{
  "status": "PASS",
  "arithmetic": "Exact rational interval arithmetic and integer-square-root enclosures; decimal strings are display-only.",
  "method": "Exact orientation-dependent exterior Crofton waste, convex-reward cap-capacity packing, and localized additive projection redundancy, using the classical four-cap strip lemma.",
  "parameters": {
    "r": "1014451/25000000000",
    "t": "301922437/312500000",
    "delta": "13432179/5000000000000",
    "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+2.6864358e-6.",
  "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, redundancy additivity, and the final monotonicity argument."
  ],
  "exact_exterior_coefficient": [
    "6363.16646518887970996040896440942366215013656",
    "6363.16646518887970996040896440942366215013656"
  ],
  "far_exterior_coefficient": [
    "5.27454645260584709569003143336642690224613061",
    "5.27454645260584709569003143336642690224613061"
  ],
  "initial_good_mass_eta0": [
    "0.0169241008",
    "0.0169241008"
  ],
  "final_good_mass_at_W_equal_delta": [
    "0.00837698170327856989263437038768479920831638409",
    "0.00837698170327856989263437038768479920831638409"
  ],
  "opposed_overlap_derivative_at_eta0": [
    "0.000142142097265912977905963450281416581529319641",
    "0.000142142097265912977905963450281416581529319641"
  ],
  "packing_reward_derivative_at_eta0": [
    "0.000314308888527502157273311775840005731933211991",
    "0.000314308888527502157273311775840005731933211991"
  ],
  "maximum_flow_for_convexity_and_branch_comparison": [
    "0.0338482016",
    "0.0338482016"
  ],
  "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": [
      "0.00004057804",
      "0.00004057804"
    ],
    "r_below_sec_lambda_minus_one": [
      "0.0823516222523939687994464107327788401221441268",
      "0.0823516222523939687994464107327788401221441268"
    ],
    "t_above_a_cos_lambda": [
      "0.0422347766680878195773751709406250427266091626",
      "0.0422347766680878195773751709406250427266091626"
    ],
    "t_below_one": [
      "0.0338482016",
      "0.0338482016"
    ],
    "w_squared_positive": [
      "0.0666318601750230876816",
      "0.0666318601750230876816"
    ],
    "eta_positive": [
      "0.0169241008",
      "0.0169241008"
    ],
    "final_eta_positive_for_W_at_most_delta": [
      "0.00837698170327856989263437038768479920831638409",
      "0.00837698170327856989263437038768479920831638409"
    ],
    "eta_positive_at_certified_initial_reward": [
      "0.00837698155508778193321792301864517125269150068",
      "0.00837698155508778193321792301864517125269150068"
    ],
    "far_radius_exceeds_near_annulus": [
      "0.366304398568436353173887325852201417331112982",
      "0.366304398568436353173887325852201417331112982"
    ],
    "far_radius_exceeds_one": [
      "0.366344976608436353173887325852201417331112982",
      "0.366344976608436353173887325852201417331112982"
    ],
    "near_coefficient_dominates_far": [
      "6357.89191873627386286471893297605723524789042",
      "6357.89191873627386286471893297605723524789042"
    ],
    "strip_numerator_positive": [
      "0.262495793491673114264650693331005357923216821",
      "0.262495793491673114264650693331005357923216821"
    ],
    "t_exceeds_w_for_neighbor_enclosing_disc": [
      "0.708020319934038612167091095629899400226620398",
      "0.708020319934038612167091095629899400226620398"
    ],
    "opposed_localized_projection_slack": [
      "0.391787323951004060466531166598692388180018345",
      "0.391787323951004060466531166598692388180018345"
    ],
    "neighbor_localized_projection_slack": [
      "0.00860948797843967064420524151211891008941664147",
      "0.00860948797843967064420524151211891008941664147"
    ],
    "neighbor_branch_dominates_on_whole_eta_interval": [
      "0.742188367266502570760092083148749312148797878",
      "0.742188367266502570760092083148749312148797878"
    ],
    "opposed_second_derivative_positive_on_whole_eta_interval": [
      "15.9886938436880576701841561004445962912779333",
      "15.9886938436880576701841561004445962912779333"
    ],
    "additive_monotonicity_slack": [
      "1.10405504235460126774193623815452988462489409E-7",
      "1.10405504235460126774193623815452988462489409E-7"
    ],
    "packing_reward_contradiction_margin": [
      "4.65776869959719519498148727627887147111207681E-14",
      "4.65776869959719519498148727627887147111207681E-14"
    ]
  },
  "opposed_overlap_bound": [
    "0.00000121290488278000823637172549562517531266387715",
    "0.00000121290488278000823637172549562517531266387715"
  ],
  "neighbor_overlap_bound": [
    "0.00000139151902159620642477868617745503788038227148",
    "0.00000139151902159620642477868617745503788038227148"
  ],
  "certified_initial_packing_reward": [
    "0.00000268643584657768699597195194981487276278871471",
    "0.00000268643584657768699597195194981487276278871471"
  ],
  "published_selected_parameter_increment": "1.0764570174055761e-6 (floating reconstruction; background only)",
  "improvement_over_published_displayed_1_076457e_6": "4024947/2500000000000"
}