Second verification run, r2_complex.py, mp.dps = 60
   N=3 tau=1 a=1.5: max relative residuals: cross closed form (Lemma 4.4) 1.7e-59; (1-k sn t)(P_l x P_l+1) 1.7e-59; P(t-D) x P(t+D) 4.3e-60; m.P=1 6.4e-61; n.V=1 6.4e-61; tangency of m(t) 1.5e-60; (V-f1)^2 factorisation 1.2e-60; r1 = dn h/(k cn h) 2.3e-61; r1 = sn(K+iK'+h) 3.5e-61; r1 = sn(K+iK'-h) 2.3e-61
      zero orders of d_l x d_l+1 in t: Pi'_+ = K+iK': 3.0, Pi'_+ = K+3iK': 3.0, Pi'_- = 3K+iK': 1.0, Pi = iK': -1.2636e-13; value at t = iK': (-9.4506903899e-63 - 0.7349601694j) (claimed -2iab c0/s0 = (0.0 - 0.7349601694j))
      vertex formulas (Lemma 3.2), max relative residual: (a) foot on chord 3.7e-61; (a) foot on tangent 6.3e-61; (b) inversion 8.7e-61; (d) polar vertex 1.8e-59; (e) dual R 1.1e-61; (e) dual P 2.4e-61
      orders at S1, S2, PA (negative = pole order, inf = identically ~0 near the point):
         A     S1:1.0  S2:1.0  PA:-1.0
         Ap    S1:1.0  S2:1.0  PA:-1.0
         Ab1   S1:1.0  S2:-1.0  PA:-2.759e-13
         Abp1  S1:1.0  S2:-1.0  PA:-2.759e-13
         As1   S1:3.0  S2:1.0  PA:-2.0
         As2   S1:1.0  S2:3.0  PA:-2.0
         Ad1   S1:-1.0  S2:1.0  PA:2.759e-13
         Apd1  S1:-1.0  S2:1.0  PA:2.759e-13
         Apol1 S1:-1.0  S2:1.0  PA:2.759e-13
         AdR1  S1:-1.0  S2:1.0  PA:2.759e-13
         AdP1  S1:-1.0  S2:1.0  PA:2.759e-13
   N=4 tau=1 a=1.5: max relative residuals: cross closed form (Lemma 4.4) 2.0e-60; (1-k sn t)(P_l x P_l+1) 2.0e-60; P(t-D) x P(t+D) 3.1e-61; m.P=1 3.5e-61; n.V=1 6.4e-61; tangency of m(t) 4.9e-61
      zero orders of d_l x d_l+1 in t: Pi'_+ = K+iK': 3.0, Pi'_+ = K+3iK': 3.0, Pi'_- = 3K+iK': 1.0, Pi = iK': -1.2636e-13; value at t = iK': (-4.781071136e-62 - 2.0j) (claimed -2iab c0/s0 = (0.0 - 2.0j))
      vertex formulas (Lemma 3.2), max relative residual: (a) foot on chord 3.1e-61; (a) foot on tangent 3.7e-61; (b) inversion 1.7e-61; (d) polar vertex 1.4e-60; (e) dual R 1.0e-61; (e) dual P 2.9e-61
      orders at S1, S2, PA (negative = pole order, inf = identically ~0 near the point):
         A     S1:1.0  S2:1.0  PA:-1.0
         Ap    S1:-1.0  S2:-1.0  PA:1.0
         Ab1   S1:-1.0  S2:-1.0  PA:1.0
         Abp1  S1:-1.0  S2:-1.0  PA:1.0
         As1   S1:1.0  S2:1.0  PA:-1.0
         As2   S1:1.0  S2:1.0  PA:-1.0
         Ad1   S1:-1.0  S2:-1.0  PA:1.0
         Apd1  S1:-1.0  S2:-1.0  PA:1.0
         Apol1 S1:-1.0  S2:-1.0  PA:1.0
         AdR1  S1:-1.0  S2:-1.0  PA:1.0
         AdP1  S1:-1.0  S2:-1.0  PA:1.0
   N=5 tau=2 a=1.5: max relative residuals: cross closed form (Lemma 4.4) 2.9e-56; (1-k sn t)(P_l x P_l+1) 2.9e-56; P(t-D) x P(t+D) 2.2e-59; m.P=1 4.4e-61; n.V=1 3.9e-61; tangency of m(t) 7.5e-59; (V-f1)^2 factorisation 3.2e-58; r1 = dn h/(k cn h) 3.1e-61; r1 = sn(K+iK'+h) 3.1e-61; r1 = sn(K+iK'-h) 3.1e-61
