{
  "status": "PASS",
  "arithmetic": "Exact rational interval arithmetic, integer-square-root enclosures, Machin pi, and alternating cosine Taylor enclosures; decimal strings are display-only.",
  "method": "Continuous rotational averaging with sharp per-point cap-occupancy bound, exact angular exterior Crofton waste, convex-reward cap-capacity packing, and localized additive projection redundancy, using the classical four-cap strip lemma.",
  "parameters": {
    "r": "44539/312500000",
    "t": "486266001/500000000",
    "delta": "84450517/10000000000000",
    "kappa": "3343199/1000000",
    "robust_missed_line_coefficient_upper": "250194151/500000000",
    "lambda": "pi/8; sine and cosine evaluated by exact radical enclosures"
  },
  "bound": "Every conventional rectifiable opaque barrier of the unit disk has length strictly greater than pi+8.4450517e-6.",
  "robust_bound": "For every finite straight set B, length(B)+0.500388302*U(B) is strictly greater than pi+8.4450517e-6, where U(B) is missed unit-disk line measure in dtheta*dp coordinates.",
  "exact_robust_missed_line_coefficient": [
    "0.500388301495003693028567362515948783723625519",
    "0.500388301495003693028567362515948783723625519"
  ],
  "scope": "Unconditional complete-barrier lower-bound improvement; the unrestricted optimum is not determined. Independent literature priority remains to be checked.",
  "analytic_dependencies": [
    "Kawamura--Moriyama--Otachi--Pach strip-overlap lemma and its normal-window-localized proof: https://doi.org/10.4230/LIPIcs.SoCG.2016.46, Section 6.",
    "Kiderlen--Pausinger disk-cap geometry: https://doi.org/10.1007/s10474-026-01606-x, Section 3.2.",
    "The accompanying manuscript proves exact orientation-position exterior waste, convex-reward capacity packing with disjoint material allocations, continuous rotational redundancy reuse, and the final monotonicity argument."
  ],
  "pi_enclosure_by_Machin_identity": [
    "3.14159265358979323846264338327950288419716940",
    "3.14159265358979323846264338327950288419716940"
  ],
  "rotation_angle_pi_over_4kappa": [
    "0.234924144030148462480295323676477446017808796",
    "0.234924144030148462480295323676477446017808796"
  ],
  "cosine_enclosure_at_rotation_angle": [
    "0.972532001065129160493899614613793797102660031",
    "0.972532001065129160493899614613793797102660031"
  ],
  "exact_exterior_coefficient": [
    "1639.49823190329311103571954349476619705892600",
    "1639.49823190329311103571954349476619705892600"
  ],
  "far_exterior_coefficient": [
    "5.18175304839211644465946224106180216925542450",
    "5.18175304839211644465946224106180216925542450"
  ],
  "initial_good_mass_eta0": [
    "0.013733999",
    "0.013733999"
  ],
  "final_good_mass_at_W_equal_delta": [
    "0.00681117533475905013852475395424315039321249098",
    "0.00681117533475905013852475395424315039321249098"
  ],
  "opposed_overlap_derivative_at_eta0": [
    "0.000164859994895088133014231804896202622584007799",
    "0.000164859994895088133014231804896202622584007799"
  ],
  "packing_reward_derivative_at_eta0": [
    "0.000364885585357478218512466427020964961359074555",
    "0.000364885585357478218512466427020964961359074555"
  ],
  "maximum_flow_for_convexity_and_branch_comparison": [
    "0.027467998",
    "0.027467998"
  ],
  "capacity_packing_reward": "R(eta)=q(4*eta/3)+q(2*eta/3). The alternative 2*q(eta) is at least q(4*eta/3), since the quadratic denominator of q increases on nonnegative arguments.",
  "positive_intervals": {
    "r_positive": [
      "0.0001425248",
      "0.0001425248"
    ],
    "r_below_sec_lambda_minus_one": [
      "0.0822496754923939687994464107327788401221441268",
      "0.0822496754923939687994464107327788401221441268"
    ],
    "t_above_a_cos_lambda": [
