=== Part 1: connected, not realizable over C, nontrivial optimal partition (r' < min(n, 2d-1)) ===
FE(N+K) = free extension (one new element in general position) of the direct sum of N and K;
FE2(N+K) = the free extension of FE(N+K).  U(k,k) = k coloops, U(1,m) = m parallel elements.
Each matroid contains N as the restriction to its first |E(N)| elements (checked below),
so it is not realizable over C.
FE(F7+U(2,2))                      n=10 d=5 conn=Y min(n,2d-1)= 9 r'= 8 nontriv=Y #layerpart=  133 dSS=   8 cert= 8(forest=True,roots=2) f(R_P)= 8 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6} {7} {8} {9}] [contains F7] OK (0.1s)
FE(F7+U(3,3))                      n=11 d=6 conn=Y min(n,2d-1)=11 r'= 9 nontriv=Y #layerpart=  217 dSS=   9 cert= 9(forest=True,roots=3) f(R_P)= 9 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6} {7} {8} {9} {10}] [contains F7] OK (0.8s)
FE(F7+U(1,2)+U(1,1))               n=11 d=5 conn=Y min(n,2d-1)= 9 r'= 8 nontriv=Y #layerpart=  133 dSS=   8 cert= 8(forest=True,roots=2) f(R_P)= 8 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6} {7,8} {9} {10}] [contains F7] OK (0.3s)
FE(F7+U(1,3)+U(1,1))               n=12 d=5 conn=Y min(n,2d-1)= 9 r'= 8 nontriv=Y #layerpart=  133 dSS=   8 cert= 8(forest=True,roots=2) f(R_P)= 8 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6} {7,8,9} {10} {11}] [contains F7] OK (0.3s)
FE(F7+U(2,2)+U(1,2))               n=12 d=6 conn=Y min(n,2d-1)=11 r'= 9 nontriv=Y #layerpart=  217 dSS=   9 cert= 9(forest=True,roots=3) f(R_P)= 9 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6} {7} {8} {9,10} {11}] [contains F7] OK (0.9s)
FE2(F7+U(3,3))                     n=12 d=6 conn=Y min(n,2d-1)=11 r'=10 nontriv=Y #layerpart=  456 dSS=  10 cert=10(forest=True,roots=2) f(R_P)=10 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6} {7} {8} {9} {10} {11}] [contains F7] OK (3.1s)
FE(AG(3,2)+U(2,2))                 n=11 d=6 conn=Y min(n,2d-1)=11 r'=10 nontriv=Y #layerpart=  532 dSS=  10 cert=10(forest=True,roots=2) f(R_P)=10 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7} {8} {9} {10}] [contains AG(3,2)] OK (3.5s)
FE(AG(3,2)+U(3,3))                 n=12 d=7 conn=Y min(n,2d-1)=12 r'=11 nontriv=Y #layerpart=  868 dSS=  11 cert=11(forest=True,roots=3) f(R_P)=11 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7} {8} {9} {10} {11}] [contains AG(3,2)] OK (12.3s)
FE(AG(3,2)+U(1,2)+U(1,1))          n=12 d=6 conn=Y min(n,2d-1)=11 r'=10 nontriv=Y #layerpart=  532 dSS=  10 cert=10(forest=True,roots=2) f(R_P)=10 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7} {8,9} {10} {11}] [contains AG(3,2)] OK (3.9s)
FE(AG(3,2)+U(1,3)+U(1,1))          n=13 d=6 conn=Y min(n,2d-1)=11 r'=10 nontriv=Y #layerpart=  532 dSS=  10 cert=10(forest=True,roots=2) f(R_P)=10 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7} {8,9,10} {11} {12}] [contains AG(3,2)] OK (4.5s)
FE(AG(3,2)+U(2,2)+U(1,2))          n=13 d=7 conn=Y min(n,2d-1)=13 r'=11 nontriv=Y #layerpart=  868 dSS=  11 cert=11(forest=True,roots=3) f(R_P)=11 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7} {8} {9} {10,11} {12}] [contains AG(3,2)] OK (13.5s)
