== A. exact rational arithmetic, random integer points ==
PASS S_3 = 0 (subset DP, exact); brute force over 2 sequences gives 0
PASS S_4 = 0 (subset DP, exact); brute force over 6 sequences gives 0
PASS S_5 = 0 (subset DP, exact); brute force over 24 sequences gives 0
PASS S_6 = 0 (subset DP, exact); brute force over 120 sequences gives 0
PASS S_7 = 0 (subset DP, exact); brute force over 720 sequences gives 0
PASS S_8 = 0 (subset DP, exact); brute force over 5040 sequences gives 0
PASS S_9 = 0 (subset DP, exact)
PASS S_10 = 0 (subset DP, exact)
PASS S_11 = 0 (subset DP, exact)
== B. modular arithmetic (subset DP) ==
PASS p = 2305843009213693951, n =  3 (n mod 4 = 3): S_n mod p = [0, 0, 0]
PASS p = 2305843009213693951, n =  4 (n mod 4 = 0): S_n mod p = [0, 0, 0]
PASS p = 2305843009213693951, n =  5 (n mod 4 = 1): S_n mod p = [0, 0, 0]
PASS p = 2305843009213693951, n =  6 (n mod 4 = 2): S_n mod p = [0, 0, 0]
PASS p = 2305843009213693951, n =  7 (n mod 4 = 3): S_n mod p = [0, 0, 0]
PASS p = 2305843009213693951, n =  8 (n mod 4 = 0): S_n mod p = [0, 0, 0]
PASS p = 2305843009213693951, n =  9 (n mod 4 = 1): S_n mod p = [0, 0, 0]
PASS p = 2305843009213693951, n = 10 (n mod 4 = 2): S_n mod p = [0, 0, 0]
PASS p = 2305843009213693951, n = 11 (n mod 4 = 3): S_n mod p = [0, 0, 0]
PASS p = 2305843009213693951, n = 12 (n mod 4 = 0): S_n mod p = [0, 0, 0]
PASS p = 2305843009213693951, n = 13 (n mod 4 = 1): S_n mod p = [0, 0, 0]
PASS p = 2305843009213693951, n = 14 (n mod 4 = 2): S_n mod p = [0, 0, 0]
PASS p = 2305843009213693951, n = 15 (n mod 4 = 3): S_n mod p = [0, 0, 0]
PASS p = 2147483647, n =  3 (n mod 4 = 3): S_n mod p = [0, 0, 0]
PASS p = 2147483647, n =  4 (n mod 4 = 0): S_n mod p = [0, 0, 0]
PASS p = 2147483647, n =  5 (n mod 4 = 1): S_n mod p = [0, 0, 0]
PASS p = 2147483647, n =  6 (n mod 4 = 2): S_n mod p = [0, 0, 0]
PASS p = 2147483647, n =  7 (n mod 4 = 3): S_n mod p = [0, 0, 0]
PASS p = 2147483647, n =  8 (n mod 4 = 0): S_n mod p = [0, 0, 0]
PASS p = 2147483647, n =  9 (n mod 4 = 1): S_n mod p = [0, 0, 0]
PASS p = 2147483647, n = 10 (n mod 4 = 2): S_n mod p = [0, 0, 0]
PASS p = 2147483647, n = 11 (n mod 4 = 3): S_n mod p = [0, 0, 0]
PASS p = 2147483647, n = 12 (n mod 4 = 0): S_n mod p = [0, 0, 0]
PASS p = 2147483647, n = 13 (n mod 4 = 1): S_n mod p = [0, 0, 0]
PASS p = 2147483647, n = 14 (n mod 4 = 2): S_n mod p = [0, 0, 0]
PASS p = 2147483647, n = 15 (n mod 4 = 3): S_n mod p = [0, 0, 0]
== C. controls (the sums must be NONZERO) ==
PASS n = 2: S_2 = ([A1A2]/[B1B2])^2 = 529/1296 is nonzero (the identity needs n >= 3)
PASS n =  3: unsigned sum nonzero; symmetric random weights zero; brackets/antisymmetric random zero; antisymmetric random/brackets zero (zero expected: n = 0,3 mod 4)
PASS n =  4: unsigned sum nonzero; symmetric random weights zero; brackets/antisymmetric random zero; antisymmetric random/brackets zero (zero expected: n = 0,3 mod 4)
PASS n =  5: unsigned sum nonzero; symmetric random weights nonzero; brackets/antisymmetric random nonzero; antisymmetric random/brackets nonzero
PASS n =  6: unsigned sum nonzero; symmetric random weights nonzero; brackets/antisymmetric random nonzero; antisymmetric random/brackets nonzero