      zero orders of d_l x d_l+1 in t: Pi'_+ = K+iK': 3.0, Pi'_+ = K+3iK': 3.0, Pi'_- = 3K+iK': 1.0, Pi = iK': -1.2636e-13; value at t = iK': (7.4323971921e-63 - 0.108943416276j) (claimed -2iab c0/s0 = (0.0 - 0.108943416276j))
      vertex formulas (Lemma 3.2), max relative residual: (a) foot on chord 3.5e-60; (a) foot on tangent 6.4e-61; (b) inversion 2.5e-61; (d) polar vertex 6.6e-57; (e) dual R 2.1e-61; (e) dual P 1.8e-60
      orders at S1, S2, PA (negative = pole order, inf = identically ~0 near the point):
         A     S1:1.0  S2:1.0  PA:-1.0
         Ap    S1:1.0  S2:1.0  PA:-1.0
         Ab1   S1:1.0  S2:-1.0  PA:-2.837e-13
         Abp1  S1:1.0  S2:-1.0  PA:-2.837e-13
         As1   S1:1.0  S2:1.0  PA:-2.0
         As2   S1:1.0  S2:1.0  PA:-2.0
         Ad1   S1:-1.0  S2:1.0  PA:2.837e-13
         Apd1  S1:-1.0  S2:1.0  PA:2.837e-13
         Apol1 S1:-1.0  S2:1.0  PA:2.837e-13
         AdR1  S1:-1.0  S2:1.0  PA:2.837e-13
         AdP1  S1:-1.0  S2:1.0  PA:2.837e-13
   N=6 tau=1 a=2.2: max relative residuals: cross closed form (Lemma 4.4) 3.7e-60; (1-k sn t)(P_l x P_l+1) 3.8e-60; P(t-D) x P(t+D) 3.1e-61; m.P=1 1.3e-60; n.V=1 1.3e-60; tangency of m(t) 4.7e-61; (V-f1)^2 factorisation 3.1e-60; r1 = dn h/(k cn h) 0; r1 = sn(K+iK'+h) 3.0e-61; r1 = sn(K+iK'-h) 1.9e-62
      zero orders of d_l x d_l+1 in t: Pi'_+ = K+iK': 3.0, Pi'_+ = K+3iK': 3.0, Pi'_- = 3K+iK': 1.0, Pi = iK': -1.2636e-13; value at t = iK': (1.07739908516e-61 - 4.64758001545j) (claimed -2iab c0/s0 = (0.0 - 4.64758001545j))
      vertex formulas (Lemma 3.2), max relative residual: (a) foot on chord 3.5e-61; (a) foot on tangent 8.3e-61; (b) inversion 4.1e-61; (d) polar vertex 9.2e-61; (e) dual R 8.0e-63; (e) dual P 3.7e-61
      orders at S1, S2, PA (negative = pole order, inf = identically ~0 near the point):
         A     S1:-1.0  S2:-1.0  PA:-1.0
         Ap    S1:1.0  S2:1.0  PA:1.0
         Ab1   S1:-1.0  S2:-1.0  PA:-1.0
         Abp1  S1:-1.0  S2:-1.0  PA:-1.0
         As1   S1:-1.0  S2:-1.0  PA:-1.0
         As2   S1:-1.0  S2:-1.0  PA:-1.0
         Ad1   S1:-1.0  S2:-1.0  PA:-1.0
         Apd1  S1:-1.0  S2:-1.0  PA:-1.0
         Apol1 S1:-1.0  S2:-1.0  PA:-1.0
         AdR1  S1:-1.0  S2:-1.0  PA:-1.0
         AdP1  S1:-1.0  S2:-1.0  PA:-1.0
   N=7 tau=3 a=1.5: max relative residuals: cross closed form (Lemma 4.4) 9.8e-54; (1-k sn t)(P_l x P_l+1) 9.8e-54; P(t-D) x P(t+D) 1.4e-58; m.P=1 8.0e-61; n.V=1 7.8e-61; tangency of m(t) 4.0e-57; (V-f1)^2 factorisation 2.6e-57; r1 = dn h/(k cn h) 1.6e-61; r1 = sn(K+iK'+h) 1.6e-61; r1 = sn(K+iK'-h) 1.6e-61
