indep_charp.py -- independent verification run 2; seed 4711

[1] S_n = 0 over GF(q) (DP), controls: unsigned sum and random symmetric weights
   GF(2) (char 2): n=3:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0)
   GF(3) (char 3): n=3:3/3 zero (ctrl unsigned nonzero 1, sym nonzero 0); n=4:3/3 zero (ctrl unsigned nonzero 1, sym nonzero 0)
   GF(4) (char 2): n=3:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=4:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=5:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0)
   GF(5) (char 5): n=3:3/3 zero (ctrl unsigned nonzero 2, sym nonzero 0); n=4:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 0); n=5:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 1); n=6:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3)
   GF(7) (char 7): n=3:3/3 zero (ctrl unsigned nonzero 1, sym nonzero 0); n=4:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 0); n=5:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 3); n=6:3/3 zero (ctrl unsigned nonzero 1, sym nonzero 2); n=7:3/3 zero (ctrl unsigned nonzero 2, sym nonzero 0); n=8:3/3 zero (ctrl unsigned nonzero 2, sym nonzero 0)
   GF(8) (char 2): n=3:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=4:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=5:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=6:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=7:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=8:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=9:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0)
   GF(9) (char 3): n=3:3/3 zero (ctrl unsigned nonzero 2, sym nonzero 0); n=4:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 0); n=5:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 2); n=6:3/3 zero (ctrl unsigned nonzero 1, sym nonzero 3); n=7:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 0); n=8:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 0); n=9:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=10:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3)
   GF(11) (char 11): n=3:3/3 zero (ctrl unsigned nonzero 1, sym nonzero 0); n=4:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 0); n=5:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=6:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=7:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 0); n=8:3/3 zero (ctrl unsigned nonzero 2, sym nonzero 0); n=9:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=10:3/3 zero (ctrl unsigned nonzero 2, sym nonzero 2); n=11:1/1 zero (ctrl unsigned nonzero 1, sym nonzero 0); n=12: no random configuration found (skipped)
   GF(13) (char 13): n=3:3/3 zero (ctrl unsigned nonzero 1, sym nonzero 0); n=4:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 0); n=5:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=6:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=7:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 0); n=8:3/3 zero (ctrl unsigned nonzero 2, sym nonzero 0); n=9:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=10:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=11:1/1 zero (ctrl unsigned nonzero 1, sym nonzero 0); n=12:1/1 zero (ctrl unsigned nonzero 1, sym nonzero 0)
   GF(16) (char 2): n=3:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=4:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=5:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=6:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=7:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=8:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=9:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=10:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=11:1/1 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=12:1/1 zero (ctrl unsigned nonzero 0, sym nonzero 0)
   GF(25) (char 5): n=3:3/3 zero (ctrl unsigned nonzero 1, sym nonzero 0); n=4:3/3 zero (ctrl unsigned nonzero 2, sym nonzero 0); n=5:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=6:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=7:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 0); n=8:3/3 zero (ctrl unsigned nonzero 2, sym nonzero 0); n=9:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 1); n=10:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=11:1/1 zero (ctrl unsigned nonzero 1, sym nonzero 0); n=12:1/1 zero (ctrl unsigned nonzero 1, sym nonzero 0)
   GF(27) (char 3): n=3:3/3 zero (ctrl unsigned nonzero 2, sym nonzero 0); n=4:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 0); n=5:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=6:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=7:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 0); n=8:3/3 zero (ctrl unsigned nonzero 2, sym nonzero 0); n=9:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=10:3/3 zero (ctrl unsigned nonzero 3, sym nonzero 3); n=11:1/1 zero (ctrl unsigned nonzero 1, sym nonzero 0); n=12:1/1 zero (ctrl unsigned nonzero 1, sym nonzero 0)
   GF(32) (char 2): n=3:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=4:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=5:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=6:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=7:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=8:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=9:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=10:3/3 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=11:1/1 zero (ctrl unsigned nonzero 0, sym nonzero 0); n=12:1/1 zero (ctrl unsigned nonzero 0, sym nonzero 0)
   note: in characteristic 2 every signed sum equals the unsigned one, and every symmetric
   weight function gives 0 for n>=3 (terms cancel in reversal pairs, 2=0); so char 2 is trivial.
   (controls are informational: in very small fields a random control can vanish by chance)

