A  Lemma 3.1: G - G^T = J+J, D1..D4 and 2 D3 = D4 + D2 on a 7x7x7 grid (degrees 1,1,2): identities hold;
   positive definite (signature 4) whenever b >= 1 and det > 0;  E(u): (1, u, 2), T(m): (1, -m, 0)
B  Lemma 2.1 on 2000 random Seifert matrices (g <= 4, general unimodular skew part): det = (-1)^g mod 4 and |det| = (-1)^(sigma/2) mod 4 always; 75 with |det| = 1, all sigma = 0 mod 8
C  Lemma 2.3 (proof implemented: symplectic basis mod 2, lift to SL_2g(Z), V = (B + A)/2) on 600 random even B with odd det: OK
D  V_8: det(V8 - V8^T) = 1 (Pfaffian 1), V8 + V8^T = E8 with det 1 and signature 8;
   all 530 admissible pairs (d, s), d <= 101 odd, |s| <= 20, realized by block sums (exact checks)
E  Remark 3.3(a): G(1,(n+19)/12,2) has det n for every n = 5 mod 12 (n <= 10^5); n = 1 mod 12 => 3 does not divide n + 4y^2;
   5 mod 8, 5 mod 12 and 9 mod 12 cover exactly the classes 5,9,13,17,21 mod 24 (all n = 1 mod 4 except 1 mod 24)
F  Remark 3.3(b): for all 332180 primes p = 1 mod 4 below 10^7 the least prime l = 3 mod 4 with (l/p) = -1 is <= 139
   (maximum 139 first at p = 9665041); 0 s
   p = 2857: least l = 11, least even y = 8, 11*283 - 4*8^2 = 2857, G(3, 71, 8): det(V-V^T)=1, det(V+V^T)=2857, sigma 4
   p = 33049: least l = 19, least even y = 6, 19*1747 - 4*6^2 = 33049, G(5, 437, 6): det(V-V^T)=1, det(V+V^T)=33049, sigma 4
G  Proposition 5.1: n + 4y^2 (n = s^2, s < 400 with prime factors = 1 mod 4, 0 <= y <= 300) never has a prime factor = 3 mod 4 (15652 values)
H  Remark 5.2 example: det(V-V^T) = 1, det(V+V^T) = 25, sigma = 4; coker(V+V^T) invariant factors [1, 1, 5, 5]
     s = 5: B = [[2, 1, 0, -1], [1, 64, -8, -10], [0, -8, 4, 1], [-1, -10, 1, 2]], V = [[1, 1, 0, 0], [0, 32, -4, -5], [0, -4, 2, 1], [-1, -5, 0, 1]]
     s = 13: B = [[2, 1, -1, -1], [1, 6, -2, 2], [-1, -2, 8, 1], [-1, 2, 1, 4]], V = [[1, 1, 0, 0], [0, 3, -1, 1], [-1, -1, 4, 1], [-1, 1, 0, 2]]
     s = 17: B = [[2, 1, 0, -1], [1, 60, 0, -9], [0, 0, 12, 1], [-1, -9, 1, 2]], V = [[1, 1, 0, 0], [0, 30, 0, -5], [0, 0, 6, 1], [-1, -4, 0, 1]]
     s = 25: B = [[2, 1, 0, -1], [1, 64, -8, -10], [0, -8, 64, 1], [-1, -10, 1, 2]], V = [[1, 1, 0, 0], [0, 32, -4, -5], [0, -4, 32, 1], [-1, -5, 0, 1]]
     s = 29: B = [[2, 1, 0, -1], [1, 68, 4, -9], [0, 4, 20, 1], [-1, -9, 1, 2]], V = [[1, 1, 0, 0], [0, 34, 2, -5], [0, 2, 10, 1], [-1, -4, 0, 1]]
