verified n = 1 mod 4, 5 <= n <= 2000000: full exact signature check for n <= 200000, block-minor check above; all passed.
routes used: {'A': 328414, 'B': 74416, 'B#': 97169}
largest auxiliary prime l used in route B: (131, 880993)
non-square n of route B# also realized directly in genus 2: 96999 values, largest even y needed: (52, 281605)

explicit data (route, l, b, c, y, m) and Seifert matrix V:
n=5: ('B', 3, 1, 6, 4, None)  V=[[1, 1, 0, 0], [0, 1, 2, 0], [0, 2, 6, 1], [0, 0, 0, 1]]
n=9: ('A', 3, 1, 1, 0, None)  V=[[1, 1, 0, 0], [0, 1, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]]
n=13: ('B', 7, 2, 15, 10, None)  V=[[1, 1, 0, 0], [0, 2, 5, 0], [0, 5, 15, 1], [0, 0, 0, 1]]
n=17: ('B', 3, 1, 7, 4, None)  V=[[1, 1, 0, 0], [0, 1, 2, 0], [0, 2, 7, 1], [0, 0, 0, 1]]
n=21: ('A', 3, 1, 2, 0, None)  V=[[1, 1, 0, 0], [0, 1, 0, 0], [0, 0, 2, 1], [0, 0, 0, 1]]
n=25: ('B#', 3, 1, 6, 4, 5)  V=[[1, 1, 0, 0, 0, 0], [0, 1, 2, 0, 0, 0], [0, 2, 6, 1, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 1]]
n=29: ('B', 3, 1, 8, 4, None)  V=[[1, 1, 0, 0], [0, 1, 2, 0], [0, 2, 8, 1], [0, 0, 0, 1]]
n=33: ('A', 3, 1, 3, 0, None)  V=[[1, 1, 0, 0], [0, 1, 0, 0], [0, 0, 3, 1], [0, 0, 0, 1]]
n=37: ('B', 19, 5, 42, 28, None)  V=[[1, 1, 0, 0], [0, 5, 14, 0], [0, 14, 42, 1], [0, 0, 0, 1]]
n=41: ('B', 3, 1, 9, 4, None)  V=[[1, 1, 0, 0], [0, 1, 2, 0], [0, 2, 9, 1], [0, 0, 0, 1]]
n=45: ('A', 3, 1, 4, 0, None)  V=[[1, 1, 0, 0], [0, 1, 0, 0], [0, 0, 4, 1], [0, 0, 0, 1]]
n=49: ('A', 7, 2, 2, 0, None)  V=[[1, 1, 0, 0], [0, 2, 0, 0], [0, 0, 2, 1], [0, 0, 0, 1]]
n=53: ('B', 3, 1, 10, 4, None)  V=[[1, 1, 0, 0], [0, 1, 2, 0], [0, 2, 10, 1], [0, 0, 0, 1]]
n=57: ('A', 3, 1, 5, 0, None)  V=[[1, 1, 0, 0], [0, 1, 0, 0], [0, 0, 5, 1], [0, 0, 0, 1]]
n=61: ('B', 7, 2, 3, 2, None)  V=[[1, 1, 0, 0], [0, 2, 1, 0], [0, 1, 3, 1], [0, 0, 0, 1]]
n=65: ('B#', 3, 1, 6, 4, 13)  V=[[1, 1, 0, 0, 0, 0], [0, 1, 2, 0, 0, 0], [0, 2, 6, 1, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 3]]
n=69: ('A', 3, 1, 6, 0, None)  V=[[1, 1, 0, 0], [0, 1, 0, 0], [0, 0, 6, 1], [0, 0, 0, 1]]
n=73: ('B', 7, 2, 12, 8, None)  V=[[1, 1, 0, 0], [0, 2, 4, 0], [0, 4, 12, 1], [0, 0, 0, 1]]
n=77: ('A', 7, 2, 3, 0, None)  V=[[1, 1, 0, 0], [0, 2, 0, 0], [0, 0, 3, 1], [0, 0, 0, 1]]
n=81: ('A', 3, 1, 7, 0, None)  V=[[1, 1, 0, 0], [0, 1, 0, 0], [0, 0, 7, 1], [0, 0, 0, 1]]
n=85: ('B#', 3, 1, 6, 4, 17)  V=[[1, 1, 0, 0, 0, 0], [0, 1, 2, 0, 0, 0], [0, 2, 6, 1, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 4]]
n=89: ('B', 3, 1, 13, 4, None)  V=[[1, 1, 0, 0], [0, 1, 2, 0], [0, 2, 13, 1], [0, 0, 0, 1]]
n=93: ('A', 3, 1, 8, 0, None)  V=[[1, 1, 0, 0], [0, 1, 0, 0], [0, 0, 8, 1], [0, 0, 0, 1]]
n=97: ('B', 7, 2, 18, 10, None)  V=[[1, 1, 0, 0], [0, 2, 5, 0], [0, 5, 18, 1], [0, 0, 0, 1]]
n=101: ('B', 3, 1, 14, 4, None)  V=[[1, 1, 0, 0], [0, 1, 2, 0], [0, 2, 14, 1], [0, 0, 0, 1]]
n=2857: ('B', 11, 3, 83, 14, None)  V=[[1, 1, 0, 0], [0, 3, 7, 0], [0, 7, 83, 1], [0, 0, 0, 1]]
n=2953: ('B', 7, 2, 120, 10, None)  V=[[1, 1, 0, 0], [0, 2, 5, 0], [0, 5, 120, 1], [0, 0, 0, 1]]
n=3001: ('B', 7, 2, 108, 2, None)  V=[[1, 1, 0, 0], [0, 2, 1, 0], [0, 1, 108, 1], [0, 0, 0, 1]]
n=3049: ('B', 11, 3, 71, 4, None)  V=[[1, 1, 0, 0], [0, 3, 2, 0], [0, 2, 71, 1], [0, 0, 0, 1]]
n=3121: ('B', 7, 2, 126, 10, None)  V=[[1, 1, 0, 0], [0, 2, 5, 0], [0, 5, 126, 1], [0, 0, 0, 1]]
n=3169: ('B', 7, 2, 114, 2, None)  V=[[1, 1, 0, 0], [0, 2, 1, 0], [0, 1, 114, 1], [0, 0, 0, 1]]

rows of an earlier draft (odd y variant), exact check:
n=2857: (b,c,y)=(3,66,3)  det(V+V^T)=2857, signature=4, Pf(V-V^T)=1  OK
n=2953: (b,c,y)=(2,107,3)  det(V+V^T)=2953, signature=4, Pf(V-V^T)=1  OK
n=3001: (b,c,y)=(2,111,5)  det(V+V^T)=3001, signature=4, Pf(V-V^T)=1  OK
n=3049: (b,c,y)=(11,18,1)  det(V+V^T)=3049, signature=4, Pf(V-V^T)=1  OK
