# r2_words.py seed=20261010
## Lemma 4.7, exhaustive ranges
k=1 tau in [-3,3] n in [-3, -2, -1, 1, 2, 3]: 42 words, 0 trivial, 0 without a trivial proper prefix  [0.0s]
k=2 tau in [-3,3] n in [-3, -2, -1, 1, 2, 3]: 1764 words, 6 trivial, 0 without a trivial proper prefix  [0.0s]
k=3 tau in [-3,3] n in [-2, -1, 1, 2]: 21952 words, 6 trivial, 0 without a trivial proper prefix  [0.0s]
k=4 tau in [-2,2] n in [-2, -1, 1, 2]: 160000 words, 164 trivial, 0 without a trivial proper prefix  [0.1s]
k=5 tau in [-2,2] n in [-1, 1]: 100000 words, 0 trivial, 0 without a trivial proper prefix  [0.1s]
k=6 tau in [-1,1] n in [-1, 1]: 46656 words, 260 trivial, 0 without a trivial proper prefix  [0.1s]
k=7 tau in [-1,1] n in [-1, 1]: 279936 words, 0 trivial, 0 without a trivial proper prefix  [0.4s]
k=8 tau in [-1,1] n in [-1, 1]: 1679616 words, 3366 trivial, 0 without a trivial proper prefix  [2.6s]
Lemma 4.7 exhaustive: 3802 trivial words, 0 violations
## Lemma 4.7, random trivial words (1 to 14 random insertions of cancelling pairs g^e g^-e, |e| <= 3)
random trivial words: 200000 (from 408784 attempts), k up to 8, histogram of k: {2: 142631, 3: 23978, 4: 27246, 5: 4282, 6: 1687, 7: 151, 8: 25}, violations 0  [4.5s]
## Lemma 4.7, random trivial words (10 to 40 random insertions of cancelling pairs g^e g^-e, |e| <= 2)
random trivial words: 100000 (from 108561 attempts), k up to 16, histogram of k: {2: 23900, 3: 7274, 4: 23262, 5: 9942, 6: 14678, 7: 7271, 8: 6832, 9: 3209, 10: 2128, 11: 883, 12: 439, 13: 116, 14: 50, 15: 11, 16: 5}, violations 0  [3.9s]
## Lemma 4.7, control: some n_j = 0 allowed
with an exponent n_j = 0: 23 trivial words (k <= 3) without a trivial proper prefix, e.g. tau=(-2, 2) n=(0, 0)
## Lemma 4.8, the range that occurs in the slice lemma: n_j = +-1 (j<k), n_k in {-1,0,1}
configurations 45264, with at least one weight of trivial word 571, with a non-zero invariant vector 0  [4.2s]
## Lemma 4.8, exhaustive: k <= 4, d_j <= 3, n_j in {+-1,+-2} (j<k), n_k in [-2,2]
configurations 28275, with at least one weight of trivial word 225, with a non-zero invariant vector 0  [0.4s]
## Lemma 4.8, random data: k <= 6, d_j <= 4 (dimension <= 2000), n_j in [-3,3] non-zero for j<k, n_k in [-3,3]
random configurations 300000, with at least one weight of trivial word 7220, with a non-zero invariant vector 0  [13.6s]
## Lemma 4.8, control: an inner exponent n_j = 0 (j < k) allowed
control configurations 1044, with a non-zero invariant vector 30 (e.g. d=(1, 1) n=(0, 0): dimension 1)
SUMMARY: Lemma 4.7 violations 0; Lemma 4.8 violations 0; controls (hypothesis dropped) with failing conclusion: 23 and 30
