{
  "schema": "apeiron-gate-specification-v1",
  "version": "1.0",
  "date": "2026-08-30",
  "scope": "frozen v7.13 Stage-0 terminal classification only",
  "authority": {
    "protocol_file": "TECHNICAL_SOURCES/RECOVERY_PROTOCOL_v7_13.json",
    "protocol_sha256": "25697b2dc6e15cfac4ca1393ea5521b31ca120f54710308c370057da98137129",
    "gate_register_file": "TECHNICAL_SOURCES/RECOVERY_GATE_REGISTER_v7_13_STAGE00.json",
    "gate_register_sha256": "1cd8d5008f0a6d599dadc81e36fb32ca86a81dec34576e8935269f1781267245",
    "result_file": "TECHNICAL_SOURCES/STAGE00_HARD_PASS_RESULT_v7_13.json",
    "result_sha256": "030f83d808d00cafbe2eba9d80385ce9dde7b7b6193ebe46f87000a748411c5f"
  },
  "representation": "IEEE-754 binary64",
  "stage_index": 0,
  "tau_exact_protocol": "0.47878124999999994315658113919198513031005859375",
  "tau_binary64": 0.47878124999999994,
  "discretization": {
    "nodes": 15361,
    "finite_difference_step_for_jacobian_vector_product": 1e-05,
    "frozen": true
  },
  "solver_budget": {
    "max_newton_steps": 6,
    "gmres_restart": 10,
    "max_gmres_cycles_per_newton": 16,
    "gmres_rtol": 0.05,
    "gmres_atol": 0.0,
    "line_search_alphas": [
      1.0,
      0.5,
      0.25,
      0.125,
      0.0625,
      0.03125,
      0.015625,
      0.0078125,
      0.00390625,
      0.001953125,
      0.0009765625,
      0.00048828125,
      0.000244140625
    ],
    "frozen": true
  },
  "gates": {
    "fixed_point": {
      "diagnostic": "fixed_point_inf",
      "operator": "<",
      "threshold": 5e-11,
      "observed": 4.6838977141305804e-11,
      "passed": true
    },
    "friedmann": {
      "diagnostic": "friedmann_constraint",
      "operator": "<",
      "threshold": 1e-10,
      "observed": 3.818888298257304e-12,
      "passed": true
    },
    "continuity": {
      "diagnostic": "continuity_residual_over_rho_scale",
      "operator": "<",
      "threshold": 1e-05,
      "observed": 3.7214611922998273e-06,
      "passed": true
    },
    "raychaudhuri": {
      "diagnostic": "raychaudhuri_residual",
      "operator": "<",
      "threshold": 2e-05,
      "observed": 1.4051164250399966e-05,
      "passed": true
    },
    "quantum_source": {
      "diagnostic": "quantum_source_identity",
      "operator": "<",
      "threshold": 1e-07,
      "observed": 3.883856960984437e-08,
      "passed": true
    },
    "wronskian": {
      "diagnostic": "wronskian",
      "operator": "<",
      "threshold": 2e-08,
      "observed": 7.432742701266731e-09,
      "passed": true
    },
    "rho": {
      "diagnostic": "min_rho",
      "operator": ">",
      "threshold": 0.0,
      "observed": 2.0729288609194362e-14,
      "passed": true
    },
    "H": {
      "diagnostic": "min_H",
      "operator": ">",
      "threshold": 0.0,
      "observed": 8.312498343027667e-08,
      "passed": true
    },
    "hyperbolicity_PX": {
      "diagnostic": "min_PX",
      "operator": ">",
      "threshold": 0.0,
      "observed": 1.0523451240627972,
      "passed": true
    },
    "hyperbolicity_kinetic_denominator": {
      "diagnostic": "min_kinetic_denominator",
      "operator": ">",
      "threshold": 0.0,
      "observed": 1.052350229025336,
      "passed": true
    },
    "finite": {
      "diagnostic": "finite",
      "operator": "is",
      "threshold": true,
      "observed": true,
      "passed": true
    },
    "finalization": {
      "diagnostic": "finalization_failure",
      "operator": "is",
      "threshold": null,
      "observed": null,
      "passed": true
    }
  },
  "classification_rule": {
    "hard_pass": "all registered numerical gates pass and finalization_failure is null",
    "recorded_classification": "HARD_PASS",
    "newton_steps": 0,
    "reclassification_or_recomputation_authorized": false
  },
  "crossing_rule": {
    "hard_pass_tau_binary64": 0.47878124999999994,
    "nonpass_tau_binary64": 0.47878125,
    "binary_endpoints_adjacent": true,
    "next_midpoint_binary64": 0.47878125,
    "further_float64_bisection_forbidden": true
  },
  "nonpass_endpoint_note": "The immutable v6.74 endpoint has its own historical gate register. Its tolerances must not be substituted for the v7.13 strict-gate table. The crossing uses its frozen NONPASS classification only.",
  "immutability": {
    "physics_frozen": true,
    "gates_frozen": true,
    "tolerances_frozen": true,
    "discretization_frozen": true,
    "solver_budget_frozen": true
  }
}