      zero orders of d_l x d_l+1 in t: Pi'_+ = K+iK': 3.0, Pi'_+ = K+3iK': 3.0, Pi'_- = 3K+iK': 1.0, Pi = iK': -1.2636e-13; value at t = iK': (-7.6639739861e-64 - 0.0159143589842j) (claimed -2iab c0/s0 = (0.0 - 0.0159143589842j))
      vertex formulas (Lemma 3.2), max relative residual: (a) foot on chord 1.6e-58; (a) foot on tangent 6.2e-61; (b) inversion 5.0e-61; (d) polar vertex 1.5e-53; (e) dual R 9.9e-62; (e) dual P 8.2e-59
      orders at S1, S2, PA (negative = pole order, inf = identically ~0 near the point):
         A     S1:1.0  S2:1.0  PA:-1.0
         Ap    S1:1.0  S2:1.0  PA:-1.0
         Ab1   S1:1.0  S2:-1.0  PA:-2.839e-13
         Abp1  S1:1.0  S2:-1.0  PA:-2.839e-13
         As1   S1:1.0  S2:1.0  PA:-2.0
         As2   S1:1.0  S2:1.0  PA:-2.0
         Ad1   S1:-1.0  S2:1.0  PA:2.839e-13
         Apd1  S1:-1.0  S2:1.0  PA:2.839e-13
         Apol1 S1:-1.0  S2:1.0  PA:2.839e-13
         AdR1  S1:-1.0  S2:1.0  PA:2.839e-13
         AdP1  S1:-1.0  S2:1.0  PA:2.839e-13
   N=8 tau=1 a=2.2: max relative residuals: cross closed form (Lemma 4.4) 2.9e-60; (1-k sn t)(P_l x P_l+1) 2.6e-60; P(t-D) x P(t+D) 2.7e-61; m.P=1 3.4e-61; n.V=1 3.5e-61; tangency of m(t) 6.6e-61; (V-f1)^2 factorisation 3.2e-60; r1 = dn h/(k cn h) 0; r1 = sn(K+iK'+h) 4.0e-63; r1 = sn(K+iK'-h) 1.2e-61
      zero orders of d_l x d_l+1 in t: Pi'_+ = K+iK': 3.0, Pi'_+ = K+3iK': 3.0, Pi'_- = 3K+iK': 1.0, Pi = iK': -1.2636e-13; value at t = iK': (-7.0340380054e-62 - 6.93534313387j) (claimed -2iab c0/s0 = (0.0 - 6.93534313387j))
      vertex formulas (Lemma 3.2), max relative residual: (a) foot on chord 8.3e-61; (a) foot on tangent 8.8e-61; (b) inversion 2.1e-61; (d) polar vertex 4.0e-61; (e) dual R 9.0e-62; (e) dual P 1.2e-61
      orders at S1, S2, PA (negative = pole order, inf = identically ~0 near the point):
         A     S1:1.0  S2:1.0  PA:-1.0
         Ap    S1:-1.0  S2:-1.0  PA:1.0
         Ab1   S1:-1.0  S2:-1.0  PA:1.0
         Abp1  S1:-1.0  S2:-1.0  PA:1.0
         As1   S1:1.0  S2:1.0  PA:-1.0
         As2   S1:1.0  S2:1.0  PA:-1.0
         Ad1   S1:-1.0  S2:-1.0  PA:1.0
         Apd1  S1:-1.0  S2:-1.0  PA:1.0
         Apol1 S1:-1.0  S2:-1.0  PA:1.0
         AdR1  S1:-1.0  S2:-1.0  PA:1.0
         AdP1  S1:-1.0  S2:-1.0  PA:1.0
   N=8 tau=3 a=1.5: max relative residuals: cross closed form (Lemma 4.4) 1.6e-58; (1-k sn t)(P_l x P_l+1) 1.6e-58; P(t-D) x P(t+D) 1.2e-59; m.P=1 8.8e-61; n.V=1 8.8e-61; tangency of m(t) 1.2e-59; (V-f1)^2 factorisation 7.3e-60; r1 = dn h/(k cn h) 0; r1 = sn(K+iK'+h) 2.3e-63; r1 = sn(K+iK'-h) 2.3e-63
