# Master integrals of the even sector of topology 40 (labels of Brammer et al., arXiv:2505.10274), the Apery K3' sector at 5PM-2SF,
# in the momentum routing given below, as returned by our
# reduction at seeding (4,4): all integration-by-parts identities generated from seeds of the ten-propagator sector with at most
# 4 dots and at most 4 numerator powers (1,092,014 equations in 951,974 distinct integrals), reduced with Kira
# as a user-defined system with the kinematic value y = 23/17 substituted.  Result: 11 master integrals.
# The worldline-exchange symmetry of the sector (u1<->u2 with k1->k2+k4, k3->k2-k3-q, k4->k1-k2) is NOT imposed in this list.
# Adding the relations it generates to the same system and reducing again leaves 9 master integrals, entries 1, 2, 3, 4, 6, 7, 8, 9, 11
# below (Sec. 5.5 of the paper and the supplementary material), the number Duhr et al. (arXiv:2503.20655, ancillary files) list
# for this sector on the maximal cut.
# Propagators P_1..P_22 (four loop momenta k_1..k_4; u_1^2 = u_2^2 = 1, u_1.u_2 = y, q^2 = -1, u_i.q = 0); P_1..P_10 define the sector,
# P_7..P_10 are the cut worldline lines, P_11..P_22 are auxiliary (numerators):
#   P_1  = k1^2
#   P_2  = (k1-k2)^2
#   P_3  = (k2-k3-q)^2
#   P_4  = (k2+k4)^2
#   P_5  = k3^2
#   P_6  = k4^2
#   P_7  = 2u1.(k1-k3)  [cut]
#   P_8  = 2u1.(k2-k3)  [cut]
#   P_9  = 2u2.k3  [cut]
#   P_10 = 2u2.(k3+k4)  [cut]
#   P_11 = k2^2
#   P_12 = (k2-q)^2
#   P_13 = 2u1.k2
#   P_14 = 2u2.k2
#   P_15 = 2u2.k1
#   P_16 = 2u1.k4
#   P_17 = (k1+q)^2
#   P_18 = (k3+q)^2
#   P_19 = (k4+q)^2
#   P_20 = (k1-k3)^2
#   P_21 = (k1-k4)^2
#   P_22 = (k3-k4)^2
# One master per line: J[n_1,...,n_22] (n_i = exponent of P_i; negative = numerator), then a short description.
J[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]   # corner integral
J[1,1,1,1,1,2,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]   # dot on P_6
J[1,1,1,1,2,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]   # dot on P_5
J[1,1,1,2,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]   # dot on P_4
J[1,2,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]   # dot on P_2
J[1,1,1,1,1,1,1,1,1,2,0,0,-1,0,0,0,0,0,0,0,0,0]   # dot on P_10; numerator P_13
J[1,1,1,1,1,1,1,1,1,2,0,0,0,-1,0,0,0,0,0,0,0,0]   # dot on P_10; numerator P_14
J[1,1,1,1,1,1,1,1,1,2,0,0,0,0,-1,0,0,0,0,0,0,0]   # dot on P_10; numerator P_15
J[1,1,1,1,1,1,1,1,1,2,0,0,0,0,0,-1,0,0,0,0,0,0]   # dot on P_10; numerator P_16
J[1,1,1,1,1,1,1,1,2,1,0,0,-1,0,0,0,0,0,0,0,0,0]   # dot on P_9; numerator P_13
J[1,1,1,1,1,1,1,1,2,1,0,0,0,0,-1,0,0,0,0,0,0,0]   # dot on P_9; numerator P_15