FE2(AG(3,2)+U(3,3))                n=13 d=7 conn=Y min(n,2d-1)=13 r'=12 nontriv=Y #layerpart= 1828 dSS=  12 cert=12(forest=True,roots=2) f(R_P)=12 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7} {8} {9} {10} {11} {12}] [contains AG(3,2)] OK (55.6s)
FE(Vamos+U(2,2))                   n=11 d=6 conn=Y min(n,2d-1)=11 r'=10 nontriv=Y #layerpart=  424 dSS=  10 cert=10(forest=True,roots=2) f(R_P)=10 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7} {8} {9} {10}] [contains Vamos] OK (2.7s)
FE(Vamos+U(3,3))                   n=12 d=7 conn=Y min(n,2d-1)=12 r'=11 nontriv=Y #layerpart=  688 dSS=  11 cert=11(forest=True,roots=3) f(R_P)=11 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7} {8} {9} {10} {11}] [contains Vamos] OK (10.1s)
FE(Vamos+U(1,2)+U(1,1))            n=12 d=6 conn=Y min(n,2d-1)=11 r'=10 nontriv=Y #layerpart=  424 dSS=  10 cert=10(forest=True,roots=2) f(R_P)=10 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7} {8,9} {10} {11}] [contains Vamos] OK (2.7s)
FE(Vamos+U(1,3)+U(1,1))            n=13 d=6 conn=Y min(n,2d-1)=11 r'=10 nontriv=Y #layerpart=  424 dSS=  10 cert=10(forest=True,roots=2) f(R_P)=10 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7} {8,9,10} {11} {12}] [contains Vamos] OK (3.2s)
FE(Vamos+U(2,2)+U(1,2))            n=13 d=7 conn=Y min(n,2d-1)=13 r'=11 nontriv=Y #layerpart=  688 dSS=  11 cert=11(forest=True,roots=3) f(R_P)=11 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7} {8} {9} {10,11} {12}] [contains Vamos] OK (11.0s)
FE2(Vamos+U(3,3))                  n=13 d=7 conn=Y min(n,2d-1)=13 r'=12 nontriv=Y #layerpart= 1468 dSS=  12 cert=12(forest=True,roots=2) f(R_P)=12 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7} {8} {9} {10} {11} {12}] [contains Vamos] OK (57.2s)
FE(nonPappus+U(2,2))               n=12 d=5 conn=Y min(n,2d-1)= 9 r'= 8 nontriv=Y #layerpart=  225 dSS=   8 cert= 8(forest=True,roots=2) f(R_P)= 8 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8} {9} {10} {11}] [contains nonPappus] OK (1.2s)
FE(nonPappus+U(3,3))               n=13 d=6 conn=Y min(n,2d-1)=11 r'= 9 nontriv=Y #layerpart=  369 dSS=   9 cert= 9(forest=True,roots=3) f(R_P)= 9 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8} {9} {10} {11} {12}] [contains nonPappus] OK (4.2s)
FE(nonPappus+U(1,2)+U(1,1))        n=13 d=5 conn=Y min(n,2d-1)= 9 r'= 8 nontriv=Y #layerpart=  225 dSS=   8 cert= 8(forest=True,roots=2) f(R_P)= 8 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8} {9,10} {11} {12}] [contains nonPappus] OK (1.8s)
FE(nonDesargues+U(2,2))            n=13 d=5 conn=Y min(n,2d-1)= 9 r'= 8 nontriv=Y #layerpart=  280 dSS=   8 cert= 8(forest=True,roots=2) f(R_P)= 8 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8,9} {10} {11} {12}] [contains nonDesargues] OK (2.3s)
Part 1: 22 of 22 cases are connected with a nontrivial optimal partition; pairwise distinct rank-size profiles (hence pairwise non-isomorphic): True
=== Part 1b: further connected non-C-realizable matroids (optimal partition may be trivial) ===
FE(F7*+U(2,2))                     n=10 d=6 conn=Y min(n,2d-1)=10 r'=10 nontriv=N #layerpart=  315 dSS=  10 cert=10(forest=True,roots=2) f(R_P)=10 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6} {7} {8} {9}] [contains F7*] OK (2.2s)