      "0.0485207937431179792291569925197795738063616666",
      "0.0485207937431179792291569925197795738063616666"
    ],
    "t_below_one": [
      "0.027467998",
      "0.027467998"
    ],
    "w_squared_positive": [
      "0.05446657499919061104",
      "0.05446657499919061104"
    ],
    "eta_positive": [
      "0.013733999",
      "0.013733999"
    ],
    "final_eta_positive_for_W_at_most_delta": [
      "0.00681117533475905013852475395424315039321249098",
      "0.00681117533475905013852475395424315039321249098"
    ],
    "eta_positive_at_certified_initial_reward": [
      "0.00681117527317818330764889896016155697430547317",
      "0.00681117527317818330764889896016155697430547317"
    ],
    "far_radius_exceeds_near_annulus": [
      "0.375225422270257998747394035936483540166916285",
      "0.375225422270257998747394035936483540166916285"
    ],
    "far_radius_exceeds_one": [
      "0.375367947070257998747394035936483540166916285",
      "0.375367947070257998747394035936483540166916285"
    ],
    "near_coefficient_dominates_far": [
      "1634.31647885490099459106008125370439488967058",
      "1634.31647885490099459106008125370439488967058"
    ],
    "strip_numerator_positive": [
      "0.313112370713568598623986446464137901833300437",
      "0.313112370713568598623986446464137901833300437"
    ],
    "t_exceeds_w_for_neighbor_enclosing_disc": [
      "0.739151250867455628286944732800196887846413442",
      "0.739151250867455628286944732800196887846413442"
    ],
    "opposed_localized_projection_slack": [
      "0.412212416136279598094388845940125273416766851",
      "0.412212416136279598094388845940125273416766851"
    ],
    "neighbor_localized_projection_slack": [
      "0.0121866417486293112503393717764246398951865730",
      "0.0121866417486293112503393717764246398951865730"
    ],
    "neighbor_branch_dominates_on_whole_eta_interval": [
      "0.915467592725691769335952217484317444146408249",
      "0.915467592725691769335952217484317444146408249"
    ],
    "opposed_second_derivative_positive_on_whole_eta_interval": [
      "15.9999851783239078183115757549362828905063786",
      "15.9999851783239078183115757549362828905063786"
    ],
    "rational_rotation_factor_positive": [
      "3.343199",
      "3.343199"
    ],
    "rotation_factor_gate_t_minus_cos_pi_over_4kappa": [
      "9.34870839506100385386206202897339968700433484E-10",
      "9.34870839506100385386206202897339968700433484E-10"
    ],
    "additive_monotonicity_slack": [
      "2.47971596345080015093756582460816486652805565E-7",
      "2.47971596345080015093756582460816486652805565E-7"
    ],
    "rotated_reward_contradiction_margin": [
      "7.51216020030550947321128335408978435742416294E-14",
      "7.51216020030550947321128335408978435742416294E-14"
    ],
    "robust_missed_line_coefficient_upper_slack": [
      "5.04996306971432637484051216276374481419222676E-10",
      "5.04996306971432637484051216276374481419222676E-10"
    ]
  },
  "opposed_overlap_bound": [
    "0.00000113977195840370384587934451324925408758205828",
    "0.00000113977195840370384587934451324925408758205828"
  ],
  "neighbor_overlap_bound": [
    "0.00000132940857468485565568314971606923792798214086",
    "0.00000132940857468485565568314971606923792798214086"
  ],
  "certified_initial_packing_reward": [
    "0.00000252603921427399386128528236940512172350429740",
    "0.00000252603921427399386128528236940512172350429740"
  ],
  "certified_initial_rotated_reward": [
    "0.00000844505177512160200305509473211283354089784357",
    "0.00000844505177512160200305509473211283354089784357"
  ],
  "published_selected_parameter_increment": "1.0764570174055761e-6 (floating reconstruction; background only)",
  "improvement_over_published_displayed_1_076457e_6": "73685947/10000000000000"
}