PASS n =  7: unsigned sum nonzero; symmetric random weights zero; brackets/antisymmetric random zero; antisymmetric random/brackets zero (zero expected: n = 0,3 mod 4)
PASS n =  8: unsigned sum nonzero; symmetric random weights zero; brackets/antisymmetric random zero; antisymmetric random/brackets zero (zero expected: n = 0,3 mod 4)
PASS n =  9: unsigned sum nonzero; symmetric random weights nonzero; brackets/antisymmetric random nonzero; antisymmetric random/brackets nonzero
PASS n = 10: unsigned sum nonzero; symmetric random weights nonzero; brackets/antisymmetric random nonzero; antisymmetric random/brackets nonzero
PASS n = 11: unsigned sum nonzero; symmetric random weights zero; brackets/antisymmetric random zero; antisymmetric random/brackets zero (zero expected: n = 0,3 mod 4)
PASS n = 12: unsigned sum nonzero; symmetric random weights zero; brackets/antisymmetric random zero; antisymmetric random/brackets zero (zero expected: n = 0,3 mod 4)
PASS n = 13: unsigned sum nonzero; symmetric random weights nonzero; brackets/antisymmetric random nonzero; antisymmetric random/brackets nonzero
PASS n = 14: unsigned sum nonzero; symmetric random weights nonzero; brackets/antisymmetric random nonzero; antisymmetric random/brackets nonzero
== D. Theorem 1.4 and Remark 5.1 ==
PASS n = 3: matrix sum = PT(123) tr(M_2 M_3) I: True
PASS n = 3: trace identity sum sgn tr(M_1 M_sigma) PT(1 sigma) = 0 (M_1 traceless)
PASS n = 3: control, general (non-traceless) M_2..M_n: matrix sum nonzero
PASS n = 4: matrix sum with traceless M_2..M_n = ['0', '0', '0', '0']
PASS n = 4: trace identity sum sgn tr(M_1 M_sigma) PT(1 sigma) = 0 (M_1 arbitrary)
PASS n = 4: control, general (non-traceless) M_2..M_n: matrix sum nonzero
PASS n = 5: matrix sum with traceless M_2..M_n = ['0', '0', '0', '0']
PASS n = 5: trace identity sum sgn tr(M_1 M_sigma) PT(1 sigma) = 0 (M_1 arbitrary)
PASS n = 5: control, general (non-traceless) M_2..M_n: matrix sum nonzero
PASS n = 6: matrix sum with traceless M_2..M_n = ['0', '0', '0', '0']
PASS n = 6: trace identity sum sgn tr(M_1 M_sigma) PT(1 sigma) = 0 (M_1 arbitrary)
PASS n = 6: control, general (non-traceless) M_2..M_n: matrix sum nonzero
PASS n = 7: matrix sum with traceless M_2..M_n = ['0', '0', '0', '0']
PASS n = 7: trace identity sum sgn tr(M_1 M_sigma) PT(1 sigma) = 0 (M_1 arbitrary)
PASS n = 7: control, general (non-traceless) M_2..M_n: matrix sum nonzero
PASS n = 8: matrix sum with traceless M_2..M_n = ['0', '0', '0', '0']
PASS n = 8: trace identity sum sgn tr(M_1 M_sigma) PT(1 sigma) = 0 (M_1 arbitrary)
PASS n = 8: control, general (non-traceless) M_2..M_n: matrix sum nonzero
PASS n = 9: matrix sum with traceless M_2..M_n = ['0', '0', '0', '0']
PASS n = 9: trace identity sum sgn tr(M_1 M_sigma) PT(1 sigma) = 0 (M_1 arbitrary)
PASS n = 9: control, general (non-traceless) M_2..M_n: matrix sum nonzero
PASS n = 3: open-chain sum = -PT(123)[A_2A_3]^2[XY]: True
PASS n = 4: Remark 5.1, open-chain sum with arbitrary X, Y = 0
PASS n = 5: Remark 5.1, open-chain sum with arbitrary X, Y = 0
PASS n = 6: Remark 5.1, open-chain sum with arbitrary X, Y = 0
PASS n = 7: Remark 5.1, open-chain sum with arbitrary X, Y = 0
PASS n = 8: Remark 5.1, open-chain sum with arbitrary X, Y = 0
== E. Proposition 5.2: sum sgn(sigma|T) PT(1 sigma) for all T in {2..n} ==