FE(F7*+U(3,3))                     n=11 d=7 conn=Y min(n,2d-1)=11 r'=11 nontriv=N #layerpart=  511 dSS=  11 cert=11(forest=True,roots=3) f(R_P)=11 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6} {7} {8} {9} {10}] [contains F7*] OK (7.8s)
FE(F7+U(1,1))                      n= 9 d=4 conn=Y min(n,2d-1)= 7 r'= 7 nontriv=N #layerpart=   70 dSS=   7 cert= 7(forest=True,roots=1) f(R_P)= 7 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8}] [contains F7] OK (0.1s)
FE(F7+U(1,2))                      n=10 d=4 conn=Y min(n,2d-1)= 7 r'= 7 nontriv=N #layerpart=   70 dSS=   7 cert= 7(forest=True,roots=1) f(R_P)= 7 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8,9}] [contains F7] OK (0.1s)
FE(F7+U(2,3))                      n=11 d=5 conn=Y min(n,2d-1)= 9 r'= 9 nontriv=N #layerpart=  231 dSS=   9 cert= 9(forest=True,roots=1) f(R_P)= 9 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8,9,10}] [contains F7] OK (1.0s)
FE(F7*+U(1,1))                     n= 9 d=5 conn=Y min(n,2d-1)= 9 r'= 9 nontriv=N #layerpart=  168 dSS=   9 cert= 9(forest=True,roots=1) f(R_P)= 9 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8}] [contains F7*] OK (0.5s)
FE(F7*+U(1,2))                     n=10 d=5 conn=Y min(n,2d-1)= 9 r'= 9 nontriv=N #layerpart=  168 dSS=   9 cert= 9(forest=True,roots=1) f(R_P)= 9 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8,9}] [contains F7*] OK (0.6s)
FE(F7*+U(2,3))                     n=11 d=6 conn=Y min(n,2d-1)=11 r'=11 nontriv=N #layerpart=  553 dSS=  11 cert=11(forest=True,roots=1) f(R_P)=11 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8,9,10}] [contains F7*] OK (6.3s)
FE(AG(3,2)+U(1,1))                 n=10 d=5 conn=Y min(n,2d-1)= 9 r'= 9 nontriv=N #layerpart=  280 dSS=   9 cert= 9(forest=True,roots=1) f(R_P)= 9 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8,9}] [contains AG(3,2)] OK (1.6s)
FE(AG(3,2)+U(1,2))                 n=11 d=5 conn=Y min(n,2d-1)= 9 r'= 9 nontriv=N #layerpart=  280 dSS=   9 cert= 9(forest=True,roots=1) f(R_P)= 9 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8,9,10}] [contains AG(3,2)] OK (1.5s)
FE(AG(3,2)+U(2,3))                 n=12 d=6 conn=Y min(n,2d-1)=11 r'=11 nontriv=N #layerpart=  924 dSS=  11 cert=11(forest=True,roots=1) f(R_P)=11 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8,9,10,11}] [contains AG(3,2)] OK (17.6s)
FE(Vamos+U(1,1))                   n=10 d=5 conn=Y min(n,2d-1)= 9 r'= 9 nontriv=N #layerpart=  226 dSS=   9 cert= 9(forest=True,roots=1) f(R_P)= 9 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8,9}] [contains Vamos] OK (1.0s)
FE(Vamos+U(1,2))                   n=11 d=5 conn=Y min(n,2d-1)= 9 r'= 9 nontriv=N #layerpart=  226 dSS=   9 cert= 9(forest=True,roots=1) f(R_P)= 9 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8,9,10}] [contains Vamos] OK (1.1s)
FE(Vamos+U(2,3))                   n=12 d=6 conn=Y min(n,2d-1)=11 r'=11 nontriv=N #layerpart=  744 dSS=  11 cert=11(forest=True,roots=1) f(R_P)=11 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8,9,10,11}] [contains Vamos] OK (12.0s)
FE(nonPappus+U(1,1))               n=11 d=4 conn=Y min(n,2d-1)= 7 r'= 7 nontriv=N #layerpart=  117 dSS=   7 cert= 7(forest=True,roots=1) f(R_P)= 7 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8,9,10}] [contains nonPappus] OK (0.4s)
