PART A  worked example: x1, x2, x3, x1-x2, x1-x3, x2-x3, x4, x1-x4 in Q^4
  ok    rank 4, 8 hyperplanes (n = 2k)
  ok    irreducible (fundamental circuits and all bipartitions agree)
  ok    multiple flats: five triple points [[0, 1, 3], [0, 2, 4], [0, 6, 7], [1, 2, 5], [3, 4, 5]] (0-based indices), no quadruple point
  ok    u = 5, v = 0, sum C(m_Z-1,2) = 5 = k+1
  ok    degrees [3, 2, 2, 2, 2, 2, 1, 1], so s = 2 hyperplanes of degree 1
  ok    formal; dim F(A) = n - k = 4
  ok    hypergraph connected (Lemma 4.1(b))
  ok    member of the class C_4
  ok    exactly one linear dependency among the 5 local relations (5 vectors in dimension 4)
  ok    its support T = the four triple points of K_4, W = the six hyperplanes of K_4
  ok    t = 4, w = 6, c = 1, s' = 2
  ok    inequalities (6) and (7): 3t >= 2w, s' - (k+1-t) = 1 >= t + c - 4 = 1 >= 1
  ok    X = intersection of the hyperplanes other than x4, x1-x4 is a line with A_X = A minus these two
  ok    X is spanned by e_4
  ok    X is modular (rank formula and explicit subspaces)
  ok    A_X is formal, A_X/X is irreducible and lies in C_3 with (u,v) = (4,0): a complete quadrilateral
  ok    the chain V > {x1=0} > {x1=x2=0} > K e_4 > 0 consists of modular elements
  ok    sizes of the steps: 1, 2, 3, 2
  ok    supersolvable (search for a modular chain from the definition)
  ok    Lemmas 4.1, 4.2 (every admissible pair), 4.3 and Theorem 5.2 hold []
  ok    worked example: all 4 derivations are logarithmic (theta(alpha_H) divisible by alpha_H, 8 forms)
  ok    worked example: homogeneous of degrees [1, 2, 2, 3], sum = number of hyperplanes = 8
  ok    worked example: det(theta_i(x_j)) = (-1) * Q(A) as polynomials (12 monomials)
  ok    chi(t) = (t-1)(t-2)^2(t-3) (Moebius function of L(A))
  ok    identity (4): sum C(d_i,2) = sum C(m_Z-1,2) = 5
  [0.0s]
PART B  sharpness examples
  ok    non-Fano: all 3 derivations are logarithmic (theta(alpha_H) divisible by alpha_H, 7 forms)
  ok    non-Fano: homogeneous of degrees [1, 3, 3], sum = number of hyperplanes = 7
  ok    non-Fano: det(theta_i(x_j)) = (2) * Q(A) as polynomials (10 monomials)
  ok    non-Fano: rank 3, irreducible, formal
  ok    non-Fano: six triple points, degrees [3, 3, 2, 2, 3, 3, 2]: no hyperplane of degree 1
  ok    non-Fano: 6 local relations in a relation space of dimension 4: two independent dependencies
  ok    non-Fano: not supersolvable (no modular element of rank 2 among 18 flats)
  ok    non-Fano: chi(t) = (t-1)(t-3)^2
  ok    rank-4 example: all 4 derivations are logarithmic (theta(alpha_H) divisible by alpha_H, 9 forms)
  ok    rank-4 example: homogeneous of degrees [1, 2, 3, 3], sum = number of hyperplanes = 9
  ok    rank-4 example: det(theta_i(x_j)) = (1) * Q(A) as polynomials (24 monomials)
  ok    rank-4 example: rank 4, 9 hyperplanes, irreducible, formal
  ok    rank-4 example: seven triple points, no quadruple point
  ok    rank-4 example: chi(t) = (t-1)(t-2)(t-3)^2
  ok    rank-4 example: not supersolvable (no chain of modular elements; 48 flats)
  ok    rank-4 example: its only modular coatom is the line of the non-Fano part, which is not supersolvable
  ok    identity (4) for exponents (1,3,3): sum C(d_i,2) = sum C(m_Z-1,2) = 6
  ok    identity (4) for exponents (1,2,3,3): sum C(d_i,2) = sum C(m_Z-1,2) = 7
  ok    x1, x2, x1+x2, x1+-x3, x1+-x4, x3+-x4: irreducible, formal, seven triple points, chi(t) = (t-1)(t-2)(t-3)^2, not supersolvable
  ok    ten hyperplanes of B_5: essential, irreducible, n = 2k, one quadruple and three triple points, sum C(m_Z-1,2) = 6 = k+1
