothing_window": [
        4.6575,
        4.664999999999999
      ],
      "raw_e_residual_rms": 0.003881953900858192,
      "max_axial_momentum_change": 0.010458955022024319,
      "direct_stopped": false,
      "direct_completed_tau": 12.0,
      "averaged_completed_tau": 11.999999999999789,
      "interpretation": "Period is measured from grouped peaks after one-outer-orbit smoothing;window variation is a method sensitivity, not a statistical confidence interval."
    }
  ],
  "controlled_comparisons": [
    {
      "hierarchyRatio": 6,
      "inclinationDegrees": 75,
      "full_newtonian_period_tau": 5.005000000000001,
      "instantaneous_quadrupole_period_tau": 4.81,
      "doubly_averaged_quadrupole_period_tau": 4.6700000000000585,
      "interpretation": "Instantaneous quadrupole versus double average changes averaging alone;full Newtonian versus instantaneous quadrupole changes multipole truncation. The two period shifts need not add linearly for long nonlinear evolution."
    },
    {
      "hierarchyRatio": 20,
      "inclinationDegrees": 75,
      "full_newtonian_period_tau": 4.675000000000001,
      "instantaneous_quadrupole_period_tau": 4.664999999999999,
      "doubly_averaged_quadrupole_period_tau": 4.6700000000000585,
      "interpretation": "Instantaneous quadrupole versus double average changes averaging alone;full Newtonian versus instantaneous quadrupole changes multipole truncation. The two period shifts need not add linearly for long nonlinear evolution."
    }
  ],
  "averaged_GR_example": {
    "inclinationDegrees": 65,
    "initialEccentricity": 0.05,
    "epsilon_GR": 0.8,
    "max_e": 0.6292782469155712,
    "direct_GR_integration": false
  },
  "limitations": [
    "Finite initial eccentricity 0.05 differs from the initially circular analytic limit",
    "Outer-orbit averaging removes real fast oscillations;compare filtered trends and raw records",
    "The direct inner two-body energy is not conserved; circular rotating Hamiltonian is the relevant invariant",
    "Near-primary exclusion stops partial records; never extend a stopped curve to the requested duration",
    "No dissipative tides or orbital migration. Full Newtonian cases retain the full companion force;instantaneous-quadrupole controls and averaged curves both truncate its expansion at quadrupole order."
  ]
}