{
  "schema_version": 1,
  "scope": "Independent bounded normalization/algebra controls only; source assumptions are conditional, no ODE/PIC rerun.",
  "source_arxiv_id": "2508.13081v1",
  "source_journal_doi": "10.1017/S0022377825100949",
  "source_journal_pdf_sha256": "1e47de8f1bbdb554c26efe029b13127cdbf4464fe6599721b64129f86b22150e",
  "script_sha256": "7db46d922e8f8c83b0d68ccf7302e74e8fa29f56884b098cf75f550e59cb649a",
  "runtime": {
    "python": "3.13.7",
    "implementation": "CPython",
    "dependencies": "Python standard library only"
  },
  "cooling_clock_cases": [
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      "a": 2.0,
      "b": 0.5,
      "Cbar": 2.0,
      "rho": 3.0,
      "pressure": 4.0,
      "Gamma_y0": 0.5,
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      "tau_pressure_over_radiated_power": 0.12830005981991685,
      "tau_out": 1.0,
      "R_from_displayed_times": 5.196152422706632,
      "R_printed_2_35": 5.196152422706632,
      "printed_to_displayed_time_ratio": 1.0,
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      "R_printed_2_35": 10.392304845413264,
      "printed_to_displayed_time_ratio": 4.0,
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      "Gamma_y0": 0.5,
      "tau_internal_energy": 0.47698969370175187,
      "tau_pressure_over_radiated_power": 0.3179931291345013,
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      "R_from_displayed_times": 4.192962712629475,
      "R_printed_2_35": 1.0482406781573688,
      "printed_to_displayed_time_ratio": 0.25,
      "R_using_pressure_over_power_convention": 6.289444068944212
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      "Cbar": 2.0,
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      "Gamma_y0": 2.0,
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      "tau_pressure_over_radiated_power": 0.3179931291345013,
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      "R_printed_2_35": 4.192962712629475,
      "printed_to_displayed_time_ratio": 4.0,
      "R_using_pressure_over_power_convention": 1.572361017236053
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    {
      "a": 3.0,
      "b": -0.5,
      "Cbar": 2.0,
      "rho": 3.0,
      "pressure": 4.0,
      "Gamma_y0": 0.5,
      "tau_internal_energy": 0.08553337321327789,
      "tau_pressure_over_radiated_power": 0.057022248808851934,
      "tau_out": 2.0,
      "R_from_displayed_times": 23.382685902179844,
      "R_printed_2_35": 5.845671475544961,
      "printed_to_displayed_time_ratio": 0.25,
      "R_using_pressure_over_power_convention": 35.07402885326976
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      "rho": 3.0,
      "pressure": 4.0,
      "Gamma_y0": 1.0,
      "tau_internal_energy": 0.08553337321327789,
      "tau_pressure_over_radiated_power": 0.057022248808851934,
      "tau_out": 1.0,
      "R_from_displayed_times": 11.691342951089922,
      "R_printed_2_35": 11.691342951089922,
      "printed_to_displayed_time_ratio": 1.0,
      "R_using_pressure_over_power_convention": 17.53701442663488
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      "a": 3.0,
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      "pressure": 4.0,
      "Gamma_y0": 2.0,
      "tau_internal_energy": 0.08553337321327789,
      "tau_pressure_over_radiated_power": 0.057022248808851934,
      "tau_out": 0.5,
      "R_from_displayed_times": 5.845671475544961,
      "R_printed_2_35": 23.382685902179844,
      "printed_to_displayed_time_ratio": 4.0,
      "R_using_pressure_over_power_convention": 8.76850721331744
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  ],
  "tilde_definition_sign_cases": [
    {
      "b": 0.0,
      "b_rational": "0",
      "exponent": 1.0,
      "exponent_rational": "1",
      "base_from_2_27": 1.3333333333333335,
      "base_from_2_36_after_own_2_35": -1.3333333333333335,
      "tilde_from_2_27": 1.3333333333333335,
      "tilde_from_2_36_after_own_2_35": -1.3333333333333335,
      "ratio": -1.0,
      "real_branch_note": "Integer power or explicitly chosen real odd-root branch."
