=== single matroids: adim(M) = r'(E) = dim(Sigma+Sigma) ===
U(2,3)                       n= 3 r=2 #layerparts=    3 r'= 3 dSS=             3 bern= 3 forest=True roots=1 f(R_P)= 3 [P={0,1,2}] minf(randomR)= 3 lemmaA=True realizable OK (0.0s)
U(3,5)                       n= 5 r=3 #layerparts=   10 r'= 5 dSS=             5 bern= 5 forest=True roots=1 f(R_P)= 5 [P={0,1,2,3,4}] minf(randomR)= 5 lemmaA=True realizable OK (0.1s)
U(4,7)                       n= 7 r=4 #layerparts=   35 r'= 7 dSS=             7 bern= 7 forest=True roots=1 f(R_P)= 7 [P={0,1,2,3,4,5,6}] minf(randomR)= 7 lemmaA=True realizable OK (0.9s)
U(3,7)                       n= 7 r=3 #layerparts=   21 r'= 5 dSS=             5 bern= 5 forest=True roots=1 f(R_P)= 5 [P={0,1,2,3,4,5,6}] minf(randomR)= 5 lemmaA=True realizable OK (0.5s)
F7-                          n= 7 r=3 #layerparts=   21 r'= 5 dSS=             5 bern= 5 forest=True roots=1 f(R_P)= 5 [P={0,1,2,3,4,5,6}] minf(randomR)= 5 lemmaA=True realizable over Q OK (0.6s)
F7-*                         n= 7 r=4 #layerparts=   47 r'= 7 dSS=             7 bern= 7 forest=True roots=1 f(R_P)= 7 [P={0,1,2,3,4,5,6}] minf(randomR)= 7 lemmaA=True realizable over Q OK (1.6s)
F7                           n= 7 r=3 #layerparts=   21 r'= 5 dSS=             5 bern= 5 forest=True roots=1 f(R_P)= 5 [P={0,1,2,3,4,5,6}] minf(randomR)= 5 lemmaA=True NOT C-realizable OK (0.4s)
F7*                          n= 7 r=4 #layerparts=   49 r'= 7 dSS=             7 bern= 7 forest=True roots=1 f(R_P)= 7 [P={0,1,2,3,4,5,6}] minf(randomR)= 7 lemmaA=True NOT C-realizable (2-sparse: r'=n) OK (1.6s)
AG(3,2)                      n= 8 r=4 #layerparts=   84 r'= 7 dSS=             7 bern= 7 forest=True roots=1 f(R_P)= 7 [P={0,1,2,3,4,5,6,7}] minf(randomR)= 7 lemmaA=True NOT C-realizable OK (3.0s)
Vamos                        n= 8 r=4 #layerparts=   66 r'= 7 dSS=             7 bern= 7 forest=True roots=1 f(R_P)= 7 [P={0,1,2,3,4,5,6,7}] minf(randomR)= 7 lemmaA=True NOT realizable over any field OK (2.8s)
Vamos*                       n= 8 r=4 #layerparts=   66 r'= 7 dSS=             7 bern= 7 forest=True roots=1 f(R_P)= 7 [P={0,1,2,3,4,5,6,7}] minf(randomR)= 7 lemmaA=True NOT realizable (V8* ~ V8) OK (2.8s)
Pappus                       n= 9 r=3 #layerparts=   36 r'= 5 dSS=             5 bern= 5 forest=True roots=1 f(R_P)= 5 [P={0,1,2,3,4,5,6,7,8}] minf(randomR)= 5 lemmaA=True realizable over Q OK (1.9s)
nonPappus                    n= 9 r=3 #layerparts=   36 r'= 5 dSS=             5 bern= 5 forest=True roots=1 f(R_P)= 5 [P={0,1,2,3,4,5,6,7,8}] minf(randomR)= 5 lemmaA=True NOT realizable over any field OK (1.7s)
