{ "status": "All exact endpoint certificates passed; not an independent audit.", "e_upper": "10174328183458584617/3742926166425600000", "e_upper_proof": "Sum through 20 plus 1/(20*20!); geometric tail bound.", "certificates": [ { "B": 1, "m": 1, "r": 2, "a": 18, "first_even_index": 2, "last_even_index": 18, "u": "6840759787/1000000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9999999997187388, "U_approx": 0.06207766483401367 }, { "B": 1, "m": 3, "r": 2, "a": 52, "first_even_index": 4, "last_even_index": 54, "u": "8990232707/1000000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9999999996648512, "U_approx": 0.4998267604882982 }, { "B": 1, "m": 5, "r": 2, "a": 88, "first_even_index": 6, "last_even_index": 92, "u": "8058760219/1000000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.999999999267402, "U_approx": 0.10512663498354334 }, { "B": 1, "m": 7, "r": 2, "a": 122, "first_even_index": 8, "last_even_index": 128, "u": "8645692609/1000000000000", "f_lt_1": true, 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20, "last_even_index": 1344, "u": "802179083/1000000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9999998963539891, "U_approx": 0.5843044323708109 }, { "B": 3, "m": 1, "r": 2, "a": 186, "first_even_index": 2, "last_even_index": 186, "u": "41902917/200000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9999999560468505, "U_approx": 0.03463225700602425 }, { "B": 3, "m": 3, "r": 2, "a": 556, "first_even_index": 4, "last_even_index": 558, "u": "217767167/1000000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9999998234707319, "U_approx": 0.08624476475752121 }, { "B": 3, "m": 19, "r": 2, "a": 3520, "first_even_index": 20, "last_even_index": 3538, "u": "218654559/1000000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9999987935741957, "U_approx": 0.08821136396494296 }, { "B": 4, "m": 1, "r": 2, "a": 360, "first_even_index": 2, "last_even_index": 360, "u": "132973689/1000000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9999998477378655, "U_approx": 0.9665403403210802 }, { "B": 4, "m": 3, "r": 2, "a": 1084, "first_even_index": 4, "last_even_index": 1086, "u": "15944463/125000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9999995190120237, "U_approx": 0.03409351578671392 }, { "B": 4, "m": 19, "r": 2, "a": 6860, "first_even_index": 20, "last_even_index": 6878, "u": "64338837/500000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9999965367956917, "U_approx": 0.3572305499833786 }, { "B": 5, "m": 1, "r": 2, "a": 554, "first_even_index": 2, "last_even_index": 554, "u": "73388877/1000000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9999996747356437, "U_approx": 0.163751547316094 }, { "B": 5, "m": 3, "r": 2, "a": 1662, "first_even_index": 4, "last_even_index": 1664, "u": "73388877/1000000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9999990241999726, "U_approx": 0.239736502323781 }, { "B": 5, "m": 19, "r": 2, "a": 10528, "first_even_index": 20, "last_even_index": 10546, "u": "36612089/500000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9999904700543348, "U_approx": 0.13306419136246786 }, { "B": 10, "m": 1, "r": 2, "a": 2794, "first_even_index": 2, "last_even_index": 2794, "u": "11811809/1000000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9999923077998839, "U_approx": 0.2628307852854306 }, { "B": 10, "m": 3, "r": 2, "a": 8382, "first_even_index": 4, "last_even_index": 8384, "u": "11811809/1000000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9999769235771612, "U_approx": 0.9913034680722593 }, { "B": 10, "m": 19, "r": 2, "a": 53092, "first_even_index": 20, "last_even_index": 53110, "u": "11793601/1000000000000", "f_lt_1": true, "U_lt_1": true, "f_approx": 0.9998196644816773, "U_approx": 0.7384580129037517 } ], "uniform_general_B": [ { "B": 1, "D_odd": 1, "a_B": 18, "endpoint_slope": 19 }, { "B": 2, "D_odd": 3, "a_B": 84, "endpoint_slope": 85 }, { "B": 3, "D_odd": 7, "a_B": 228, "endpoint_slope": 229 }, { "B": 4, "D_odd": 13, "a_B": 460, "endpoint_slope": 461 }, { "B": 5, "D_odd": 19, "a_B": 714, "endpoint_slope": 715 }, { "B": 10, "D_odd": 83, "a_B": 3668, "endpoint_slope": 3669 } ], "uniform_single_value": "a(m)=2*ceil((879*m-50)/100), endpoint a(m)+m-1; every positive odd m.", "single_value_uniform_comparisons_passed": true, "integer_zero_range_exclusions": [ { "r": 1, "profile_at_zero_for_a_18m_already_too_large": true }, { "r": 3, "profile_at_zero_for_a_18m_already_too_large": true }, { "r": 4, "profile_at_zero_for_a_18m_already_too_large": true }, { "r": 5, "profile_at_zero_for_a_18m_already_too_large": true }, { "r": 6, "profile_at_zero_for_a_18m_already_too_large": true }, { "r": 7, "profile_at_zero_for_a_18m_already_too_large": true }, { "r": 8, "profile_at_zero_for_a_18m_already_too_large": true } ], "r_at_least_9": "Degree condition forces a>19m, so cannot improve a<=18m.", "optimization_scope": "Same factorial denominator normalization and absolute-value exponential profile; integer zero-range r. No global obstruction for other constructions is asserted." }