
== A. Theorem 1.2: Aff(Z5,2) and Aff(Z5,3) (Section 3) ==
PASS  A1 Aff(Z5,2): a|>b = 2b - a
PASS  A1 Aff(Z5,3): a|>b = 3a + 3b
PASS  A1 both are racks and quandles
PASS  A1 r and s satisfy the YBE
PASS  A1 Aff(Z5,3) is the dual of Aff(Z5,2)
PASS  A2 Aff(Z5,3) commutative, Aff(Z5,2) not (0|>1 = 2, 1|>0 = 4)
PASS  A2 none of the 120 bijections is a rack isomorphism
PASS  A2 none of the 120 bijections is a solution isomorphism
PASS  A3 D_r, D_s bijective; printed inverses correct
PASS  A3 D_r r12 D_r^-1 = S1 x id, D_r r23 D_r^-1 = S2 x id
PASS  A3 D_s s12 D_s^-1 = S1' x id, D_s s23 D_s^-1 = S2' x id
PASS  A4 theta_W is a bijection of F5^2
PASS  A4 the twelve rows of Table 1 hold and agree with the printed table
PASS  A4 theta_W S_i = S_i' theta_W on all of F5^2 (i = 1, 2)
PASS  A4 theta = D_s^-1 (theta_W x id) D_r is a B3-set isomorphism
PASS  A5 C_s^-1 as printed; F = C_s^-1 (u -> u^3) C_r
PASS  A5 C_r r C_r^-1 = (u,l) -> (3u,l), C_s s C_s^-1 = (u,l) -> (2u,l)
PASS  A5 F is a bijection and F r F^-1 = s
PASS  A6 (F, Phi, Psi) with Psi = F12^-1 theta, Phi = F23^-1 theta: bijections, (dt1)-(dt3), F r F^-1 = s
PASS  A7 Remark 2.4: transposition (0 1) relabels r into a different map, by a twist
PASS  A8 Lemma 2.5: (tau F tau, R Psi R, R Phi R) twists tau r tau into tau s tau
PASS  A8 tau r tau(a,b) = (b, a <| b) with a <| b = b |> a (right rack convention)
   cycle type of r and s on X^2: [1, 1, 1, 1, 1, 4, 4, 4, 4, 4] [1, 1, 1, 1, 1, 4, 4, 4, 4, 4]
   B3-orbit sizes on X^3: [1, 1, 1, 1, 1, 24, 24, 24, 24, 24] [1, 1, 1, 1, 1, 24, 24, 24, 24, 24]
PASS  A9 all B3-set isomorphisms W_2 -> W_3: exactly 4, the multiples c*theta_W

== B. Theorem 1.3: Lemmas 4.1-4.3, Proposition 4.5, explicit twists ==
PASS  B1 Lemma 4.1 for p = 5, 7: Aff(Z_p,t) = Aff(Z_p,s) iff t = s (all bijections)
PASS  B2 Lemma 4.2 for p in {5,7,11,13}, all t != 0,1 with 1+t+t^2 != 0
PASS  B3 Lemma 4.3: P S_i^-T = S_i(1/t) P (exact, rational functions), det P = 1+t+t^2
PASS  B4 Proposition 4.5 on all 46 pairs (p,t), p <= 61 prime, ord(-t) in {3,4,5}: t^2 != 1, 1+t+t^2 != 0, S1^k = 1, (S1S2)^3 scalar, |G_t| | (p-1)12k/(6-k), p does not divide |G_t|
   |G_t| for (5,2),(7,3),(11,2),(11,7),(13,4),(13,5): [96, 24, 600, 600, 24, 96]
PASS  B5 Theorem 1.3 for (p,t) = (5,2),(7,3),(11,2),(11,7),(13,4),(13,5): F(u,l) = (u^(p-2), l) extends to a twist (dt1)-(dt3) checked literally
   twist-related pairs {t, 1/t}: [(5, 2, 3), (7, 3, 5), (11, 2, 6), (11, 7, 8), (13, 4, 10), (13, 5, 8)]
PASS  B6 Remark 5.1 scan, p in {5,7,11,13}: Aff(Z_p,t), Aff(Z_p,1/t) twist-related iff ord(-t) in {3,4,5}

