{"_provenance": "22x22 rational IBP connection d/ds M(s) = [A0(s) + eps*A1(s)] M(s) for the threshold kite at squared masses (1,4,9,3,0) on the lines D1 = k1^2 - 1, D2 = (k1-p)^2 - 4, D3 = k2^2 - 9, D4 = (k2-p)^2 - 3, D5 = (k1-k2)^2, s = p^2 (the same family as kite-threshold-connection.json with the fourth squared mass 2 -> 3): the same 22 masters in the same order, the same nonzero pattern; built by the same Kira reduction and parsed to integer coefficient lists by the same parser (entry = (P0/P0den + eps*P1/P1den)/(Q/Qden), ascending in s).  This is the ANALYTIC FORMULA (exact rationals), not an evaluated result.  Singular points (real roots of every Q): s = 0, 4 - 2 sqrt3, 1, 12 - 6 sqrt3, 4 + 2 sqrt3, 9, 12 + 6 sqrt3, 25, 297/8 -- the negative real axis is pole-free; the nearest nonzero singularity to s = 0 is 4 - 2 sqrt3 = 0.5359 (the (1,4,9,2,0) connection has 3 - 2 sqrt2 = 0.1716 and the apparent singularity 85/4).", "masters": [[1, 0, 1, 0, 0], [0, 1, 1, 0, 0], [1, 1, 1, 0, 0], [1, 0, 0, 1, 0], [1, 0, 1, 1, 0], [0, 1, 0, 1, 0], [1, 1, 0, 1, 0], [0, 1, 1, 1, 0], [1, 1, 1, 1, 0], [1, 0, 1, 0, 1], [0, 1, 1, 0, 1], [-1, 1, 1, 0, 1], [0, 1, 1, -1, 1], [1, 1, 1, 0, 1], [1, 0, 0, 1, 1], [1, 0, -1, 1, 1], [1, -1, 0, 1, 1], [1, 0, 1, 1, 1], [0, 1, 0, 1, 1], [1, 1, 0, 1, 1], [0, 1, 1, 1, 1], [1, 1, 1, 1, 1]], "TOP_idx": 21, "entries": [{"i": 2, "j": 0, "P0": [-3, -1], "P0den": 1, "P1": [3, 1], "P1den": 1, "Q": [0, 9, -10, 1], "Qden": 1}, {"i": 2, "j": 1, "P0": [3, -1], "P0den": 1, "P1": [-3, 1], "P1den": 1, "Q": [0, 9, -10, 1], "Qden": 1}, {"i": 2, "j": 2, "P0": [-9, 5], "P0den": 1, "P1": [9, 0, -1], "P1den": 1, "Q": [0, 9, -10, 1], "Qden": 1}, {"i": 4, "j": 0, "P0": [6, -1], "P0den": 1, "P1": [-6, 1], "P1den": 1, "Q": [0, 36, -24, 1], "Qden": 1}, {"i": 4, "j": 3, "P0": [-6, -1], "P0den": 1, "P1": [6, 1], "P1den": 1, "Q": [0, 36, -24, 1], "Qden": 1}, {"i": 4, "j": 4, "P0": [-36, 12], "P0den": 1, "P1": [36, 0, -1], "P1den": 1, "Q": [0, 36, -24, 1], "Qden": 1}, {"i": 6, "j": 3, "P0": [-3, -1], "P0den": 1, "P1": [3, 1], "P1den": 1, "Q": [0, 9, -10, 1], "Qden": 1}, {"i": 6, "j": 5, "P0": [3, -1], "P0den": 1, "P1": [-3, 1], "P1den": 1, "Q": [0, 9, -10, 1], "Qden": 1}, {"i": 6, "j": 6, "P0": [-9, 5], "P0den": 1, "P1": [9, 0, -1], "P1den": 1, "Q": [0, 9, -10, 1], "Qden": 1}, {"i": 7, "j": 1, "P0": [6, -1], "P0den": 1, "P1": [-6, 1], "P1den": 1, "Q": [0, 