PASS n = 3: nonzero exactly for T = [(2, 3)] (expected [(2, 3)])
PASS n = 4: all 8 sets T give 0
PASS n = 5: all 16 sets T give 0
PASS n = 6: all 32 sets T give 0
PASS n = 7: all 64 sets T give 0
PASS n = 8: all 128 sets T give 0
== F. Lemma 3.1, Corollary 3.2, Lemma 4.2, Corollary 1.3 ==
PASS n = 3: shuffle identity for 2 pairs (u, v)
PASS n = 3: sum_sigma PT(1 sigma) sigma = sum_Q PT(12Q) l[2Q] coefficientwise
PASS n = 3: Lemma 4.2 (trace form of the numerator) and tr M_v = 0
PASS n = 4: shuffle identity for 12 pairs (u, v)
PASS n = 4: sum_sigma PT(1 sigma) sigma = sum_Q PT(12Q) l[2Q] coefficientwise
PASS n = 4: Lemma 4.2 (trace form of the numerator) and tr M_v = 0
PASS n = 5: shuffle identity for 72 pairs (u, v)
PASS n = 5: sum_sigma PT(1 sigma) sigma = sum_Q PT(12Q) l[2Q] coefficientwise
PASS n = 5: Lemma 4.2 (trace form of the numerator) and tr M_v = 0
PASS n = 6: shuffle identity for 480 pairs (u, v)
PASS n = 6: sum_sigma PT(1 sigma) sigma = sum_Q PT(12Q) l[2Q] coefficientwise
PASS n = 6: Lemma 4.2 (trace form of the numerator) and tr M_v = 0
PASS n = 7: shuffle identity for 400 pairs (u, v)
PASS n = 7: sum_sigma PT(1 sigma) sigma = sum_Q PT(12Q) l[2Q] coefficientwise
PASS n = 7: Lemma 4.2 (trace form of the numerator) and tr M_v = 0
PASS n = 4, B1 = B2: bracket polynomial D*S_n = 0
PASS n = 4, B1 = B2 and B3 = -2 B4: bracket polynomial D*S_n = 0
PASS n = 4, generic: bracket polynomial D*S_n = 0
PASS n = 5, B1 = B2: bracket polynomial D*S_n = 0
PASS n = 5, B1 = B2 and B3 = -2 B4: bracket polynomial D*S_n = 0
PASS n = 5, generic: bracket polynomial D*S_n = 0
PASS n = 6, B1 = B2: bracket polynomial D*S_n = 0
PASS n = 6, B1 = B2 and B3 = -2 B4: bracket polynomial D*S_n = 0
PASS n = 6, generic: bracket polynomial D*S_n = 0
PASS n = 7, B1 = B2: bracket polynomial D*S_n = 0
PASS n = 7, B1 = B2 and B3 = -2 B4: bracket polynomial D*S_n = 0
PASS n = 7, generic: bracket polynomial D*S_n = 0
== G. small prime fields F_p (the B_i must be pairwise independent, so n <= p + 1) ==
PASS F_3, n =  3: S_n = [0, 0, 0]
PASS F_3, n =  4: S_n = [0, 0, 0]
PASS F_5, n =  3: S_n = [0, 0, 0]
PASS F_5, n =  4: S_n = [0, 0, 0]
PASS F_5, n =  5: S_n = [0, 0, 0]
PASS F_5, n =  6: S_n = [0, 0, 0]
PASS F_7, n =  3: S_n = [0, 0, 0]
PASS F_7, n =  4: S_n = [0, 0, 0]
PASS F_7, n =  5: S_n = [0, 0, 0]
PASS F_7, n =  6: S_n = [0, 0, 0]
PASS F_7, n =  7: S_n = [0, 0, 0]
PASS F_7, n =  8: S_n = [0, 0, 0]
PASS F_11, n =  3: S_n = [0, 0, 0]
PASS F_11, n =  4: S_n = [0, 0, 0]
PASS F_11, n =  5: S_n = [0, 0, 0]
PASS F_11, n =  6: S_n = [0, 0, 0]
PASS F_11, n =  7: S_n = [0, 0, 0]
PASS F_11, n =  8: S_n = [0, 0, 0]
PASS F_11, n =  9: S_n = [0, 0, 0]
PASS F_11, n = 10: S_n = [0, 0, 0]
PASS F_11, n = 11: S_n = [0, 0, 0]
PASS F_11, n = 12: S_n = [0, 0, 0]
PASS F_13, n =  3: S_n = [0, 0, 0]
PASS F_13, n =  4: S_n = [0, 0, 0]
PASS F_13, n =  5: S_n = [0, 0, 0]
PASS F_13, n =  6: S_n = [0, 0, 0]
PASS F_13, n =  7: S_n = [0, 0, 0]
PASS F_13, n =  8: S_n = [0, 0, 0]
PASS F_13, n =  9: S_n = [0, 0, 0]
PASS F_13, n = 10: S_n = [0, 0, 0]
PASS F_13, n = 11: S_n = [0, 0, 0]
PASS F_13, n = 12: S_n = [0, 0, 0]
time 4.2 s
ALL CHECKS PASSED
python3 lead/check_bkrg.py  4.20s user 0.02s system 99% cpu 4.228 total
