# r2_misc.py
## (M1) Lemma 5.2(a)
word length <= 2, series truncated after degree 4: 17 elements, 21 monomials, rank modulo p = 17  (linearly independent)
word length <= 3, series truncated after degree 6: 53 elements, 83 monomials, rank modulo p = 53  (linearly independent)
word length <= 4, series truncated after degree 8: 161 elements, 325 monomials, rank modulo p = 161  (linearly independent)
word length <= 5, series truncated after degree 9: 485 elements, 763 monomials, rank modulo p = 453  (truncation too short to separate them)
word length <= 5, series truncated after degree 10: 485 elements, 1275 monomials, rank modulo p = 485  (linearly independent)
word length <= 5, series truncated after degree 8: 485 elements, 437 monomials, rank modulo p = 357  (truncation too short to separate them)
## (M4) Remark 6.2
mu(a - a^-1) up to degree 2 = 2*X + -1*XX   [so mu(S(a)) = -i X + higher terms]
mu(b - b^-1) up to degree 2 = 2*Y + -1*YY
mu([a - a^-1, b - b^-1]) up to degree 2 = 4*XY + -4*YX   [= (2i)^2 mu([S(a),S(b)]) = -4 mu([S(a),S(b)]); so mu([S(a),S(b)]) = -[X,Y] + higher terms]
## (M2) the group of triples of Section 7.1
associativity on 281250 triples of elements: True; inverses (-i,-j,-k+2ji): True
no non-trivial element of the box has a trivial power x^n, n <= 12: True; x^n = (ni, nj, nk + ji n(n-1)) confirmed
commutators [x,y] = (0,0,2(j i' - j' i)): confirmed on the box; all commutators central: True; third coordinates of commutators found: [-16, -12, -10, -8, -6, -4] ... (all even: True; the value 2 occurs: True)
(i,j,k) -> (i, j, k mod 2) is a homomorphism onto Z^2 x Z_2: True; its kernel {(0,0,2n)} is the commutator subgroup
## (M3) Example 4.10
T_13|24  coefficient at f_a f_b f_A f_B: +1.000+0.000i ; at f_a f_B f_A f_b: +1.000+0.000i
D        coefficient at f_a f_b f_A f_B: +1.000+0.000i ; at f_a f_B f_A f_b: -1.000+0.000i
T_12|34  coefficient at f_a f_b f_A f_B: +0.000+0.000i ; at f_a f_B f_A f_b: +0.000+0.000i
T_14|23  coefficient at f_a f_b f_A f_B: +0.000+0.000i ; at f_a f_B f_A f_b: +0.000+0.000i
max |D + (signed sum of the 24 permutations of f_a f_A f_b f_B)| = 3.33e-16
