p = 1000000007; sqrt2 -> 59713600, sqrt3 -> 82062379  (p = -1 mod 24, so both are squares)
== bracket (a,b,c) = (1, sqrt2, sqrt3), degree of g <= 10
  f = x+y+z              deg 1: dim ker_p = 11, powers of f = 11  -> CERTIFIED equal
  f = xy+z^2+x           deg 2: dim ker_p = 6, powers of f = 6  -> CERTIFIED equal
  f = x+yz               deg 2: dim ker_p = 6, powers of f = 6  -> CERTIFIED equal
  f = x^2y+z^3+y         deg 3: dim ker_p = 4, powers of f = 4  -> CERTIFIED equal
  f = xyz+x+y+z          deg 3: dim ker_p = 4, powers of f = 4  -> CERTIFIED equal
  f = x^2+y^2+z^2        deg 2: dim ker_p = 6, powers of f = 6  -> CERTIFIED equal
  f = xy+yz+zx           deg 2: dim ker_p = 6, powers of f = 6  -> CERTIFIED equal
  f = random deg2 #0     deg 2: dim ker_p = 6, powers of f = 6  -> CERTIFIED equal
  f = random deg2 #1     deg 2: dim ker_p = 6, powers of f = 6  -> CERTIFIED equal
  f = random deg2 #2     deg 2: dim ker_p = 6, powers of f = 6  -> CERTIFIED equal
  f = random deg3 #0     deg 3: dim ker_p = 4, powers of f = 4  -> CERTIFIED equal
  f = random deg3 #1     deg 3: dim ker_p = 4, powers of f = 4  -> CERTIFIED equal
  f = random deg3 #2     deg 3: dim ker_p = 4, powers of f = 4  -> CERTIFIED equal
  f = x^(1, 0, 0): dim ker_p = 11, multiples of primitive (np.int64(1), np.int64(0), np.int64(0)) in degree <= 10: 11 -> CERTIFIED equal
  f = x^(2, 0, 0): dim ker_p = 11, multiples of primitive (np.int64(1), np.int64(0), np.int64(0)) in degree <= 10: 11 -> CERTIFIED equal
  f = x^(1, 1, 0): dim ker_p = 6, multiples of primitive (np.int64(1), np.int64(1), np.int64(0)) in degree <= 10: 6 -> CERTIFIED equal
  f = x^(2, 3, 1): dim ker_p = 2, multiples of primitive (np.int64(2), np.int64(3), np.int64(1)) in degree <= 10: 2 -> CERTIFIED equal
  f = x^(1, 2, 2): dim ker_p = 3, multiples of primitive (np.int64(1), np.int64(2), np.int64(2)) in degree <= 10: 3 -> CERTIFIED equal
  all certified for (1,sqrt2,sqrt3): True
== controls with rational / Q-rank-2 brackets (degree of g <= 10); larger centralizers expected where a monomial pair exists
  (1,2,3)              f = x+y+z : dim ker_p = 16 (upper bound for the true centralizer dim; powers of f: 11)
  (1,2,3)              f = x     : dim ker_p = 21 (upper bound for the true centralizer dim; powers of f: 11)
  (1,2,3)              f = y     : dim ker_p = 26 (upper bound for the true centralizer dim; powers of f: 11)
  (1,2,3)              f = z     : dim ker_p = 18 (upper bound for the true centralizer dim; powers of f: 11)
  (1,1,sqrt2)          f = x+y+z : dim ker_p = 11 (upper bound for the true centralizer dim; powers of f: 11)
  (1,1,sqrt2)          f = x     : dim ker_p = 11 (upper bound for the true centralizer dim; powers of f: 11)
  (1,1,sqrt2)          f = y     : dim ker_p = 36 (upper bound for the true centralizer dim; powers of f: 11)
  (1,1,sqrt2)          f = z     : dim ker_p = 11 (upper bound for the true centralizer dim; powers of f: 11)
  (1,sqrt2,-1-sqrt2)   f = x+y+z : dim ker_p = 11 (upper bound for the true centralizer dim; powers of f: 11)
  (1,sqrt2,-1-sqrt2)   f = x     : dim ker_p = 11 (upper bound for the true centralizer dim; powers of f: 11)
  (1,sqrt2,-1-sqrt2)   f = y     : dim ker_p = 11 (upper bound for the true centralizer dim; powers of f: 11)
  (1,sqrt2,-1-sqrt2)   f = z     : dim ker_p = 11 (upper bound for the true centralizer dim; powers of f: 11)
  (sqrt2,-2,1)         f = x+y+z : dim ker_p = 11 (upper bound for the true centralizer dim; powers of f: 11)
  (sqrt2,-2,1)         f = x     : dim ker_p = 11 (upper bound for the true centralizer dim; powers of f: 11)
  (sqrt2,-2,1)         f = y     : dim ker_p = 11 (upper bound for the true centralizer dim; powers of f: 11)
  (sqrt2,-2,1)         f = z     : dim ker_p = 11 (upper bound for the true centralizer dim; powers of f: 11)
== Theorem D bracket on R^4, lambda_ij = theta^e_ij, e = (0,1,2,3,4,5), degree of g <= 7
  prime q = 1000000663, theta -> 141770888 (theta^6 = 2 mod q)
  f = x1+x2+x3+x4      deg 1: dim ker_q = 8, powers of f = 8 -> CERTIFIED equal
  f = x1x2+x3+x4^2     deg 2: dim ker_q = 4, powers of f = 4 -> CERTIFIED equal
  f = x1x3+x2x4        deg 2: dim ker_q = 4, powers of f = 4 -> CERTIFIED equal
  f = x1+x2x3x4        deg 3: dim ker_q = 3, powers of f = 3 -> CERTIFIED equal
  f = random deg2 #0   deg 2: dim ker_q = 4, powers of f = 4 -> CERTIFIED equal
  f = random deg2 #1   deg 2: dim ker_q = 4, powers of f = 4 -> CERTIFIED equal
  f = x^(1, 0, 0, 0): dim ker_q = 8, expected 8 -> CERTIFIED equal
  f = x^(1, 1, 0, 0): dim ker_q = 4, expected 4 -> CERTIFIED equal
  f = x^(0, 1, 1, 1): dim ker_q = 3, expected 3 -> CERTIFIED equal
  all certified for Theorem D bracket: True
total 0.6s