  ok    ten hyperplanes of B_5: NOT formal (local relations span dimension 4 of 5; three hyperplanes of degree 0)
  ok    ten hyperplanes of B_5: not supersolvable (121 flats)
  ok    eight hyperplanes in Q^4: essential, formal, n = 2k, one quadruple and two triple points, sum C(m_Z-1,2) = 5 = k+1
  ok    eight hyperplanes in Q^4: reducible (the last hyperplane is a factor of rank 1), not supersolvable
  [0.1s]
PART C  hypergraph statements behind Lemma 4.3 (exhaustive for small k)
  ok    (i) k = 2: 1 covering families of 1 triples on 3 points containing {0,1,2}; without a triple carrying two points of degree 1: 0
  ok    (i) k = 3: 3 covering families of 2 triples on 5 points containing {0,1,2}; without a triple carrying two points of degree 1: 0
  ok    (i) k = 4: 63 covering families of 3 triples on 7 points containing {0,1,2}; without a triple carrying two points of degree 1: 0
  ok    (i) k = 5: 4125 covering families of 4 triples on 9 points containing {0,1,2}; without a triple carrying two points of degree 1: 0
  ok    (ii) v = 1, k = 3: 4 connected covering families (one 4-set {0,1,2,3}, 1 triples); without such a triple: 0
  ok    (ii) v = 1, k = 4: 96 connected covering families (one 4-set {0,1,2,3}, 2 triples); without such a triple: 0
  ok    (ii) v = 1, k = 5: 6000 connected covering families (one 4-set {0,1,2,3}, 3 triples); without such a triple: 0
  ok    (ii) v = 0, k = 3: 6 connected covering families containing {0,1,2}, all admissible, none has a triple with two points of degree 1: the K_4 configuration, (t,c) = (4,0)
  ok    (ii) v = 0, k = 4: 6750 connected covering families containing {0,1,2}; admissible: 450; without such a triple: 0; subfamilies T tested: 450, inequalities (6), (7) violated: 0, (t,c) = (4,0): 0, T = all triples: never
  [0.5s]
PART D  Lemmas 4.1-4.3, Lemma 5.1 and Theorem 5.2 on all members of the classes in two ground sets
  ok    13 lines with normals in {0,+-1}^3: 1716 subsets of size 6; members of C_3: 306, by (u,v): [((1, 1), 240), ((4, 0), 66)]; members failing a lemma or not supersolvable: 0
  ok    13 lines with normals in {0,+-1}^3: 1287 subsets of size 5; members of C'_3: 402, by (u,v): [((2, 0), 402)]; members failing a lemma or not supersolvable: 0
  [1.3s]
PART E  Appendices A and B in low degrees (exact ranks)
  ok    worked example: dim M_d = coefficient of t^d in P(t)/(1-t)^k, formula (11), for d <= 3: [4, 9, 14, 19]
  ok    worked example: dim (M_Z)_d = coefficient of t^d in (m(1-t) - 2 + t + t^(m-1))/(1-t)^k, formula (12), for all multiple Z, d <= 3
  ok    worked example: M_d -> sum_Z (M_Z)_d is injective for d <= 3; dimensions of the cokernel: [1, 1, 1, 1]
  ok    non-Fano: dim M_d = coefficient of t^d in P(t)/(1-t)^k, formula (11), for d <= 4: [4, 6, 6, 6, 6]
  ok    non-Fano: dim (M_Z)_d = coefficient of t^d in (m(1-t) - 2 + t + t^(m-1))/(1-t)^k, formula (12), for all multiple Z, d <= 4
  ok    non-Fano: M_d -> sum_Z (M_Z)_d is injective for d <= 4; dimensions of the cokernel: [2, 0, 0, 0, 0]
  ok    rank-4 example: dim M_d = coefficient of t^d in P(t)/(1-t)^k, formula (11), for d <= 3: [5, 12, 19, 26]
  ok    rank-4 example: dim (M_Z)_d = coefficient of t^d in (m(1-t) - 2 + t + t^(m-1))/(1-t)^k, formula (12), for all multiple Z, d <= 3
  ok    rank-4 example: M_d -> sum_Z (M_Z)_d is injective for d <= 3; dimensions of the cokernel: [2, 2, 2, 2]
  ok    contrast, four generic planes in Q^3: not formal; M_0 has dimension 1 and there is no multiple Z, so the map of Appendix A is not injective in degree 0 (injectivity in degree 0 is formality)
  [1.4s]
TOTAL failures: 0
