{
  "slug": "2026-09-10-hidden-dissipation",
  "hidden_waiting": {
    "base": {
      "clock_factors": [
        1.0,
        2.0,
        1.0,
        0.5
      ],
      "clockwise": 9.0,
      "counterclockwise": 1.0,
      "stationary_probabilities": [
        0.22222222222222224,
        0.1111111111111111,
        0.2222222222222222,
        0.4444444444444444
      ],
      "full_entropy_rate": 15.624708105502005,
      "visible_edge_current_max": 0.0,
      "waiting_KL_AB_BA": 0.5983187366808262,
      "waiting_KL_BA_AB": 1.0782990047277865,
      "crossing_AB_rate": 1.8222222222222222,
      "crossing_BA_rate": 1.8222222222222222,
      "observable_path_entropy_rate": 3.0551701065668055,
      "mean_H_wait_AB": 0.05182926829268293,
      "mean_H_wait_BA": 0.19817073170731708
    },
    "equal_hidden_clocks": {
      "clock_factors": [
        1.0,
        1.0,
        1.0,
        1.0
      ],
      "clockwise": 9.0,
      "counterclockwise": 1.0,
      "stationary_probabilities": [
        0.25,
        0.25,
        0.25,
        0.25
      ],
      "full_entropy_rate": 17.577796618689757,
      "visible_edge_current_max": 0.0,
      "waiting_KL_AB_BA": 0.0,
      "waiting_KL_BA_AB": 0.0,
      "crossing_AB_rate": 2.05,
      "crossing_BA_rate": 2.05,
      "observable_path_entropy_rate": 0.0,
      "mean_H_wait_AB": 0.1,
      "mean_H_wait_BA": 0.1
    },
    "equilibrium_control": {
      "clock_factors": [
        1.0,
        2.0,
        1.0,
        0.5
      ],
      "clockwise": 1.0,
      "counterclockwise": 1.0,
      "stationary_probabilities": [
        0.22222222222222227,
        0.11111111111111112,
        0.2222222222222222,
        0.44444444444444453
      ],
      "full_entropy_rate": 2.465190328815662e-32,
      "visible_edge_current_max": 1.1102230246251565e-16,
      "waiting_KL_AB_BA": 0.0,
      "waiting_KL_BA_AB": 0.0,
      "crossing_AB_rate": 0.22222222222222227,
      "crossing_BA_rate": 0.2222222222222222,
      "observable_path_entropy_rate": 0.0,
      "mean_H_wait_AB": 0.625,
      "mean_H_wait_BA": 0.625
    },
    "control_path_checks": [
      {
        "options": {
          "plus": 1.0,
          "minus": 1.0
        },
        "duration": 100.0,
        "sample_observed_logratio": 1.1368683772161603e-13,
        "full_entropy_rate": 2.465190328815662e-32
      },
      {
        "options": {
          "clocks": [
            1.0,
            1.0,
            1.0,
            1.0
          ]
        },
        "duration": 100.0,
        "sample_observed_logratio": 2.0463630789890885e-12,
        "full_entropy_rate": 17.577796618689757
      }
    ],
    "independent_path_filtering": {
      "replicates": 12,
      "duration_each": 400.0,
      "mean_observed_rate": 3.04172229504436,
      "standard_error_across_independent_runs": 0.03616790303990032
    },
    "limits": "Chosen4state model with instantaneous mapping and second-order semi-Markov observed dynamics. Exact filtered forward/reverse path probabilities check the waiting-time expression. Zero visible flux or zero time-reversal KLD is not proof of equilibrium. No experiment fitted."
  },
  "equilibrium_preprocessing": {
    "KL_psi_x_psi_y": 0.28239262052223957,
    "KL_psi_y_psi_x": 0.4729671218185052,
    "mean_exit_time": 0.5,
    "each_direction_probability": 0.25,
    "naive_last_seen_KLD_rate_with_direction_weights": 0.3776798711703724,
    "physical_entropy_rate_of_underlying_equilibrium_model": 2.234078735489194e-31,
    "finite_equilibrium_realization_states": 18,
    "source_numeric_discrepancy": "Comment displays~1.5 for its sum; with its four probabilities1/4 and mean time1/2, explicitly summing its written expression gives0.5*(KLxy+KLyx), the value reported here. No cause inferred; positivity counterexample unaffected.",
    "timeline": {
      "raw": [
        0,
        3,
        3,
        1,
        3,
        3,
        3,
        0
      ],
      "process_reversed": [
        0,
        0,
        0,
        0,
        1,
        1,
        1,
        0
      ],
      "reverse_processed": [
        0,
        1,
        1,
        1,
        1,
        0,
        0,
        0
      ]
    },
    "limits": "Do not interpret the positive naive KLD as physical entropy. Last-visited-state map depends on history and does not commute with reversal; a pointwise lumping does."
  },
  "OU": {
    "parameters": {
      "lambda": 1.0,
      "omega": 3.0,
      "D": 1.0,
      "dt": 0.002,
      "samples_per_set": 250000,
      "training_seed": 42101,
      "independent_test_seed": 42102
    },
    "continuous_entropy_rate": 18.0,
    "population_at_sampling_interval": [
      17.963808432740375,
      17.646804615183182
    ],
    "scalar_gaussian_seven_time_reversal_covariance_check": true,
    "fitted_linear_field_weights": [
      0.0360429032574412,
      3.003259760727107,
      -2.9485855151277454,
      -0.025874999554029523
    ],
    "true_short_time_weights": [
      0.0,
      3.0,
      -3.0,
      0.0
    ],
    "training_TUR_ratio": 18.36439276898756,
    "independent_test_TUR_ratio": 18.06092063148299,
    "independent_test_calibrated_current_mean_rate": 18.26838463577642,
    "half_turn_interval": 1.0471975511965976,
    "half_turn_sampled_entropy_rate": 8.045275398202566e-33,
    "hidden_independent_cycle": [
      {
        "hidden_speed_scale": 0.1,
        "visible_entropy_rate": 0.0,
        "full_entropy_rate": 1.757779661868976
      },
      {
        "hidden_speed_scale": 1.0,
        "visible_entropy_rate": 0.0,
        "full_entropy_rate": 17.57779661868975
      },
      {
        "hidden_speed_scale": 10.0,
        "visible_entropy_rate": 0.0,
        "full_entropy_rate": 175.77796618689754
      }
    ],
    "limits": "OU simulation uses exact Gaussian time steps. Linear basis optimized from training mean/covariance and tested on an independently generated trajectory; not2026NN reproduction. Finite-data objectives are estimates, not guaranteed confidence bounds. Population sampled path KLD and finite-step linear TUR objective differ; both approach full rate at shortdt. Arbitrary sampling intervals need not give monotonic loss because grids are not nested. Only x is a stationary scalar Gaussian process with reversible path law; assuming the full specificOU model could add inference from correlations, which is model information. The independent hidden cycle proves nonidentifiability when additional hidden processes are allowed."
  }
}