FE(nonPappus+U(1,2))               n=12 d=4 conn=Y min(n,2d-1)= 7 r'= 7 nontriv=N #layerpart=  117 dSS=   7 cert= 7(forest=True,roots=1) f(R_P)= 7 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8,9,10,11}] [contains nonPappus] OK (0.5s)
Part 1b: 16 matroids, pairwise distinct rank-size profiles: True
=== Part 2: the named matroids and realizable controls ===
F7                                 n= 7 d=3 conn=Y min(n,2d-1)= 5 r'= 5 nontriv=N #layerpart=   21 dSS=   5 cert= 5(forest=True,roots=1) f(R_P)= 5 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6}] not C-realizable OK (0.0s)
F7*                                n= 7 d=4 conn=Y min(n,2d-1)= 7 r'= 7 nontriv=N #layerpart=   49 dSS=   7 cert= 7(forest=True,roots=1) f(R_P)= 7 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6}] not C-realizable OK (0.1s)
AG(3,2)                            n= 8 d=4 conn=Y min(n,2d-1)= 7 r'= 7 nontriv=N #layerpart=   84 dSS=   7 cert= 7(forest=True,roots=1) f(R_P)= 7 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7}] not C-realizable OK (0.1s)
Vamos                              n= 8 d=4 conn=Y min(n,2d-1)= 7 r'= 7 nontriv=N #layerpart=   66 dSS=   7 cert= 7(forest=True,roots=1) f(R_P)= 7 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7}] not realizable OK (0.1s)
nonPappus                          n= 9 d=3 conn=Y min(n,2d-1)= 5 r'= 5 nontriv=N #layerpart=   36 dSS=   5 cert= 5(forest=True,roots=1) f(R_P)= 5 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8}] not realizable OK (0.1s)
nonPappus*                         n= 9 d=6 conn=Y min(n,2d-1)= 9 r'= 9 nontriv=N #layerpart=  198 dSS=   9 cert= 9(forest=True,roots=3) f(R_P)= 9 f(R)>=r' on 24 random R: True P=[{0} {1} {2} {3} {4} {5} {6} {7} {8}] not realizable OK (0.9s)
nonDesargues                       n=10 d=3 conn=Y min(n,2d-1)= 5 r'= 5 nontriv=N #layerpart=   45 dSS=   5 cert= 5(forest=True,roots=1) f(R_P)= 5 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8,9}] not realizable OK (0.1s)
F7-                                n= 7 d=3 conn=Y min(n,2d-1)= 5 r'= 5 nontriv=N #layerpart=   21 dSS=   5 cert= 5(forest=True,roots=1) f(R_P)= 5 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6}] realizable over Q OK (0.0s)
Pappus                             n= 9 d=3 conn=Y min(n,2d-1)= 5 r'= 5 nontriv=N #layerpart=   36 dSS=   5 cert= 5(forest=True,roots=1) f(R_P)= 5 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6,7,8}] realizable over Q OK (0.0s)
U(3,6)                             n= 6 d=3 conn=Y min(n,2d-1)= 5 r'= 5 nontriv=N #layerpart=   15 dSS=   5 cert= 5(forest=True,roots=1) f(R_P)= 5 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5}] realizable OK (0.0s)
F7+U(2,3)                          n=10 d=5 conn=N min(n,2d-1)= 9 r'= 8 nontriv=Y #layerpart=   63 dSS=   8 cert= 8(forest=True,roots=2) f(R_P)= 8 f(R)>=r' on 24 random R: True P=[{0,1,2,3,4,5,6} {7,8,9}] not C-realizable, disconnected OK (0.2s)
=== Part 3: the inequality of Corollary 3.2 can be strict for general fans (Proposition 5.1) ===
general fan F = union of the 4 coordinate 3-planes R^(star of v), v in V(K4), in R^6: dim(F+F) = 5; min over the 64 coordinate subspaces U of 2dim(U+F)-dim U = 6 (attained at U = 0); min over 3000 random rational U = 6; OK
total time 257s
ALL OK
