[1] formulas, coincidences, classification of a <= b <= 120
  area formula = number of cells for all 477 admissible benzels with a, b <= 30
  (n,2n), (n,2n-1), (n,2n-2): same cell set and same invariant for 2 <= n <= 30
  the map (i,j,k) -> (i,k,j) sends B(a,b) onto B(b,a) for a, b <= 24
  I < 0 (excluded): 249
  K1 residue 1 (DLPY Thm 1.1): 1220
  K2 residue 2 (DFLPY Thm 1.1): 1239
  K3 b = 2a (DLPY Thm 1.1 via (a,2a-2)): 59
  K4 I = 0 (Kim-Propp Thm 3): 7
  R7 (open): 944
  R7 equals {(d+3J, 2d+3J)}; |R7_d| for d = 3..11: [2, 5, 9, 14, 20, 27, 35, 44, 54]
  number of R7 benzels with b <= 45, 60: 107 206
[2] exact tiling numbers of small benzels against Propp's tables
  18 values of T103 and 18 values of T013 (a <= b <= 10) agree with the tables;
  in particular T103(3,3) = 0 and T013(3,3) = 3, so "right stone" = {p+e1,p+e2,p+e3} as in the note
[3] Theorem 1.2: the tiling of B(4D-3, 5D-3) for 2 <= D <= 40
  partition (Lemma 3.1), rho-symmetry, H = B(t,2t), the run formulas of Lemma 3.2 (cases (i)-(iii)), bones,
  the centre-residue stones of H (Lemma 3.3), exact cover, #R = I/3 = t(t-1)/2: checked for 2 <= D <= 40
  (largest: (157,197), 29952 cells)
[4] stored certificates (type-103 tilings) and the coverage statements of Theorem 1.3 / Table 1
  certificates/original: 136 files, 0 failures
  certificates/verification_run_B: 87 files, 0 failures
  distinct benzels: 223; in R7: 219; with I = 0: 4; other: 0
  d = 3: |R7_d| = 2, certified 2, missing []
  d = 4: |R7_d| = 5, certified 5, missing []
  d = 5: |R7_d| = 9, certified 9, missing []
  d = 6: |R7_d| = 14, certified 14, missing []
  d = 7: |R7_d| = 20, certified 20, missing []
  d = 8: |R7_d| = 27, certified 22, missing [(77, 85), (80, 88), (83, 91), (86, 94), (89, 97)]
  R7 with max(a,b) <= 45: 107 benzels, missing []
  R7 with max(a,b) <= 60: 206 benzels, missing []
  certified R7 benzels with b > 60: 13, all with b - a in {7, 8} (8 with b - a = 8); largest area: (5958, (74, 82))
  R7 in the window max(a,b) <= 14: [(6, 9), (9, 12), (7, 11), (10, 14), (8, 13)] (all certified); certified on the row (n,2n-3): 26 benzels, n = 6..31
[5] identities of Section 5 and Remark 3.4(c)
  2(I(a+x,b+y) - I(a,b)) = 3(y-x)(2d+y-x) - (x+y) for residue-0 (a, a+d), d < 40, x, y < 16;
  if x - y <= 2d and the change is >= 0 then y > x; for (x,y) = (3k+1,3k+2) the change is 3(d-k)
  (4k+1, 5k+2) lies on b = 2a - 3k with 2k <= (a-2)+1, and has k(k-1)/2 right stones (2 <= k < 60)
time 4.5 s
ALL CHECKS PASSED