36, -24, 1], "Qden": 1}, {"i": 7, "j": 5, "P0": [-6, -1], "P0den": 1, "P1": [6, 1], "P1den": 1, "Q": [0, 36, -24, 1], "Qden": 1}, {"i": 7, "j": 7, "P0": [-36, 12], "P0den": 1, "P1": [36, 0, -1], "P1den": 1, "Q": [0, 36, -24, 1], "Qden": 1}, {"i": 8, "j": 2, "P0": [6, -1], "P0den": 1, "P1": [-6, 1], "P1den": 1, "Q": [0, 36, -24, 1], "Qden": 1}, {"i": 8, "j": 4, "P0": [-3, -1], "P0den": 1, "P1": [3, 1], "P1den": 1, "Q": [0, 9, -10, 1], "Qden": 1}, {"i": 8, "j": 6, "P0": [-6, -1], "P0den": 1, "P1": [6, 1], "P1den": 1, "Q": [0, 36, -24, 1], "Qden": 1}, {"i": 8, "j": 7, "P0": [3, -1], "P0den": 1, "P1": [-3, 1], "P1den": 1, "Q": [0, 9, -10, 1], "Qden": 1}, {"i": 8, "j": 8, "P0": [-648, 864, -285, 17], "P0den": 1, "P1": [648, -576, 0, 34, -2], "P1den": 1, "Q": [0, 324, -576, 285, -34, 1], "Qden": 1}, {"i": 10, "j": 1, "P0": [2], "P0den": 1, "P1": [-2], "P1den": 1, "Q": [25, -26, 1], "Qden": 1}, {"i": 10, "j": 10, "P0": [-15, 14, 3], "P0den": 1, "P1": [15, -1, -4], "P1den": 1, "Q": [0, 25, -26, 1], "Qden": 1}, {"i": 10, "j": 11, "P0": [-5, -3], "P0den": 1, "P1": [5, 3], "P1den": 1, "Q": [0, 25, -26, 1], "Qden": 1}, {"i": 10, "j": 12, "P0": [5, -3], "P0den": 1, "P1": [-5, 3], "P1den": 1, "Q": [0, 25, -26, 1], "Qden": 1}, {"i": 11, "j": 1, "P0": [16], "P0den": 1, "P1": [-16], "P1den": 1, "Q": [25, -26, 1], "Qden": 1}, {"i": 11, "j": 10, "P0": [30, -19, 4, 1], "P0den": 1, "P1": [-30, 123, -12, -1], "P1den": 1, "Q": [0, 25, -26, 1], "Qden": 1}, {"i": 11, "j": 11, "P0": [-40, -24], "P0den": 1, "P1": [40, 24], "P1den": 1, "Q": [0, 25, -26, 1], "Qden": 1}, {"i": 11, "j": 12, "P0": [15, 2, -1], "P0den": 1, "P1": [-15, -2, 1], "P1den": 1, "Q": [0, 25, -26, 1], "Qden": 1}, {"i": 12, "j": 1, "P0": [2], "P0den": 1, "P1": [-2], "P1den": 1, "Q": [25, -26, 1], "Qden": 1}, {"i": 12, "j": 10, "P0": [60, -39, -20, 1], "P0den": 1, "P1": [-60, 52, 19, -1], "P1den": 1, "Q": [0, 25, -26, 1], "Qden": 1}, {"i": 12, "j": 11, "P0": [-30, 23, -1], "P0den": 1, "P1": [30, -23, 1], "P1den": 1, "Q": [0, 25, -26, 1], "Qden": 1}, {"i": 12, "j": 12, "P0": [5, -3], "P0den": 1, "P1": [-5, 3], "P1den": 1, "Q": [0, 25, -26, 1], "Qden": 1}, {"i": 13, "j": 1, "P0": [4], "P0den": 1, "P1": [-4], "P1den": 1, "Q": [-225, 259, -35, 1], "Qden": 1}, {"i": 13, "j": 9, "P0": [-3, -1], "P0den": 1, "P1": [3, 1], "P1den": 1, "Q": [0, 9, -10, 1], "Qden": 1}, {"i": 13, "j": 10, "P0": [-30, 28, 6], "P0den": 