     s = 37: B = [[2, 1, -1, 0], [1, 20, -1, -2], [-1, -1, 10, 1], [0, -2, 1, 4]], V = [[1, 1, 0, 0], [0, 10, -1, -1], [-1, 0, 5, 1], [0, -1, 0, 2]]
     s = 41: B = [[2, 1, 0, -1], [1, 48, 10, -6], [0, 10, 28, 1], [-1, -6, 1, 2]], V = [[1, 1, 0, 0], [0, 24, 5, -3], [0, 5, 14, 1], [-1, -3, 0, 1]]
     s = 53: B = [[2, 1, -1, 0], [1, 28, -1, -2], [-1, -1, 14, 1], [0, -2, 1, 4]], V = [[1, 1, 0, 0], [0, 14, -1, -1], [-1, 0, 7, 1], [0, -1, 0, 2]]
     s = 61: B = [[2, 1, -1, 1], [1, 18, 8, 0], [-1, 8, 36, 1], [1, 0, 1, 4]], V = [[1, 1, 0, 1], [0, 9, 4, 0], [-1, 4, 18, 1], [0, 0, 0, 2]]
     s = 65: B = [[2, 1, 0, -1], [1, 64, -8, -10], [0, -8, 424, 1], [-1, -10, 1, 2]], V = [[1, 1, 0, 0], [0, 32, -4, -5], [0, -4, 212, 1], [-1, -5, 0, 1]]
     s = 73: B = [[2, 1, 0, -1], [1, 16, -6, 3], [0, -6, 40, 1], [-1, 3, 1, 6]], V = [[1, 1, 0, 0], [0, 8, -3, 1], [0, -3, 20, 1], [-1, 2, 0, 3]]
     s = 85: B = [[2, 1, 0, -1], [1, 64, -8, -10], [0, -8, 724, 1], [-1, -10, 1, 2]], V = [[1, 1, 0, 0], [0, 32, -4, -5], [0, -4, 362, 1], [-1, -5, 0, 1]]
     s = 89: B = [[2, 1, -1, -1], [1, 18, -6, 6], [-1, -6, 48, 1], [-1, 6, 1, 8]], V = [[1, 1, 0, 0], [0, 9, -3, 3], [-1, -3, 24, 1], [-1, 3, 0, 4]]
     s = 97: B = [[2, 1, 0, -1], [1, 28, -6, 3], [0, -6, 26, 1], [-1, 3, 1, 8]], V = [[1, 1, 0, 0], [0, 14, -3, 1], [0, -3, 13, 1], [-1, 2, 0, 4]]
     s = 101: B = [[2, 1, -1, 1], [1, 20, 8, -2], [-1, 8, 56, 1], [1, -2, 1, 6]], V = [[1, 1, 0, 1], [0, 10, 4, -1], [-1, 4, 28, 1], [0, -1, 0, 3]]
   own search (forms [[2,1,u,x],[1,2b,y,z],[u,y,2c,1],[x,z,1,2d]]): genus-2 Seifert matrices of det s^2, sigma 4 for s = 5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101; not found in this box: []
   every such cokernel is non-cyclic, as Remark 5.2 predicts: last invariant factors [(5, [5, 5]), (13, [13, 13]), (17, [17, 17]), (25, [5, 125]), (29, [29, 29]), (37, [37, 37]), (41, [41, 41]), (53, [53, 53]), (61, [61, 61]), (65, [5, 845]), (73, [73, 73]), (85, [5, 1445]), (89, [89, 89]), (97, [97, 97]), (101, [101, 101])]
   case (c): coker = Z/p + Z/(n/p) for n = 25, 169, 289, 625, 841
I  two-bridge knots: every one of the 4930 non-squares n = 1 mod 4, n <= 20001, is det S(n,q) with |sigma| = 4 (least such q is at most 106, attained at n = 16185);
   for the 35 odd squares n = s^2 <= 5041 every S(n,q) has sigma = 0 mod 8 (2 s)
ALL CHECKS PASSED (12 s)
