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