File size: 7,583 Bytes
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"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"
}
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