  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] nodal  k=3 s=3 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 4 (want 4); a*deg J_r = 12 >= 2*1; bound k m a = 9 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] nodal  k=3 s=3 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 4 (want 4); a*deg J_r = 12 >= 2*1; bound k m a = 9 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] conj   k=3 s=3 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 4 (want 4); a*deg J_r = 4 >= 1*1; bound k m a = 3 -> OK
  [5.1 mu=1] conj   k=3 s=3 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 4 (want 4); a*deg J_r = 4 >= 1*1; bound k m a = 3 -> OK
  [5.1 mu=1] nodal  k=4 s=4 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 6 (want 6); a*deg J_r = 18 >= 2*1; bound k m a = 12 -> OK
  [5.1 mu=1] conj   k=3 s=3 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 4 (want 4); a*deg J_r = 4 >= 1*1; bound k m a = 3 -> OK
  [5.1 mu=1] conj   k=4 s=4 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 6 (want 6); a*deg J_r = 6 >= 1*1; bound k m a = 4 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] conj   k=3 s=3 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 4 (want 4); a*deg J_r = 4 >= 1*1; bound k m a = 3 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] conj   k=3 s=3 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 4 (want 4); a*deg J_r = 4 >= 1*1; bound k m a = 3 -> OK
  [5.1 mu=1] conj   k=3 s=3 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 4 (want 4); a*deg J_r = 4 >= 1*1; bound k m a = 3 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] nodal  k=3 s=3 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 4 (want 4); a*deg J_r = 12 >= 2*1; bound k m a = 9 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [5.1 mu=1] conj   k=3 s=3 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 4 (want 4); a*deg J_r = 4 >= 1*1; bound k m a = 3 -> OK
  [5.1 mu=1] conj   k=3 s=3 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 4 (want 4); a*deg J_r = 4 >= 1*1; bound k m a = 3 -> OK
  [5.1 mu=1] conj   k=3 s=3 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 4 (want 4); a*deg J_r = 4 >= 1*1; bound k m a = 3 -> OK
  [5.1 mu=1] conj   k=2 s=2 r=m_D=1: orders [1, 0]; J_r(special point)=0; J_r|_D != 0: True; deg J_r = 2 (want 2); a*deg J_r = 2 >= 1*1; bound k m a = 2 -> OK
  [3.7] nodal quartic V(Q1,Q2): maximal minors at the node [Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1), Fraction(0, 1)] (all zero: True); bound (N-1)(k-1)a/2 = 4 >= 1
  numerical inequalities for k, r, f < 300: OK
  Lemma 3.7 constant within the quadric budget (real, non-real) for N=3..7: [(3, True, True), (4, True, True), (5, True, True), (6, False, False), (7, False, False)]
  Remark 8.2 genus (N, p_a, 2^N): [(3, 1, 8), (4, 5, 16), (5, 17, 32), (6, 49, 64), (7, 129, 128), (8, 321, 256)]
  Lemma 3.6(b) bound a-w versus Castelnuovo genus bound (w, a, a-w, pi): [(2, 2, 0, 0), (2, 3, 1, 1), (3, 3, 0, 0), (3, 4, 1, 1), (3, 5, 2, 2), (4, 4, 0, 0), (4, 5, 1, 1), (4, 6, 2, 2), (4, 7, 3, 3), (5, 5, 0, 0), (5, 6, 1, 1), (5, 7, 2, 2), (5, 8, 3, 3), (5, 9, 4, 4), (6, 6, 0, 0), (6, 7, 1, 1), (6, 8, 2, 2), (6, 9, 3, 3), (6, 10, 4, 4), (6, 11, 5, 5)]
  case-table arithmetic of Prop 1.7, Thm 1.5, Prop 8.1: [True, True, True, True, True, True, True, True, True] -> OK
  grid example: V(F_1..F_N) = k^N real points for N, k <= 4: OK
SUMMARY Prop 5.1 Case mu=1 examples: 30 run, failures 0; Lemma 3.7 node example: OK; numbers: OK; grids: OK