    },
    {
      "b": 0.5,
      "b_rational": "1/2",
      "exponent": 2.0,
      "exponent_rational": "2",
      "base_from_2_27": 0.6666666666666667,
      "base_from_2_36_after_own_2_35": -0.6666666666666667,
      "tilde_from_2_27": 0.44444444444444453,
      "tilde_from_2_36_after_own_2_35": 0.44444444444444453,
      "ratio": 1.0,
      "real_branch_note": "Integer power or explicitly chosen real odd-root branch."
    },
    {
      "b": 0.3333333333333333,
      "b_rational": "1/3",
      "exponent": 1.5,
      "exponent_rational": "3/2",
      "base_from_2_27": 0.888888888888889,
      "base_from_2_36_after_own_2_35": -0.888888888888889,
      "tilde_from_2_27": 0.8380524814062786,
      "tilde_from_2_36_after_own_2_35": null,
      "real_branch_note": "Negative base with an even-denominator reduced rational exponent: no real power value. This is not replaced by a complex coefficient."
    },
    {
      "b": -0.5,
      "b_rational": "-1/2",
      "exponent": 0.6666666666666666,
      "exponent_rational": "2/3",
      "base_from_2_27": 2.0,
      "base_from_2_36_after_own_2_35": -2.0,
      "tilde_from_2_27": 1.5874010519681994,
      "tilde_from_2_36_after_own_2_35": 1.5874010519681994,
      "ratio": 1.0,
      "real_branch_note": "Integer power or explicitly chosen real odd-root branch."
    },
    {
      "b": 0.6666666666666666,
      "b_rational": "2/3",
      "exponent": 3.0,
      "exponent_rational": "3",
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      "base_from_2_36_after_own_2_35": -0.4444444444444445,
      "tilde_from_2_27": 0.08779149519890263,
      "tilde_from_2_36_after_own_2_35": -0.08779149519890263,
      "ratio": -1.0,
      "real_branch_note": "Integer power or explicitly chosen real odd-root branch."
    },
    {
      "b": 0.75,
      "b_rational": "3/4",
      "exponent": 4.0,
      "exponent_rational": "4",
      "base_from_2_27": 0.33333333333333337,
      "base_from_2_36_after_own_2_35": -0.33333333333333337,
      "tilde_from_2_27": 0.012345679012345685,
      "tilde_from_2_36_after_own_2_35": 0.012345679012345685,
      "ratio": 1.0,
      "real_branch_note": "Integer power or explicitly chosen real odd-root branch."
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  ],
  "sound_Alfven_normalization": {
    "assumed_beta": "8*pi*P/B^2",
    "gamma": 1.6666666666666667,
    "beta": 2.0,
    "cs_over_VA": 1.2909944487358056,
    "gamma_cs_over_VA": 2.151657414559676,
    "printed_sqrt_gamma_beta": 1.8257418583505538,
    "exact_to_printed_ratio": 1.1785113019775793,
    "interpretation": "Order-one coefficient/convention check, not a refutation of the beta-half-power scaling."