      zero orders of d_l x d_l+1 in t: Pi'_+ = K+iK': 3.0, Pi'_+ = K+3iK': 3.0, Pi'_- = 3K+iK': 1.0, Pi = iK': -1.2636e-13; value at t = iK': (7.78500650968e-63 - 0.283962659822j) (claimed -2iab c0/s0 = (0.0 - 0.283962659822j))
      vertex formulas (Lemma 3.2), max relative residual: (a) foot on chord 3.6e-61; (a) foot on tangent 2.9e-61; (b) inversion 6.5e-61; (d) polar vertex 1.2e-58; (e) dual R 1.0e-61; (e) dual P 1.8e-61
      orders at S1, S2, PA (negative = pole order, inf = identically ~0 near the point):
         A     S1:1.0  S2:1.0  PA:-1.0
         Ap    S1:-1.0  S2:-1.0  PA:1.0
         Ab1   S1:-1.0  S2:-1.0  PA:1.0
         Abp1  S1:-1.0  S2:-1.0  PA:1.0
         As1   S1:1.0  S2:1.0  PA:-1.0
         As2   S1:1.0  S2:1.0  PA:-1.0
         Ad1   S1:-1.0  S2:-1.0  PA:1.0
         Apd1  S1:-1.0  S2:-1.0  PA:1.0
         Apol1 S1:-1.0  S2:-1.0  PA:1.0
         AdR1  S1:-1.0  S2:-1.0  PA:1.0
         AdP1  S1:-1.0  S2:-1.0  PA:1.0
   N=10 tau=3 a=1.5: max relative residuals: cross closed form (Lemma 4.4) 1.2e-60; (1-k sn t)(P_l x P_l+1) 1.2e-60; P(t-D) x P(t+D) 9.9e-61; m.P=1 3.9e-61; n.V=1 3.1e-61; tangency of m(t) 3.3e-61; (V-f1)^2 factorisation 3.2e-60; r1 = dn h/(k cn h) 5.6e-61; r1 = sn(K+iK'+h) 5.6e-62; r1 = sn(K+iK'-h) 5.7e-61
      zero orders of d_l x d_l+1 in t: Pi'_+ = K+iK': 3.0, Pi'_+ = K+3iK': 3.0, Pi'_- = 3K+iK': 1.0, Pi = iK': -1.2636e-13; value at t = iK': (4.08246102037e-62 - 1.18912341112j) (claimed -2iab c0/s0 = (0.0 - 1.18912341112j))
      vertex formulas (Lemma 3.2), max relative residual: (a) foot on chord 2.5e-61; (a) foot on tangent 1.2e-60; (b) inversion 8.3e-61; (d) polar vertex 4.0e-60; (e) dual R 1.0e-61; (e) dual P 5.3e-61
      orders at S1, S2, PA (negative = pole order, inf = identically ~0 near the point):
         A     S1:-1.0  S2:-1.0  PA:-1.0
         Ap    S1:1.0  S2:1.0  PA:1.0
         Ab1   S1:-1.0  S2:-1.0  PA:-1.0
         Abp1  S1:-1.0  S2:-1.0  PA:-1.0
         As1   S1:-1.0  S2:-1.0  PA:-1.0
         As2   S1:-1.0  S2:-1.0  PA:-1.0
         Ad1   S1:-1.0  S2:-1.0  PA:-1.0
         Apd1  S1:-1.0  S2:-1.0  PA:-1.0
         Apol1 S1:-1.0  S2:-1.0  PA:-1.0
         AdR1  S1:-1.0  S2:-1.0  PA:-1.0
         AdP1  S1:-1.0  S2:-1.0  PA:-1.0
   N=9 tau=2 a=2.6: max relative residuals: cross closed form (Lemma 4.4) 2.1e-60; (1-k sn t)(P_l x P_l+1) 1.8e-60; P(t-D) x P(t+D) 3.4e-61; m.P=1 3.1e-61; n.V=1 7.0e-61; tangency of m(t) 9.5e-61; (V-f1)^2 factorisation 3.4e-60; r1 = dn h/(k cn h) 2.6e-60; r1 = sn(K+iK'+h) 2.3e-60; r1 = sn(K+iK'-h) 2.9e-60
      zero orders of d_l x d_l+1 in t: Pi'_+ = K+iK': 3.0, Pi'_+ = K+3iK': 3.0, Pi'_- = 3K+iK': 1.0, Pi = iK': -1.2636e-13; value at t = iK': (-3.47824608262e-61 - 7.37262898951j) (claimed -2iab c0/s0 = (0.0 - 7.37262898951j))