1, "P1": [30, -27, -7], "P1den": 1, "Q": [0, -225, 259, -35, 1], "Qden": 1}, {"i": 13, "j": 11, "P0": [15, -7], "P0den": 1, "P1": [-15, 7], "P1den": 1, "Q": [0, -225, 259, -35, 1], "Qden": 1}, {"i": 13, "j": 12, "P0": [-15, -5], "P0den": 1, "P1": [15, 5], "P1den": 1, "Q": [0, -225, 259, -35, 1], "Qden": 1}, {"i": 13, "j": 13, "P0": [-9, 5], "P0den": 1, "P1": [9, 0, -1], "P1den": 1, "Q": [0, 9, -10, 1], "Qden": 1}, {"i": 14, "j": 3, "P0": [2], "P0den": 1, "P1": [-2], "P1den": 1, "Q": [4, -8, 1], "Qden": 1}, {"i": 14, "j": 14, "P0": [6, -31, 3], "P0den": 1, "P1": [-6, 35, -4], "P1den": 1, "Q": [0, 4, -8, 1], "Qden": 1}, {"i": 14, "j": 15, "P0": [2, -3], "P0den": 1, "P1": [-2, 3], "P1den": 1, "Q": [0, 4, -8, 1], "Qden": 1}, {"i": 14, "j": 16, "P0": [-2, -3], "P0den": 1, "P1": [2, 3], "P1den": 1, "Q": [0, 4, -8, 1], "Qden": 1}, {"i": 15, "j": 3, "P0": [-16], "P0den": 1, "P1": [16], "P1den": 1, "Q": [4, -8, 1], "Qden": 1}, {"i": 15, "j": 14, "P0": [-60, 276, -35, 1], "P0den": 1, "P1": [60, -308, 43, -1], "P1den": 1, "Q": [0, 4, -8, 1], "Qden": 1}, {"i": 15, "j": 15, "P0": [-16, 24], "P0den": 1, "P1": [16, -24], "P1den": 1, "Q": [0, 4, -8, 1], "Qden": 1}, {"i": 15, "j": 16, "P0": [12, 32, -1], "P0den": 1, "P1": [-12, -32, 1], "P1den": 1, "Q": [0, 4, -8, 1], "Qden": 1}, {"i": 16, "j": 3, "P0": [-2], "P0den": 1, "P1": [2], "P1den": 1, "Q": [4, -8, 1], "Qden": 1}, {"i": 16, "j": 14, "P0": [-30, 83, -17, 1], "P0den": 1, "P1": [30, -87, 18, -1], "P1den": 1, "Q": [0, 4, -8, 1], "Qden": 1}, {"i": 16, "j": 15, "P0": [-6, 11, -1], "P0den": 1, "P1": [6, -11, 1], "P1den": 1, "Q": [0, 4, -8, 1], "Qden": 1}, {"i": 16, "j": 16, "P0": [2, 3], "P0den": 1, "P1": [-2, -3], "P1den": 1, "Q": [0, 4, -8, 1], "Qden": 1}, {"i": 17, "j": 3, "P0": [-8, 4], "P0den": 1, "P1": [8, -4], "P1den": 1, "Q": [144, -384, 232, -32, 1], "Qden": 1}, {"i": 17, "j": 9, "P0": [6, -1], "P0den": 1, "P1": [-6, 1], "P1den": 1, "Q": [0, 36, -24, 1], "Qden": 1}, {"i": 17, "j": 14, "P0": [-60, 208, -83, 6], "P0den": 1, "P1": [60, -220, 87, -7], "P1den": 1, "Q": [0, 144, -384, 232, -32, 1], "Qden": 1}, {"i": 17, "j": 15, "P0": [-12, 24, -7], "P0den": 1, "P1": [12, -24, 7], "P1den": 1, "Q": [0, 144, -384, 232, -32, 1], "Qden": 1}, {"i": 17, "j": 16, "P0": [12, 0, -5], "P0den": 1, "P1": [-12, 0, 5], "P1den": 1, "Q": [0, 144, -384, 232, -32, 1], "Qden": 1}, {"i": 17, "j": 17, "P0": [-36, 12], "P0den": 1, "P1": [36, 