A. Totaro exponents
  ok    formula of Theorem 0.1 reproduces the printed table l = 0..18
  ok    n in [8,400] with u(n) < [(n+1)/2]: [9, 11]
        n in [8,400] with u(n) = [(n+1)/2]: [8, 10, 12, 13]
  ok    binom(n+1,2)+1 < 2^(floor(n/2)+1) for 13 <= n <= 400
  ok    n^2+5n+8 < 2(n^2+n+2) for 5 <= n <= 2000 (growth factor < 2)
  ok    u(n) >= n - floor(log2(binom(n+1,2)+1)) >= [(n+1)/2] for 13 <= n <= 400
B. Term numbers
  ok    n= 8 m=18 r=[m/4]=4 conjectured term 5 u(n)=4  ([(2n+2)/4] = [(n+1)/2])
  ok       n=8: j=4, m-j-1-n = n+1-j = 5 = conjectured term
  ok    n= 9 m=20 r=[m/4]=5 conjectured term 5 u(n)=4  ([(2n+2)/4] = [(n+1)/2])
  ok       n=9: j=5, m-j-1-n = n+1-j = 5 = conjectured term
  ok    n=10 m=22 r=[m/4]=5 conjectured term 6 u(n)=5  ([(2n+2)/4] = [(n+1)/2])
  ok    n=11 m=24 r=[m/4]=6 conjectured term 6 u(n)=5  ([(2n+2)/4] = [(n+1)/2])
  ok       n=11: j=6, m-j-1-n = n+1-j = 6 = conjectured term
  ok    n=12 m=26 r=[m/4]=6 conjectured term 7 u(n)=6  ([(2n+2)/4] = [(n+1)/2])
  ok       n=9 with j=4 (computer-assisted input): term 6
  ok    n=6: j = u(n)+1 = 3, term 4
  ok    n=7: j = u(n)+1 = 4, term 4
C. g.c.d. step of Proposition 3.2(i)
  ok    gcd([L:K], 2^a) divides 2^(j-1) whenever v2[L:K] <= j-1, a <= j (j <= 12)
D. Lemma 3.1 through Chern characters
  ok    ch(2 l_i - l_{i-1}) = ch(h^{D-i}) = (1-e^{-h})^{D-i}, 1 <= i <= n-1, n <= 12
  ok    identity (2): 2^{j-1} l_{j-1} = l_0 + sum_{i<j} 2^{i-1} h^{D-i}, j <= 12
  ok    identity (2) with j = [m/4], D = m-2 is the formula printed in Karpenko 2017, m <= 58
E. Lemma 4.1: explicit forms (entries are signed monomials in t_1..t_2n)
  ok    n=8: dim chi = 16, dim phi' = 17
        phi' (n=8) = <1, -t15, -t16, t13*t15*t16, t14*t15*t16, -t11*t13*t14*t15*t16, -t12*t13*t14*t15*t16, t9*t11*t12*t13*t14*t15*t16, t10*t11*t12*t13*t14*t15*t16, -t7*t9*t10*t11*t12*t13*t14*t15*t16, -t8*t9*t10*t11*t12*t13*t14*t15*t16, t5*t7*t8*t9*t10*t11*t12*t13*t14*t15*t16, t6*t7*t8*t9*t10*t11*t12*t13*t14*t15*t16, -t3*t5*t6*t7*t8*t9*t10*t11*t12*t13*t14*t15*t16, -t4*t5*t6*t7*t8*t9*t10*t11*t12*t13*t14*t15*t16, t1*t3*t4*t5*t6*t7*t8*t9*t10*t11*t12*t13*t14*t15*t16, t2*t3*t4*t5*t6*t7*t8*t9*t10*t11*t12*t13*t14*t15*t16>
  ok    n=9: dim chi = 18, dim phi' = 19
  ok    n=11: dim chi = 22, dim phi' = 23
F. Gamma levels on OGr(n,2n+1), implementation of this note
        n=1: {'chiO': Fraction(1, 1), 'chi_en': Fraction(1, 1), 'lead_ok': True, 'rank': 2, 'pivots_one': True, 'en_in_K': True, 'gens_integral': True, 'level': 1, 'gcd_chi_gammaD': 1}
  ok    n=1: chi(O) = chi(e_n) = 1, leading terms of the generators e_1, 2e_j
  ok    n=1: K lattice of rank 2^n with all pivots 1; ch(e_n) and products integral
  ok    n=1: largest k with e_n in gamma^k = 1 (recorded 1; Prop. 3.2 bound 1)
  ok    n=1: gcd of chi on gamma^D = 1 = 2^u(1)
