== Part 1: exact centralizer dimensions ==
4.10a (1,2,3)              f=x+y+z     D=6: dim=8  powers-of-f=7  all kernel elts dependent on f (df^dg=0): False  [0.0s]
4.10a (1,2,3)              f=xy+z^2+x  D=6: dim=5  powers-of-f=4  all kernel elts dependent on f (df^dg=0): False  [0.0s]
4.10b (1,1,sqrt2)          f=x+y+z     D=6: dim=7  powers-of-f=7  all kernel elts dependent on f (df^dg=0): True  [0.0s]
4.10b (1,1,sqrt2)          f=xy+z^2+x  D=6: dim=4  powers-of-f=4  all kernel elts dependent on f (df^dg=0): True  [0.0s]
4.10c (1,sqrt2,-1-sqrt2)   f=x+y+z     D=6: dim=7  powers-of-f=7  all kernel elts dependent on f (df^dg=0): True  [0.0s]
4.10c (1,sqrt2,-1-sqrt2)   f=xy+z^2+x  D=6: dim=4  powers-of-f=4  all kernel elts dependent on f (df^dg=0): True  [0.0s]
4.10d (sqrt2,-2,1)         f=x+y+z     D=6: dim=7  powers-of-f=7  all kernel elts dependent on f (df^dg=0): True  [0.0s]
4.10d (sqrt2,-2,1)         f=xy+z^2+x  D=6: dim=4  powers-of-f=4  all kernel elts dependent on f (df^dg=0): True  [0.0s]
4.10e (1,sqrt2,sqrt3)      f=x+y+z     D=6: dim=7  powers-of-f=7  all kernel elts dependent on f (df^dg=0): True  [0.0s]
4.10e (1,sqrt2,sqrt3)      f=xy+z^2+x  D=6: dim=4  powers-of-f=4  all kernel elts dependent on f (df^dg=0): True  [0.0s]
Theorem D bracket f=x1+x2+x3+x4   D=4: dim=5 powers-of-f=5 dependent=True [0.0s]
Theorem D bracket f=x1x2+x3+x4^2  D=4: dim=3 powers-of-f=3 dependent=True [0.0s]
== Part 2: random sparse f, certified centralizer = C[f] in degree <= D ==
prime for K4: 1000000009 s mod p: 548614189 check s^4-10s^2+1 = 0 mod p: True
Theorem-A bracket, 150 random f (2-4 terms, deg<=3), D=6: {'certified': 148, 'mismatch': 2} [1s]
  mismatch f=[((0, 0, 1), 2), ((0, 0, 2), -1)] modp-dim=7 powers=4 -> exact dim=7, every kernel element has df^dg=0: True
  mismatch f=[((1, 0, 0), -3), ((3, 0, 0), 4)] modp-dim=7 powers=3 -> exact dim=7, every kernel element has df^dg=0: True
Theorem-A bracket, all binomials x^a +- x^b (deg<=3), D=6: {'certified': 324, 'mismatch': 18}
  mismatch f=[((1, 0, 0), 1), ((2, 0, 0), 1)] modp-dim=7 powers=4 -> exact dim=7, all df^dg=0: True
  mismatch f=[((1, 0, 0), 1), ((2, 0, 0), -1)] modp-dim=7 powers=4 -> exact dim=7, all df^dg=0: True
  mismatch f=[((1, 0, 0), 1), ((3, 0, 0), 1)] modp-dim=7 powers=3 -> exact dim=7, all df^dg=0: True
  mismatch f=[((1, 0, 0), 1), ((3, 0, 0), -1)] modp-dim=7 powers=3 -> exact dim=7, all df^dg=0: True
  mismatch f=[((0, 1, 0), 1), ((0, 2, 0), 1)] modp-dim=7 powers=4 -> exact dim=7, all df^dg=0: True
  mismatch f=[((0, 1, 0), 1), ((0, 2, 0), -1)] modp-dim=7 powers=4 -> exact dim=7, all df^dg=0: True
  (all binomial mismatches re-examined exactly; any with dependence False would be a counterexample)
control (1,2,3), f=x+y+z: modp dim = 8 powers = 7 exact (dim, all dependent) = (8, False)
control (1,2,3), f=x: exact (dim, all dependent) = (10, False)
prime for K6: 1000000663 theta mod p: 141770888 check: True
Theorem-D bracket, 60 random f (2-4 terms, deg<=2), D=4: {'certified': 60, 'mismatch': 0} [0s]