[2] Theorem 1.4(a) over GF(q) with random traceless matrices (general-matrix control informational;
    it can vanish by chance in small fields, e.g. a separate probe gave 4/5 nonzero at GF(8), n=8)
   GF(8) char 2: n=3: equals PT(123)tr(M2M3)I: True; n=4: zero=True, general-matrix control nonzero=True; n=5: zero=True, general-matrix control nonzero=True; n=6: zero=True, general-matrix control nonzero=True; n=7: zero=True, general-matrix control nonzero=True; n=8: zero=True, general-matrix control nonzero=False
   GF(16) char 2: n=3: equals PT(123)tr(M2M3)I: True; n=4: zero=True, general-matrix control nonzero=True; n=5: zero=True, general-matrix control nonzero=True; n=6: zero=True, general-matrix control nonzero=True; n=7: zero=True, general-matrix control nonzero=True; n=8: zero=True, general-matrix control nonzero=True
   GF(9) char 3: n=3: equals PT(123)tr(M2M3)I: True; n=4: zero=True, general-matrix control nonzero=True; n=5: zero=True, general-matrix control nonzero=True; n=6: zero=True, general-matrix control nonzero=True; n=7: zero=True, general-matrix control nonzero=True; n=8: zero=True, general-matrix control nonzero=True
   GF(3) char 3: n=3: equals PT(123)tr(M2M3)I: True; n=4: zero=True, general-matrix control nonzero=True
   GF(5) char 5: n=3: equals PT(123)tr(M2M3)I: True; n=4: zero=True, general-matrix control nonzero=True; n=5: zero=True, general-matrix control nonzero=True; n=6: zero=True, general-matrix control nonzero=True
   GF(7) char 7: n=3: equals PT(123)tr(M2M3)I: True; n=4: zero=True, general-matrix control nonzero=True; n=5: zero=True, general-matrix control nonzero=True; n=6: zero=True, general-matrix control nonzero=True; n=7: zero=True, general-matrix control nonzero=True; n=8: zero=True, general-matrix control nonzero=True
   GF(11) char 11: n=3: equals PT(123)tr(M2M3)I: True; n=4: zero=True, general-matrix control nonzero=True; n=5: zero=True, general-matrix control nonzero=True; n=6: zero=True, general-matrix control nonzero=True; n=7: zero=True, general-matrix control nonzero=True; n=8: zero=True, general-matrix control nonzero=True

[3] Lemma 2.2 (multilinear Ree) over F_p: annihilator of Sh_C, dimension and expansion (2.1)
   p=2: m=2: dim=1 (=(m-1)!=1), expansion ok=True; m=3: dim=2 (=(m-1)!=2), expansion ok=True; m=4: dim=6 (=(m-1)!=6), expansion ok=True; m=5: dim=24 (=(m-1)!=24), expansion ok=True; m=6: dim=120 (=(m-1)!=120), expansion ok=True
   p=3: m=2: dim=1 (=(m-1)!=1), expansion ok=True; m=3: dim=2 (=(m-1)!=2), expansion ok=True; m=4: dim=6 (=(m-1)!=6), expansion ok=True; m=5: dim=24 (=(m-1)!=24), expansion ok=True; m=6: dim=120 (=(m-1)!=120), expansion ok=True
   p=5: m=2: dim=1 (=(m-1)!=1), expansion ok=True; m=3: dim=2 (=(m-1)!=2), expansion ok=True; m=4: dim=6 (=(m-1)!=6), expansion ok=True; m=5: dim=24 (=(m-1)!=24), expansion ok=True; m=6: dim=120 (=(m-1)!=120), expansion ok=True
   p=7: m=2: dim=1 (=(m-1)!=1), expansion ok=True; m=3: dim=2 (=(m-1)!=2), expansion ok=True; m=4: dim=6 (=(m-1)!=6), expansion ok=True; m=5: dim=24 (=(m-1)!=24), expansion ok=True; m=6: dim=120 (=(m-1)!=120), expansion ok=True
   contrast (not multilinear): over F_p the word x^p is orthogonal to all proper shuffles but is not a Lie polynomial,
   p=2: binomials [0]; p=3: binomials [0, 0]; p=5: binomials [0, 0, 0, 0]  -> Ree's theorem itself needs char 0 or multilinearity, as the paper says.

FAILURES: 0 []
elapsed 4.1s
