S1  Lemma 2.3(a)
  ok    S1  300/300 random (lambda, E): isotropic E has dim <= n-s, contains K when dim = n-s, and extends to an isotropic subspace of dim n-s containing E and K
S2  Proposition 4.2(b), all nondegenerate integer 4x4 skew matrices with entries in [-B,B]
  ok    S2 B=2  13312 nondegenerate matrices; pair found and valid: 13312 (zero entry: 9504, mixed row: 3808); neither case: 0
  ok    S2 B=3  108756 nondegenerate matrices; pair found and valid: 108756 (zero entry: 64116, mixed row: 44640); neither case: 0
S3  Proposition 4.2(a): rational constants, monomial systems in every dimension
  ok    S3  400/400 random rational lambda; (n, s) cases covered: 28; example sizes: n=2,s=0:24/24, n=2,s=1:31/31, n=3,s=0:31/31, n=3,s=1:30/30, n=4,s=0:22/22, n=4,s=1:12/12, n=4,s=2:15/15, n=5,s=0:20/20, n=5,s=1:12/12, n=5,s=2:14/14, n=6,s=0:12/12, n=6,s=1:14/14 ...
S4  Theorem B, rho = 2
  ok    S4 count  578 primitive nu in [-4,4]^3, criterion holds for 402
  ok    S4 (i)  criterion = existence of two independent vectors of nu^perp in {0..8}^3: 578/578
  ok    S4 (ii)  criterion = existence of commuting monomials with exponents in {0..4}^3 for k = u + sqrt2 w: 578/578
S5  Theorem B, rho = 1
  ok    S5  578/578 primitive kernel vectors
S6  commuting pairs of monomials (Section 7(D) of the note)
  ok    S6 Theorem A (1, sqrt2, sqrt3)  0 pairs (expected 0); first: None
  ok    S6 Example 4.1(a) (1,2,3), k = (2,3,1)  191 pairs (expected 191); first: ((0, 0, 1), (2, 3, 0))
  ok    S6 Example 4.1(b) (1, sqrt2, -1-sqrt2), k = (sqrt2, -1-sqrt2, 1)  0 pairs (expected 0); first: None
  ok    S6 Example 4.1(c) (1, 1, sqrt2), k = (1, sqrt2, 1)  86 pairs (expected 86); first: ((0, 1, 0), (1, 0, 1))
  ok    S6 Example 4.1(d) (sqrt2, -2, 1), k = (-2, 1, sqrt2)  0 pairs (expected 0); first: None
  ok    S6 Theorem D, degree <= 4  0 pairs (expected 0)
  ok    S6 CONTROL lambda_ij = 1 on R^4, degree <= 4  156 pairs (expected 156)
TOTAL failures: 0
running time: 11 s
