A. Lemma 3.1 identities proved symbolically: G - G^T = J+J, det(G + G^T) = (4b-1)(4c-1) - 4y^2,
   leading minors 2, 4b-1, D3 with 2 D3 = D4 + D2;  E(u): det(E+E^T) = u;  T(m): det(T+T^T) = -m.
B. Lemma 3.2 followed literally for all 249501 non-squares n = 1 mod 4, 5 <= n <= 1000001;
   largest prime l (least prime = t mod 4m not dividing s) = 287024779 at n = 889681.
C. Theorem 1.1 for every n = 1 mod 4, 5 <= n <= 1000001: cases (a)/(b)/(c) = 162118/87760/122; genus 2 in cases (a), (b), genus 3 in case (c);
   det(V - V^T) = 1, |det(V + V^T)| = n, signature 4 checked exactly for every matrix.
D. Proposition 5.1 (sanity check): for the 122 squares s^2, s <= 1000 with all prime factors of s
   = 1 mod 4, and 0 <= y <= 300, no s^2 + 4y^2 has a prime factor = 3 mod 4 (so no G(b,c,y) has det s^2).
E. V_E8 is a Seifert matrix with det(V+V^T) = 1 and signature 8.  Theorem 1.2: all 530 admissible
   pairs (d, s), d odd <= 101, |s| <= 20, realized by the block sums of the proof and checked exactly;
   the other 541 pairs violate Murasugi's congruence or are (1, s) with s = 4 mod 8.
F. Explicit knots (2-bridge Seifert matrices from the even continued fraction; |sigma| also by the
   floor-sum formula); calibration on 3_1, ..., 7_7 passed.
   n = 5: 5_1 = C(5) [det 5, sigma 4]  -> det 5, signature 4
   n = 9: 3_1 = C(3) [det 3, sigma 2] # 3_1 = C(3) [det 3, sigma 2]  -> det 9, signature 4
   n = 13: 7_3 = C(4 3) [det 13, sigma 4]  -> det 13, signature 4
   n = 17: 7_5 = C(3 2 2) [det 17, sigma 4]  -> det 17, signature 4
   n = 17: 8_2 = C(5 1 2) [det 17, sigma 4]  -> det 17, signature 4
   n = 21: 9_4 = C(5 4) [det 21, sigma 4]  -> det 21, signature 4
   n = 25: 5_1 = C(5) [det 5, sigma 4] # 4_1 = C(2 2) [det 5, sigma 0]  -> det 25, signature 4
   n = 29: 9_7 = C(3 4 2) [det 29, sigma 4]  -> det 29, signature 4
   n = 33: 9_10 = C(3 3 3) [det 33, sigma 4]  -> det 33, signature 4
   n = 33: 9_11 = C(4 1 2 2) [det 33, sigma 4]  -> det 33, signature 4
   n = 37: 9_13 = C(3 2 1 3) [det 37, sigma 4]  -> det 37, signature 4
   n = 41: 9_18 = C(3 2 2 2) [det 41, sigma 4]  -> det 41, signature 4
   n = 41: 9_20 = C(3 1 2 1 2) [det 41, sigma 4]  -> det 41, signature 4
   n = 45: 9_23 = C(2 2 1 2 2) [det 45, sigma 4]  -> det 45, signature 4
   n = 49: 5_2 = C(3 2) [det 7, sigma 2] # 5_2 = C(3 2) [det 7, sigma 2]  -> det 49, signature 4
   S(2857,652) = C(4 2 1 1 1 1 1 1 1 3 3): det 2857, |sigma| = 4 (19 crossings).
   n = 2857 (prime, = 1 mod 24): Lemma 3.2 literally: G(12858,32408,40826); least l = 11: G(3,71,8) [(11)(283) - 4*8^2 = 2857];
      minimal-crossing 2-bridge knot with |sigma| = 4: S(2857,652), 19 crossings, Conway notation C(4 2 1 1 1 1 1 1 1 3 3)
   n = 33049 (prime, = 1 mod 24): Lemma 3.2 literally: G(5,489,32); least l = 19: G(5,437,6) [(19)(1747) - 4*6^2 = 33049];
      minimal-crossing 2-bridge knot with |sigma| = 4: S(33049,9229), 24 crossings, Conway notation C(3 1 1 2 1 1 2 2 1 1 2 3 1 1 2)
G. genus-2 Seifert matrices with det s^2 and signature 4 checked exactly for s in [5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89].
All 3115300 checks passed (15 s).