== C. Remark 5.1: racks not twist-related to their duals ==
PASS  C  p=7, t=2, dual 1/t=4: #Hom(O,A) = 49, #Hom(O,B) = 7 (printed 49, 7)
PASS  C  p=11, t=3, dual 1/t=4: #Hom(O,A) = 121, #Hom(O,B) = 11 (printed 121, 11)
PASS  C  p=13, t=2, dual 1/t=7: #Hom(O,A) = 169, #Hom(O,B) = 13 (printed 169, 13)
PASS  C  (7,2) lies outside Lemma 4.2: 1 + t + t^2 = 0 mod 7
PASS  C  G_t = GL2(F_p) for (11,3) and (13,2): orders 13200 and 26208

== D. Proposition 5.2: all racks of order n <= 6 ==
   n=1: labelled racks 1, classes 1, quandle classes 1, twist classes 1, group sizes [], unordered twist pairs 0 (certified 0), not dual 0, quandle pairs 0, self-dual classes 1 (0.0s)
PASS  D  n=1: counts (1, 1, 1, 1, 0, 0) as printed; every pair certified (explicit twist, no isomorphism among 1 bijections); every rack twist-related to its dual
   n=2: labelled racks 2, classes 2, quandle classes 1, twist classes 2, group sizes [], unordered twist pairs 0 (certified 0), not dual 0, quandle pairs 0, self-dual classes 2 (0.0s)
PASS  D  n=2: counts (2, 2, 1, 2, 0, 0) as printed; every pair certified (explicit twist, no isomorphism among 2 bijections); every rack twist-related to its dual
   n=3: labelled racks 13, classes 6, quandle classes 3, twist classes 6, group sizes [], unordered twist pairs 0 (certified 0), not dual 0, quandle pairs 0, self-dual classes 6 (0.0s)
PASS  D  n=3: counts (13, 6, 3, 6, 0, 0) as printed; every pair certified (explicit twist, no isomorphism among 6 bijections); every rack twist-related to its dual
   n=4: labelled racks 114, classes 19, quandle classes 7, twist classes 19, group sizes [], unordered twist pairs 0 (certified 0), not dual 0, quandle pairs 0, self-dual classes 19 (0.0s)
PASS  D  n=4: counts (114, 19, 7, 19, 0, 0) as printed; every pair certified (explicit twist, no isomorphism among 24 bijections); every rack twist-related to its dual
   n=5: labelled racks 1708, classes 74, quandle classes 22, twist classes 73, group sizes [2], unordered twist pairs 1 (certified 1), not dual 0, quandle pairs 1, self-dual classes 72 (0.2s)
PASS  D  n=5: counts (1708, 74, 22, 73, 1, 0) as printed; every pair certified (explicit twist, no isomorphism among 120 bijections); every rack twist-related to its dual
PASS  D  n=5: the only pair is {Aff(Z5,2), Aff(Z5,3)}; they are the two non-self-dual classes
   n=6: labelled racks 36538, classes 353, quandle classes 73, twist classes 338, group sizes [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3], unordered twist pairs 16 (certified 16), not dual 15, quandle pairs 5, self-dual classes 351 (7.1s)
PASS  D  n=6: counts (36538, 353, 73, 338, 16, 15) as printed; every pair certified (explicit twist, no isomorphism among 720 bijections); every rack twist-related to its dual
PASS  D  n=6: 13 groups of size 2 and one of size 3; 5 of the 16 pairs are pairs of quandles; all three racks of the size-3 group are self-dual

== E. Remark 5.3: B4-sets on X^4 ==
   p=5, t=2, 1/t=3: equal B4-orbit sizes True; B4-sets isomorphic: False
PASS  E  Aff(Z5,2) vs Aff(Z5,3): B4-sets on X^4 not isomorphic
   witness orbit (start point, size, #Hom into A, #Hom into B): ((0, 0, 0, 1), 125, 25, 5)
PASS  E  a witness orbit with unequal Hom counts exists
   p=7, t=3, 1/t=5: equal B4-orbit sizes True; B4-sets isomorphic: True
PASS  E  Aff(Z7,3) vs Aff(Z7,5): B4-sets on X^4 isomorphic

45 of 45 checks passed (9s).
ALL CHECKS PASS
