{
  "kind": "synthetic mathematical verification; not empirical replication",
  "utc": "2026-11-05T10:58:33.630443+00:00",
  "python": "3.12.14 (main, Aug 25 2026, 14:01:42) [MSC v.1944 64 bit (AMD64)]",
  "platform": "Windows-11-10.0.26300-SP0",
  "units": "total-carbon atom amount units; time in years; isotope delta in per mil",
  "script_sha256": "2bb6636114102157a492fb81930522583cbe12dd7b7ef757690165afd4d1ca0b",
  "conditional_mixing": [
    {
      "delta_n": -13.2,
      "linear_fraction": -0.0,
      "exact_total_amount_fraction": -0.0
    },
    {
      "delta_n": -13,
      "linear_fraction": 0.013333333333333286,
      "exact_total_amount_fraction": 0.013331151182041142
    },
    {
      "delta_n": -8,
      "linear_fraction": 0.25999999999999995,
      "exact_total_amount_fraction": 0.259957448049958
    },
    {
      "delta_n": 0,
      "linear_fraction": 0.4714285714285714,
      "exact_total_amount_fraction": 0.47135141679387044
    },
    {
      "delta_n": 1.6,
      "linear_fraction": 0.49999999999999994,
      "exact_total_amount_fraction": 0.4999181693268356
    }
  ],
  "delta_i_interval_minus13point4_to_minus13point0": {
    "at_delta_n_minus8": [
      0.25,
      0.27
    ],
    "at_delta_n_minus13": [
      0,
      0.02666666666666669
    ],
    "at_delta_n_zero": [
      0.4642857142857143,
      0.4785714285714286
    ]
  },
  "exact_constant_net_signature_curve": {
    "fit": {
      "intercept": -13.200333052183773,
      "slope": 4020.1935059366274,
      "pearson_r": 0.9999999999792032,
      "pearson_r2": 0.9999999999584064
    },
    "intercept_error": -0.0003330521837732192,
    "min_delta": -8.73344364228823,
    "max_delta": -6.499999999999728,
    "chord_error": -0.00033092686523161774,
    "analytical_chord_error": -0.00033092686491780073,
    "maximum_deviation_from_paper_curve": 0.00011030895489483328
  },
  "signed_flux_counterexample": {
    "fossil_input": 10.0,
    "natural_input": 95.0,
    "natural_output": 100.0,
    "fossil_delta": -28.0,
    "natural_input_delta": -20.0,
    "natural_output_delta": -21.14003084422722,
    "total_growth": 5.0,
    "natural_net_flux": -5.0,
    "net_delta": -13.199999999999656,
    "natural_net_equivalent_delta": -42.795156416855164,
    "emission_over_net_growth": 2.0,
    "counterfactual_growth_if_fossil_removed_and_other_fluxes_fixed": -5.0,
    "isotope13_balance_residual": 2.7755575615628914e-17,
    "isotope12_balance_residual": -9.769962616701378e-15
  },
  "origin_tag_counterexample": {
    "atmospheric_fossil_tag_share": 0.05,
    "atmosphere_delta": -8.5,
    "fossil_input": 10,
    "natural_input": 95,
    "total_output": 100,
    "fossil_output": 5,
    "natural_output": 95,
    "fossil_net_growth": 5,
    "natural_net_growth": 0,
    "total_growth": 5,
    "fossil_net_fraction": 1,
    "fossil_tag_and_emission_delta": -28.0,
    "natural_stock_and_output_delta": -7.473451212397286,
    "natural_input_delta": -6.694270952891879,
    "natural_net_isotope13_flux": 0.0008095096651536578,
    "observed_net_delta": -13.199999999999212,
    "statement": "Identical nonfractionating per-atom removal for both provenance tags. Natural net carbon is zero while natural net isotope flux is nonzero; a natural residual delta cannot be assigned."
  },
  "constant_intercept_does_not_fix_f": {
    "start": [
      0.0,
      -13.2,
      -13.2000000000001
    ],
    "end": [
      0.26,
      -7.999999999999999,
      -7.998849692889887
    ],
    "statement": "N signature changes within approximately [-13.2,-8]; intercept fixed; f need not remain zero"
  },
  "near_linear_not_constant_net_signature": {
    "regression": {
      "intercept": -13.201184598642811,
      "slope": 2010.4224704857083,
      "pearson_r": 0.9999793071089909,
      "pearson_r2": 0.9999586146461776
    },
    "delta_perturbation_amplitude": 0.005,
    "net_delta_min": -15.82261823214781,
    "net_delta_max": -10.561011211531412
  },
  "variable_source_OLS_not_flux_mean": {
    "N": [
      1.0,
      2.0,
      3.0
    ],
    "delta": [
      -6.999999999999784,
      -8.500024879654733,
      -15.66781929427663
    ],
    "regression": {
      "intercept": -16.924289193074937,
      "slope": 10.693649130160182,
      "pearson_r": 0.8008911528757451,
      "pearson_r2": 0.64142663875464
    },
    "exact_integrated_net_delta": -20.001105903066694
  },
  "same_operator_two_box_return": {
    "k": 0.2631578947368421,
    "return_rate": 0.02631578947368421,
    "mean_initial_residence": 3.8000000000000003,
    "eigenvalues": [
      0,
      -0.2894736842105263
    ],
    "long_run_atmospheric_fraction": 0.0909090909090909,
    "rows": [
      [
        0,
        1.0,
        0.0,
        1.0
      ],
      [
        1,
        0.7715068128769704,
        0.22849318712302957,
        0.7686205265937358
      ],
      [
        3.8,
        0.3935191669982541,
        0.6064808330017459,
        0.3678794411714424
      ],
      [
        10,
        0.14119416307697313,
        0.8588058369230269,
        0.07196474394172299
      ],
      [
        30,
        0.09106294271685073,
        0.9089370572831492,
        0.00037269996622361624
      ],
      [
        60,
        0.09090911694650752,
        0.9090908830534925,
        1.389052648230847e-07
      ],
      [
        100,
        0.09090909090933463,
        0.9090909090906654,
        3.725613894868347e-12
      ]
    ],
    "statement": "No isotope-specific rates. Eq15 omits ell*B_f; local residence does not determine system perturbation response."
  },
  "two_box_with_slow_deep_loss": {
    "deep_loss_rate": 0.001,
    "eigenvalues": [
      -0.0009088046005139294,
      -0.2895648796100124
    ],
    "timescales": [
      1100.3465425180502,
      3.453457481952941
    ]
  },
  "eq15_linear_ramp": [
    {
      "tau": 3.8,
      "Nf": 38.661851428571424,
      "net_fossil": 0.22582857142857193,
      "f_if_total_growth5": 0.045165714285714385,
      "predicted_delta_under_paper_mixture": -13.677485714285716
    },
    {
      "tau": 50,
      "Nf": 375.91503982274446,
      "net_fossil": 2.8816992035451108,
      "f_if_total_growth5": 0.5763398407090221,
      "predicted_delta_under_paper_mixture": -21.645097610635332
    },
    {
      "tau": 60,
      "Nf": 421.63439661361394,
      "net_fossil": 3.372760056439768,
      "f_if_total_growth5": 0.6745520112879536,
      "predicted_delta_under_paper_mixture": -23.118280169319306
    }
  ],
  "two_monotone_curves": {
    "intercept": -0.13677239245541903,
    "slope": 0.45618199618241495,
    "pearson_r": 0.6613063200874065,
    "pearson_r2": 0.43732604898754734
  },
  "literal_mass_number_ratio_delta_for_atom_delta_minus8point5": 74.125
}