{
  "conventions": {
    "RW_tail": "R_N=1+C1/sqrt(N)+C2/N; C1^2=-4 i w; Re C1<0; C2=-3/4-4 i w",
    "Teukolsky_tail": "C2=s-3/4-4 i w; not used in the zero-tail independent checks",
    "backward_relation": "R_(j-1)=-gamma_j/(beta_j+alpha_j R_j)",
    "forward_ratio": "R_j=a_(j+1)/a_j",
    "frequency": "w=GM omega; exp(-i omega t); c=1 with G retained on restoring units",
    "inverted_fraction": "F_v=beta_v+alpha_v R_v+gamma_v/R_(v-1,left), with no left term at v=0",
    "residual_normalization": "absolute sum divided by sum of absolute term magnitudes; zero/nonfinite scale rejected",
    "zero_tail": "R_N=0, equivalently a_(N+1)=0"
  },
  "exact_checks": {
    "RW_Teukolsky_adjacent_products": "0",
    "RW_tail_C1_equation": "0",
    "RW_tail_C2_equation": "0",
    "RW_tail_constant": "0",
    "RW_tail_half_power": "0",
    "Teukolsky_tail_C1_equation": "0",
    "Teukolsky_tail_C2_equation": "0",
    "Teukolsky_tail_constant": "0",
    "Teukolsky_tail_half_power": "0",
    "coefficient_rescaling_alpha": "0",
    "coefficient_rescaling_gamma": "0",
    "forward_ratio_next_row_index": "0",
    "raw_D_alpha_specialization": "0",
    "raw_D_beta_specialization": "0",
    "raw_D_gamma_specialization": "0"
  },
  "exact_negative_controls": {
    "RW_C2_used_for_spin_minus_two_Teukolsky": {
      "rejected": true,
      "residual": "-2"
    },
    "forward_ratio_unshifted_row_index": {
      "rejected": true,
      "residual": "-2/21"
    },
    "forward_tail_used_as_negative_backward_ratio": {
      "rejected": true,
      "residual": "2"
    }
  },
  "limitations": [
    "The two recurrences encode the same Schwarzschild spectral representation. Separate formulas, ratio-versus-denominator evaluation, greater depth and precision test implementation and truncation, not an independent continuum method.",
    "An inversion index rearranges the same spectral equation where its denominators are nonzero; it is not a theorem assigning the nth root. Explicit seeds and disjoint windows select these two cases.",
    "All numerical divisions require nonzero denominators. Exceptional frequencies, algebraically special modes, pure-imaginary/Stokes branch cases and root searches outside these windows are excluded.",
    "A two-term asymptotic tail is not exact. Differences between tails and depths are empirical convergence evidence for these cases, not a universal error bound.",
    "The reference uncertainty indicator is ten times a finite depth drift, floored at 1e-88; it is not a rigorous error bound.",
    "Reference roots are solved at 100 digits and their absolute residual is checked before serialization. Stored reference roots and their recomputed normalized residuals use the declared 90-digit decimals.",
    "Stored non-seam recurrence residuals verify construction and decimal serialization. The seam is independently matched; the far boundary is an explicitly imposed finite zero-tail condition, not the exact infinite endpoint.",
    "A small root-finder residual alone is insufficient. Coarse converged roots failing independent checks remain present and labeled below target accuracy.",
    "The displayed error floor is a conservative plotting convention based on decimal precision and reference drift, not measured physical uncertainty.",
    "No table of externally tabulated frequencies, qnm package, angular solver, Kerr root, or time-domain evolution is used."
  ],
  "numerical_negative_controls": {
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    "Teukolsky_tail_inserted_into_RW": {
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      "rejected": true,
      "rejection_threshold": "0.0100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
    },
    "dominant_tail_branch_selected": {
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    "fundamental_substituted_for_first_overtone": {
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      "rejected": true,
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    "inversion_index_ignored_at_generic_probe": {
      "nonzero_witness": "4.44696739371981020029188017345087213348171736373503123187313533494524274054522084842479798",
      "rejected": true,
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    "perturbed_root_fails_deep_fraction": {
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      "rejected": true,
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    "poisoned_far_boundary": {
      "nonzero_witness": "0.200052938992914746526119819133411693141863952307032041825893461824045972221102225644612424",
      "rejected": true,
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    "poisoned_initial_coefficient": {
      "nonzero_witness": "0.000499750124937531234382808595702148925537231384307846076961519240379810094952523738130933520",
      "rejected": true,
      "rejection_threshold": "0.0000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000"
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    "poisoned_seam_coefficient": {
      "nonzero_witness": "0.000194252008760239000400949429457195435117856083980876798279017380741711136284755858823573935",
      "rejected": true,
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    "solver_residual_alone_false_positive": {
      "nonzero_witness": "0.0000000168065087359429926401803322403924780149332975209136153536799280682196054876062481741068680",
      "rejected": true,
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  },
  "parameters": {
    "cases": [
      {
        "imag_window": [
          "-0.11",
          "-0.07"
        ],
        "inversion": 0,
        "n": 0,
        "name": "fundamental",
        "real_window": [
          "0.36",
          "0.39"
        ],
        "seeds": [
          [
            "0.37",
            "-0.09"
          ],
          [
            "0.38",
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          ]
        ]
      },
      {
        "imag_window": [
          "-0.30",
          "-0.25"
        ],
        "inversion": 1,
        "n": 1,
        "name": "first_overtone",
        "real_window": [
          "0.33",
          "0.36"
        ],
        "seeds": [
          [
            "0.34",
            "-0.27"
          ],
          [
            "0.35",
            "-0.28"
          ]
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    ],
    "coefficient_prefix_last_index": 65,
    "coefficient_seam_index": 12,
    "coefficient_stored_digits": 90,
    "depths": [
      64,
      128,
      256,
      512,
      1024
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    "ell": 2,
    "independent_check_depths": [
      2048,
      4096
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    "independent_check_dps": 100,
    "maxsteps": 50,
    "nonseam_self_consistency_tolerance": "1e-85",
    "precisions": [
      30,
      50,
      80
    ],
    "reference_depths": [
      4096,
      8192
    ],
    "reference_dps": 100,
    "reference_solver_tolerance": "1e-90",
    "solver_tolerance_rule": "10^(-(dps-8)) absolute; explicitly reevaluated at serialized root",
    "spin": -2,
    "tails": [
      "zero",
      "nollert_C1_C2"
    ],
    "target_accuracy_tolerance": "1e-25"
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