{
  "schema": "classical-reconstruction-boyer-circular-mean-drift/v1",
  "status": "CIRCULAR_MEAN_ACTION_ATTRACTOR_ONLY_STOCHASTIC_STABILITY_UNPAID",
  "source": {
    "arxiv": "2603.13448",
    "path": "research/boyer-2026/hydrogen/Hydrogen3.tex",
    "sha256": "d05b076cdc904cf43a71cc4aa92d9557016e96fbfee48a777321e05c34098e57"
  },
  "derivation": {
    "source_equations": [
      "P_gain=(e^2/(M gamma))(omega^3/c^3) J_rad C",
      "P_loss=M c^2 (M c^3/e^2) [e^2/(Jc)]^8/[1-(e^2/(Jc))^2] C",
      "omega=(M gamma c^3/e^2)[e^2/(Jc)]^3",
      "dU/dJ=omega"
    ],
    "simplified_ratio": "P_gain/P_loss = J_rad/J",
    "normalized_variable": "j=J/J_rad",
    "normalized_mean_drift": "dj/ds = 1/j - 1",
    "fixed_point_j": 1.0,
    "root_scan_interval": [
      1.0,
      1.0
    ],
    "meaning_of_s": "positive rescaled time absorbing the unsummed mode-dependent prefactor; not a physical relaxation time"
  },
  "trajectories": [
    {
      "j0": 0.25,
      "j_final": 0.9999999992697857,
      "abs_error_final": 7.302143334442235e-10,
      "monotone_to_one": true
    },
    {
      "j0": 0.5,
      "j_final": 0.9999999993749236,
      "abs_error_final": 6.250764350568261e-10,
      "monotone_to_one": true
    },
    {
      "j0": 0.8,
      "j_final": 0.999999999662494,
      "abs_error_final": 3.375060231292082e-10,
      "monotone_to_one": true
    },
    {
      "j0": 1.2,
      "j_final": 1.0000000005034981,
      "abs_error_final": 5.034981320761744e-10,
      "monotone_to_one": true
    },
    {
      "j0": 1.5,
      "j_final": 1.000000001699134,
      "abs_error_final": 1.6991339446548182e-09,
      "monotone_to_one": true
    },
    {
      "j0": 2.0,
      "j_final": 1.0000000056027964,
      "abs_error_final": 5.602796449011294e-09,
      "monotone_to_one": true
    },
    {
      "j0": 4.0,
      "j_final": 1.0000001241981173,
      "abs_error_final": 1.241981173460971e-07,
      "monotone_to_one": true
    }
  ],
  "unpaid_physics": [
    "No diffusion coefficient or covariance for action increments is derived.",
    "No stochastic evolution equation for radial action/eccentricity is supplied.",
    "The calculation constrains a circular orbit and freezes orbital parameters over a short interval.",
    "Mean restoring drift does not establish recurrence, ergodicity, or survival against eccentric excursions.",
    "No physical relaxation time is claimed from the normalized integration."
  ],
  "claim": "The published circular gain/loss algebra implies a restoring mean action drift toward J=J_rad=hbar from both sides, but only on the constrained circular mean-balance manifold."
}