      vertex formulas (Lemma 3.2), max relative residual: (a) foot on chord 2.6e-61; (a) foot on tangent 1.1e-60; (b) inversion 5.2e-61; (d) polar vertex 1.1e-60; (e) dual R 3.5e-61; (e) dual P 1.1e-60
      orders at S1, S2, PA (negative = pole order, inf = identically ~0 near the point):
         A     S1:1.0  S2:1.0  PA:-1.0
         Ap    S1:1.0  S2:1.0  PA:-1.0
         Ab1   S1:1.0  S2:-1.0  PA:-1.099e-12
         Abp1  S1:1.0  S2:-1.0  PA:-1.099e-12
         As1   S1:1.0  S2:1.0  PA:-2.0
         As2   S1:1.0  S2:1.0  PA:-2.0
         Ad1   S1:-1.0  S2:1.0  PA:1.099e-12
         Apd1  S1:-1.0  S2:1.0  PA:1.099e-12
         Apol1 S1:-1.0  S2:1.0  PA:1.099e-12
         AdR1  S1:-1.0  S2:1.0  PA:1.099e-12
         AdP1  S1:-1.0  S2:1.0  PA:1.099e-12
   N=12 tau=5 a=1.5: max relative residuals: cross closed form (Lemma 4.4) 2.5e-55; (1-k sn t)(P_l x P_l+1) 2.5e-55; P(t-D) x P(t+D) 2.4e-58; m.P=1 4.7e-61; n.V=1 6.2e-61; tangency of m(t) 6.1e-59; (V-f1)^2 factorisation 1.5e-57; r1 = dn h/(k cn h) 3.1e-61; r1 = sn(K+iK'+h) 3.1e-61; r1 = sn(K+iK'-h) 3.1e-61
      zero orders of d_l x d_l+1 in t: Pi'_+ = K+iK': 3.0, Pi'_+ = K+3iK': 3.0, Pi'_- = 3K+iK': 1.0, Pi = iK': -1.2636e-13; value at t = iK': (-2.12364642654e-63 - 0.0416557247266j) (claimed -2iab c0/s0 = (0.0 - 0.0416557247266j))
      vertex formulas (Lemma 3.2), max relative residual: (a) foot on chord 1.8e-59; (a) foot on tangent 4.1e-61; (b) inversion 1.7e-61; (d) polar vertex 2.3e-55; (e) dual R 1.1e-61; (e) dual P 9.0e-60
      orders at S1, S2, PA (negative = pole order, inf = identically ~0 near the point):
         A     S1:1.0  S2:1.0  PA:-1.0
         Ap    S1:-1.0  S2:-1.0  PA:1.0
         Ab1   S1:-1.0  S2:-1.0  PA:1.0
         Abp1  S1:-1.0  S2:-1.0  PA:1.0
         As1   S1:1.0  S2:1.0  PA:-1.0
         As2   S1:1.0  S2:1.0  PA:-1.0
         Ad1   S1:-1.0  S2:-1.0  PA:1.0
         Apd1  S1:-1.0  S2:-1.0  PA:1.0
         Apol1 S1:-1.0  S2:-1.0  PA:1.0
         AdR1  S1:-1.0  S2:-1.0  PA:1.0
         AdP1  S1:-1.0  S2:-1.0  PA:1.0

Comparison with Table 2 of the note (bounds: order >= -1 at the listed pole sets, >= 0 elsewhere;
 for odd N the S1-functions vanish on S2 and the S2-functions on S1; A*_1 >= -2 at PA, >= -1 for even N):
   N=3 tau=1: A*_1 order at PA = -2.0
   N=4 tau=1: A*_1 order at PA = -1.0
   N=5 tau=2: A*_1 order at PA = -2.0
   N=6 tau=1: A*_1 order at PA = -1.0
   N=7 tau=3: A*_1 order at PA = -2.0
   N=8 tau=1: A*_1 order at PA = -1.0
   N=8 tau=3: A*_1 order at PA = -1.0
   N=10 tau=3: A*_1 order at PA = -1.0
   N=9 tau=2: A*_1 order at PA = -2.0
   N=12 tau=5: A*_1 order at PA = -1.0
violations: none