0, -1], "P1den": 1, "Q": [0, 36, -24, 1], "Qden": 1}, {"i": 19, "j": 3, "P0": [-10, 4], "P0den": 1, "P1": [10, -4], "P1den": 1, "Q": [36, -112, 93, -18, 1], "Qden": 1}, {"i": 19, "j": 14, "P0": [-30, 167, -77, 6], "P0den": 1, "P1": [30, -183, 82, -7], "P1den": 1, "Q": [0, 36, -112, 93, -18, 1], "Qden": 1}, {"i": 19, "j": 15, "P0": [6, -5], "P0den": 1, "P1": [-6, 5], "P1den": 1, "Q": [0, -36, 76, -17, 1], "Qden": 1}, {"i": 19, "j": 16, "P0": [6, 19, -7], "P0den": 1, "P1": [-6, -19, 7], "P1den": 1, "Q": [0, 36, -112, 93, -18, 1], "Qden": 1}, {"i": 19, "j": 18, "P0": [3, -1], "P0den": 1, "P1": [-3, 1], "P1den": 1, "Q": [0, 9, -10, 1], "Qden": 1}, {"i": 19, "j": 19, "P0": [-9, 5], "P0den": 1, "P1": [9, 0, -1], "P1den": 1, "Q": [0, 9, -10, 1], "Qden": 1}, {"i": 20, "j": 1, "P0": [22, 4], "P0den": 1, "P1": [-22, -4], "P1den": 1, "Q": [900, -1536, 685, -50, 1], "Qden": 1}, {"i": 20, "j": 10, "P0": [60, -110, 70, 6], "P0den": 1, "P1": [-60, 278, -81, -7], "P1den": 1, "Q": [0, 900, -1536, 685, -50, 1], "Qden": 1}, {"i": 20, "j": 11, "P0": [-30, -69, -5], "P0den": 1, "P1": [30, 69, 5], "P1den": 1, "Q": [0, 900, -1536, 685, -50, 1], "Qden": 1}, {"i": 20, "j": 12, "P0": [30, 3, -7], "P0den": 1, "P1": [-30, -3, 7], "P1den": 1, "Q": [0, 900, -1536, 685, -50, 1], "Qden": 1}, {"i": 20, "j": 18, "P0": [-6, -1], "P0den": 1, "P1": [6, 1], "P1den": 1, "Q": [0, 36, -24, 1], "Qden": 1}, {"i": 20, "j": 20, "P0": [-36, 12], "P0den": 1, "P1": [36, 0, -1], "P1den": 1, "Q": [0, 36, -24, 1], "Qden": 1}, {"i": 21, "j": 1, "P0": [5832, 645, -298, 9], "P0den": 1, "P1": [-5832, -645, 298, -9], "P1den": 1, "Q": [2405700, -4437828, 2404989, -398703, 26603, -769, 8], "Qden": 1}, {"i": 21, "j": 2, "P0": [-60, 1], "P0den": 1, "P1": [60, -1], "P1den": 1, "Q": [-10692, 7416, -489, 8], "Qden": 1}, {"i": 21, "j": 3, "P0": [-13608, 15144, -4484, 305, -9], "P0den": 1, "P1": [13608, -15144, 4484, -305, 9], "P1den": 1, "Q": [-384912, 1464480, -1842552, 937200, -190553, 16962, -633, 8], "Qden": 1}, {"i": 21, "j": 4, "P0": [69, -1], "P0den": 1, "P1": [-69, 1], "P1den": 1, "Q": [-2673, 3042, -377, 8], "Qden": 1}, {"i": 21, "j": 6, "P0": [114, -8], "P0den": 1, "P1": [-114, 8], "P1den": 1, "Q": [-10692, 7416, -489, 8], "Qden": 1}, {"i": 21, "j": 7, "P0": [-42, 8], "P0den": 1, "P1": [42, -8], "P1den": 1, "Q": [-2673, 3042, -377, 8], "Qden": 1}, {"i": 21, "j": 8, "P0": [5346, -5337, 891, -16], "P0den": 