        n=2: {'chiO': Fraction(1, 1), 'chi_en': Fraction(1, 1), 'lead_ok': True, 'rank': 4, 'pivots_one': True, 'en_in_K': True, 'gens_integral': True, 'level': 2, 'gcd_chi_gammaD': 1}
  ok    n=2: chi(O) = chi(e_n) = 1, leading terms of the generators e_1, 2e_j
  ok    n=2: K lattice of rank 2^n with all pivots 1; ch(e_n) and products integral
  ok    n=2: largest k with e_n in gamma^k = 2 (recorded 2; Prop. 3.2 bound 2)
  ok    n=2: gcd of chi on gamma^D = 1 = 2^u(2)
        n=3: {'chiO': Fraction(1, 1), 'chi_en': Fraction(1, 1), 'lead_ok': True, 'rank': 8, 'pivots_one': True, 'en_in_K': True, 'gens_integral': True, 'level': 2, 'gcd_chi_gammaD': 2}
  ok    n=3: chi(O) = chi(e_n) = 1, leading terms of the generators e_1, 2e_j
  ok    n=3: K lattice of rank 2^n with all pivots 1; ch(e_n) and products integral
  ok    n=3: largest k with e_n in gamma^k = 2 (recorded 2; Prop. 3.2 bound 2)
  ok    n=3: gcd of chi on gamma^D = 2 = 2^u(3)
        n=4: {'chiO': Fraction(1, 1), 'chi_en': Fraction(1, 1), 'lead_ok': True, 'rank': 16, 'pivots_one': True, 'en_in_K': True, 'gens_integral': True, 'level': 4, 'gcd_chi_gammaD': 2}
  ok    n=4: chi(O) = chi(e_n) = 1, leading terms of the generators e_1, 2e_j
  ok    n=4: K lattice of rank 2^n with all pivots 1; ch(e_n) and products integral
  ok    n=4: largest k with e_n in gamma^k = 4 (recorded 4; Prop. 3.2 bound 4)
  ok    n=4: gcd of chi on gamma^D = 2 = 2^u(4)
        n=5: {'chiO': Fraction(1, 1), 'chi_en': Fraction(1, 1), 'lead_ok': True, 'rank': 32, 'pivots_one': True, 'en_in_K': True, 'gens_integral': True, 'level': 4, 'gcd_chi_gammaD': 2}
  ok    n=5: chi(O) = chi(e_n) = 1, leading terms of the generators e_1, 2e_j
  ok    n=5: K lattice of rank 2^n with all pivots 1; ch(e_n) and products integral
  ok    n=5: largest k with e_n in gamma^k = 4 (recorded 4; Prop. 3.2 bound 4)
  ok    n=5: gcd of chi on gamma^D = 2 = 2^u(5)
        n=6: {'chiO': Fraction(1, 1), 'chi_en': Fraction(1, 1), 'lead_ok': True, 'rank': 64, 'pivots_one': True, 'en_in_K': True, 'gens_integral': True, 'level': 4, 'gcd_chi_gammaD': 4}
  ok    n=6: chi(O) = chi(e_n) = 1, leading terms of the generators e_1, 2e_j
  ok    n=6: K lattice of rank 2^n with all pivots 1; ch(e_n) and products integral
  ok    n=6: largest k with e_n in gamma^k = 4 (recorded 4; Prop. 3.2 bound 4)
  ok    n=6: gcd of chi on gamma^D = 4 = 2^u(6)
        n=7: {'chiO': Fraction(1, 1), 'chi_en': Fraction(1, 1), 'lead_ok': True, 'rank': 128, 'pivots_one': True, 'en_in_K': True, 'gens_integral': True, 'level': 5, 'gcd_chi_gammaD': 8}
  ok    n=7: chi(O) = chi(e_n) = 1, leading terms of the generators e_1, 2e_j
  ok    n=7: K lattice of rank 2^n with all pivots 1; ch(e_n) and products integral
  ok    n=7: largest k with e_n in gamma^k = 5 (recorded 5; Prop. 3.2 bound 4)
  ok    n=7: gcd of chi on gamma^D = 8 = 2^u(7)
TOTAL failures: 0
