{
  "compound_identity": {
    "h": [
      0.02,
      0.01,
      0.005,
      0.0025
    ],
    "errors": [
      0.006243278576986646,
      0.0015769036915523503,
      0.00039623812080580946,
      9.93015408996616e-05
    ],
    "loglog_slope_expected_2": 1.9915686739147795
  },
  "predictions": {
    "P1_compound_identity_error_is_O_h2": {
      "observed_slope": 1.9915686739147795,
      "expected": 2.0
    },
    "P2_shortest_path_min_entry_scales_alpha3": {
      "observed_log_slope": 3.000328903279753,
      "expected": 3.0,
      "note": "For the bidirectional 6-cycle, the worst off-diagonal entries require three edges; the leading path term is (alpha*t)^3/3!."
    },
    "P3_stronger_uniform_coupling_improves_projective_contraction": {
      "alphas": [
        0.05,
        0.1,
        0.2,
        0.4,
        0.8
      ],
      "q_bounds": [
        0.9999573853840754,
        0.9996603554483036,
        0.9973244232616174,
        0.9799237269249854,
        0.8752531714166867
      ],
      "q_at_alpha_0.05": 0.9999573853840754,
      "q_at_alpha_0.8": 0.8752531714166867
    }
  },
  "rows": [
    {
      "alpha": 0.05,
      "min_P": 9.58951945520055e-06,
      "q_birkhoff_bound": 0.9999573853840754,
      "max_sampled_ratio": 0.9904368818740336,
      "path_minentry_at_t0.08": 1.9693227495635823e-08,
      "path_prediction_alpha3_t3_over_6": 1.0666666666666668e-08,
      "repeated_window_ratios": [
        0.8944291654498862,
        0.8975874342752008,
        0.9006513184643506,
        0.9035844819373794,
        0.9063929759773381
      ]
    },
    {
      "alpha": 0.1,
      "min_P": 7.680825185932804e-05,
      "q_birkhoff_bound": 0.9996603554483036,
      "max_sampled_ratio": 0.944372279716586,
      "path_minentry_at_t0.08": 1.5754771052248844e-07,
      "path_prediction_alpha3_t3_over_6": 8.533333333333334e-08,
      "repeated_window_ratios": [
        0.6886449585037385,
        0.797978357156317,
        0.8769247795188834,
        0.8890655437792796,
        0.8965605690316094
      ]
    },
    {
      "alpha": 0.2,
      "min_P": 0.0006174201745380509,
      "q_birkhoff_bound": 0.9973244232616174,
      "max_sampled_ratio": 0.8915108767982187,
      "path_minentry_at_t0.08": 1.260442183468729e-06,
      "path_prediction_alpha3_t3_over_6": 6.826666666666667e-07,
      "repeated_window_ratios": [
        0.6805728376856377,
        0.8257056192834158,
        0.8964865403937533,
        0.8791229156627779,
        0.8671507262264395
      ]
    },
    {
      "alpha": 0.4,
      "min_P": 0.0050348059863144735,
      "q_birkhoff_bound": 0.9799237269249854,
      "max_sampled_ratio": 0.8004953597589071,
      "path_minentry_at_t0.08": 1.0085473630854028e-05,
      "path_prediction_alpha3_t3_over_6": 5.461333333333334e-06,
      "repeated_window_ratios": [
        0.7738700322543988,
        0.6876036475171393,
        0.6977015394129747,
        0.7319873434335363,
        0.7234811407745301
      ]
    },
    {
      "alpha": 0.8,
      "min_P": 0.043452021571022015,
      "q_birkhoff_bound": 0.8752531714166867,
      "max_sampled_ratio": 0.6798605371956746,
      "path_minentry_at_t0.08": 8.074577006188443e-05,
      "path_prediction_alpha3_t3_over_6": 4.369066666666667e-05,
      "repeated_window_ratios": [
        0.19163812173118022,
        0.4905163641804326,
        0.5379220870005692,
        0.5286077901331985,
        0.5274295384889923
      ]
    }
  ],
  "baseline_signed_generator": {
    "min_transition_entry": -0.1462517985915507,
    "valid_positive_output_sample_ratio_max": 2.491604424899665,
    "metzler_violation": 0.8
  },
  "interpretation": "Positive irreducible compound dynamics produced strictly positive finite-window transitions and geometric Hilbert contraction. The signed baseline has a negative transition entry, so the positive-cone theorem is unavailable."
}