(A) invariance of the Gaussian-parity bracket on all chord diagrams with at most 7 chords
    n = 0:    1 classes; moves R1     0, R2     0, R3     0
    n = 1:    1 classes; moves R1     1, R2     0, R3     0
    n = 2:    2 classes; moves R1     2, R2     2, R3     0
    n = 3:    5 classes; moves R1     6, R2     7, R3     6
    n = 4:   17 classes; moves R1    19, R2    24, R3    19
    n = 5:   79 classes; moves R1    89, R2   100, R3   100
    n = 6:  554 classes; moves R1   607, R2   661, R3   708
    n = 7: 5283 classes; moves R1  5712, R2  5905, R3  6901
    total: 20869 moves (R1 6436, R2 6699, R3 7734), violations: 0
(C) negative controls
    false third move (one pair exchanged) changes the bracket in 1483 of 7734 cases
    false parity (odd = linked with >= 2 chords) changes the bracket in 1037 of 2348 moves
(B) invariance of the mixed-parity bracket on all two-component diagrams with at most 6 vertices
    n = 0:    1 classes; moves R1     0, R2     0, R3     0
    n = 1:    2 classes; moves R1     1, R2     0, R3     0
    n = 2:    5 classes; moves R1     5, R2     3, R3     0
    n = 3:   14 classes; moves R1    16, R2    15, R3    12
    n = 4:   54 classes; moves R1    63, R2    65, R3    53
    n = 5:  273 classes; moves R1   316, R2   316, R3   308
    n = 6: 1962 classes; moves R1  2202, R2  2174, R3  2327
    total: 7876 moves (R1 2603, R2 2573, R3 2700), violations: 0
(D) parts (iii) and (iv) by enumeration
    chord diagrams with 0 chords and bracket {K1}: 0
    chord diagrams with 1 chords and bracket {K1}: 0
    chord diagrams with 2 chords and bracket {K1}: 0
    chord diagrams with 3 chords and bracket {K1}: 0
    chord diagrams with 4 chords and bracket {K1}: 0
    chord diagrams with 5 chords and bracket {K1}: 0
    chord diagrams with 6 chords and bracket {K1}: 0
    chord diagrams with 7 chords and bracket {K1}: 2 (K1, K2)
    two-component diagrams (X, 1 2 3 4 5 6), X an arrangement of e, e, 1, ..., 6: 20160 arrangements;
    isomorphism classes with bracket {D1}: 2 (D1, D2)
(E) chord diagrams with 7 chords satisfying Proposition 4.1 with a non-isomorphic third-move image
    8 classes
    4 of them share one bracket; all 7-chord classes with this bracket: 4
        a b a c d b e c f g d g e f
        a b a c d b e f e c g d g f
        a b a c d b e f e g d g c f
        a b a c d e f b g c f d g e
    2 of them share one bracket; all 7-chord classes with this bracket: 2
        a b a c d b e f e d g c g f
        a b a c d b e f e g c g d f
    2 of them share one bracket; all 7-chord classes with this bracket: 2
        a b a c d c e b f g d g e f
        a b a c d e f b f c g d g e
RESULT: no violation of invariance