  },
  "steady_state_scaling_positive_control": {
    "cases": 24,
    "all_identities_pass": true,
    "retained_assumptions": [
      "Nearly uniform layer density, A=rhoX/rhoin~rhoout/rhoin",
      "Alfvenic outflow with its order-one factor suppressed",
      "Continuity and Ohm-law balances",
      "Local ohmic-radiative power balance",
      "Outflow-time/cooling-time closure",
      "Common order-one coefficients as in displayed 4.7/4.9"
    ]
  },
  "domain_cases": [
    {
      "a": 2.0,
      "b": 0.5,
      "beta_in": 1e-06,
      "calR": 100.0,
      "S_system": 100000000.0,
      "kappa": 0.3098386676965934,
      "A": 1.000001,
      "L_over_Lsystem": 3098.3835785823553,
      "vin_over_VA": 1.7965223563359808e-06,
      "delta_over_Lsystem": 0.00556630980111768,
      "delta_over_L": 1.7965205598154208e-06,
      "rate_over_classical_system_SP": 0.01796522356335981,
      "fits_system": false,
      "sub_Alfvenic_inflow": true,
      "aspect_ratio_below_0_1": true,
      "brems_domain_bounds": {
        "calR_for_L_below_system": 309838.3578582356,
        "calR_for_rate_above_system_SP": 309838.04802018753,
        "calR_at_delta_equals_L": 30983866769659.34,
        "calR_at_vin_equals_VA": 30983804802018.754
      }
    },
    {
      "a": 2.0,
      "b": 0.5,
      "beta_in": 1.0,
      "calR": 100.0,
      "S_system": 100000000.0,
      "kappa": 0.3098386676965934,
      "A": 2.0,
      "L_over_Lsystem": 0.001549193338482967,
      "vin_over_VA": 0.003593041119630842,
      "delta_over_Lsystem": 2.7831576837137407e-06,
      "delta_over_L": 0.0017965205598154208,
      "rate_over_classical_system_SP": 35.93041119630842,
      "fits_system": true,
      "sub_Alfvenic_inflow": true,
      "aspect_ratio_below_0_1": true,
      "brems_domain_bounds": {
        "calR_for_L_below_system": 0.1549193338482967,
        "calR_for_rate_above_system_SP": 0.07745966692414835,
        "calR_at_delta_equals_L": 30983866.76965934,
        "calR_at_vin_equals_VA": 7745966.692414835
      }
    },
    {
      "a": 2.0,
      "b": 0.5,
      "beta_in": 1.0,
      "calR": 10000000000.0,
      "S_system": 1000000.0,
      "kappa": 0.3098386676965934,
      "A": 2.0,
      "L_over_Lsystem": 1.549193338482967e-11,
      "vin_over_VA": 359.3041119630842,
      "delta_over_Lsystem": 2.7831576837137404e-09,
      "delta_over_L": 179.65205598154208,
      "rate_over_classical_system_SP": 359304.1119630842,
      "fits_system": true,
      "sub_Alfvenic_inflow": false,
      "aspect_ratio_below_0_1": false,
      "brems_domain_bounds": {
        "calR_for_L_below_system": 0.1549193338482967,
        "calR_for_rate_above_system_SP": 0.07745966692414835,
        "calR_at_delta_equals_L": 309838.6676965934,
        "calR_at_vin_equals_VA": 77459.66692414835
      }
    },
    {
      "a": 2.0,
      "b": 0.5,
      "beta_in": 100.0,
      "calR": 10.0,
      "S_system": 10000000.0,
      "kappa": 0.3098386676965934,
      "A": 101.0,
      "L_over_Lsystem": 3.0677095811543906e-06,
      "vin_over_VA": 1.814485765413575,
      "delta_over_Lsystem": 5.511203334086615e-08,
      "delta_over_L": 0.017965205598154205,
      "rate_over_classical_system_SP": 5737.9078006608715,
      "fits_system": true,
      "sub_Alfvenic_inflow": false,
      "aspect_ratio_below_0_1": true,
      "brems_domain_bounds": {
        "calR_for_L_below_system": 3.0677095811543904e-05,
        "calR_for_rate_above_system_SP": 3.0373362189647425e-07,
        "calR_at_delta_equals_L": 30983.866769659344,
        "calR_at_vin_equals_VA": 3.037336218964743
      }
    }
  ],
  "exclusions": [
    "No claim that the posted figures use incorrect Gamma or units",
    "No universal invalidity claim from an unspecified temperature unit",
    "No physical prediction for formal cases outside sheet assumptions",
    "A positive stationary scaling is not a thermal stability proof",
    "No manuscript, publication, credentials or original source mutation"
  ]
}