1, "P1": [-10692, 10674, -1782, 32], "P1den": 1, "Q": [-96228, 173664, -89253, 12378, -569, 8], "Qden": 1}, {"i": 21, "j": 9, "P0": [5346, -5337, 891, -16], "P0den": 1, "P1": [-5346, 5337, -891, 16], "P1den": 1, "Q": [0, -96228, 173664, -89253, 12378, -569, 8], "Qden": 1}, {"i": 21, "j": 10, "P0": [53460, -101214, 60768, -6721, -114, 9], "P0den": 1, "P1": [-53460, 164772, -84330, 9214, 57, -9], "P1den": 1, "Q": [0, 2405700, -4437828, 2404989, -398703, 26603, -769, 8], "Qden": 1}, {"i": 21, "j": 11, "P0": [-26730, -1287, -2616, 594, -17], "P0den": 1, "P1": [26730, 1287, 2616, -594, 17], "P1den": 1, "Q": [0, 2405700, -4437828, 2404989, -398703, 26603, -769, 8], "Qden": 1}, {"i": 21, "j": 12, "P0": [26730, -16209, 681, 300, -10], "P0den": 1, "P1": [-26730, 16209, -681, -300, 10], "P1den": 1, "Q": [0, 2405700, -4437828, 2404989, -398703, 26603, -769, 8], "Qden": 1}, {"i": 21, "j": 13, "P0": [69, -1], "P0den": 1, "P1": [-138, 2], "P1den": 1, "Q": [-2673, 3042, -377, 8], "Qden": 1}, {"i": 21, "j": 14, "P0": [-53460, 293076, -293013, 92251, -10117, 444, -9], "P0den": 1, "P1": [53460, -313272, 310257, -94846, 10351, -435, 9], "P1den": 1, "Q": [0, -384912, 1464480, -1842552, 937200, -190553, 16962, -633, 8], "Qden": 1}, {"i": 21, "j": 15, "P0": [10692, -18072, 6681, -528, 17], "P0den": 1, "P1": [-10692, 18072, -6681, 528, -17], "P1den": 1, "Q": [0, 384912, -1079568, 762984, -174216, 16337, -625, 8], "Qden": 1}, {"i": 21, "j": 16, "P0": [10692, 12060, -20679, 6243, -370, 10], "P0den": 1, "P1": [-10692, -12060, 20679, -6243, 370, -10], "P1den": 1, "Q": [0, -384912, 1464480, -1842552, 937200, -190553, 16962, -633, 8], "Qden": 1}, {"i": 21, "j": 17, "P0": [-540, 9], "P0den": 1, "P1": [1080, -18], "P1den": 1, "Q": [-10692, 7416, -489, 8], "Qden": 1}, {"i": 21, "j": 18, "P0": [5346, -5337, 891, -16], "P0den": 1, "P1": [-5346, 5337, -891, 16], "P1den": 1, "Q": [0, -96228, 173664, -89253, 12378, -569, 8], "Qden": 1}, {"i": 21, "j": 19, "P0": [-168, 32], "P0den": 1, "P1": [336, -64], "P1den": 1, "Q": [-2673, 3042, -377, 8], "Qden": 1}, {"i": 21, "j": 20, "P0": [342, -24], "P0den": 1, "P1": [-684, 48], "P1den": 1, "Q": [-10692, 7416, -489, 8], "Qden": 1}, {"i": 21, "j": 21, "P0": [297, -8], "P0den": 1, "P1": [-297], "P1den": 1, "Q": [0, -297, 8], "Qden": 1}], "note": "A(eps,s)=A0+eps*A1; entry = (P0/P0den + eps*P1/P1den)/(Q/Qden), coeff lists ascending in s"}