| { |
| "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" |
| } |
|
|