nonPappus*                   n= 9 r=6 #layerparts=  198 r'= 9 dSS=             9 bern= 9 forest=True roots=3 f(R_P)= 9 [P={0} {1} {2} {3} {4} {5} {6} {7} {8}] minf(randomR)= 9 lemmaA=True NOT realizable; 2-sparse (r'=n=9) OK (15.4s)
Desargues                    n=10 r=3 #layerparts=   45 r'= 5 dSS=             5 bern= 5 forest=True roots=1 f(R_P)= 5 [P={0,1,2,3,4,5,6,7,8,9}] minf(randomR)= 5 lemmaA=True realizable over Q OK (2.7s)
nonDesargues                 n=10 r=3 #layerparts=   45 r'= 5 dSS=             5 bern= 5 forest=True roots=1 f(R_P)= 5 [P={0,1,2,3,4,5,6,7,8,9}] minf(randomR)= 5 lemmaA=True NOT realizable over any field OK (2.7s)
F7+U(2,3)                    n=10 r=5 #layerparts=   63 r'= 8 dSS=             8 bern= 8 forest=True roots=2 f(R_P)= 8 [P={0,1,2,3,4,5,6} {7,8,9}] minf(randomR)= 9 lemmaA=True NOT C-realizable, disconnected OK (5.0s)
T(AG(3,2))                   n= 8 r=3 #layerparts=   28 r'= 5 dSS=             5 bern= 5 forest=True roots=1 f(R_P)= 5 [P={0,1,2,3,4,5,6,7}] minf(randomR)= 5 lemmaA=True rank-3 truncation of AG(3,2) OK (0.8s)
=== random sparse paving matroids (mostly non-realizable for larger n) ===
SP(3,7;5)                    n= 7 r=3 #layerparts=   21 r'= 5 dSS=             5 bern= 5 forest=True roots=1 f(R_P)= 5 [P={0,1,2,3,4,5,6}] minf(randomR)= 5 lemmaA=True random sparse paving OK (0.3s)
SP(4,8;6)                    n= 8 r=4 #layerparts=   68 r'= 7 dSS=             7 bern= 7 forest=True roots=1 f(R_P)= 7 [P={0,1,2,3,4,5,6,7}] minf(randomR)= 7 lemmaA=True random sparse paving OK (1.6s)
SP(4,8;9)                    n= 8 r=4 #layerparts=   74 r'= 7 dSS=             7 bern= 7 forest=True roots=1 f(R_P)= 7 [P={0,1,2,3,4,5,6,7}] minf(randomR)= 7 lemmaA=True random sparse paving OK (1.5s)
SP(4,8;9)                    n= 8 r=4 #layerparts=   74 r'= 7 dSS=             7 bern= 7 forest=True roots=1 f(R_P)= 7 [P={0,1,2,3,4,5,6,7}] minf(randomR)= 7 lemmaA=True random sparse paving OK (1.7s)
SP(4,9;12)                   n= 9 r=4 #layerparts=  108 r'= 7 dSS=             7 bern= 7 forest=True roots=1 f(R_P)= 7 [P={0,1,2,3,4,5,6,7,8}] minf(randomR)= 7 lemmaA=True random sparse paving OK (3.5s)
SP(4,9;15)                   n= 9 r=4 #layerparts=  114 r'= 7 dSS=             7 bern= 7 forest=True roots=1 f(R_P)= 7 [P={0,1,2,3,4,5,6,7,8}] minf(randomR)= 7 lemmaA=True random sparse paving OK (3.5s)
SP(5,9;10)                   n= 9 r=5 #layerparts=  176 r'= 9 dSS=             9 bern= 9 forest=True roots=1 f(R_P)= 9 [P={0,1,2,3,4,5,6,7,8}] minf(randomR)= 9 lemmaA=True random sparse paving OK (6.6s)
SP(5,10;12)                  n=10 r=5 #layerparts=  270 r'= 9 dSS=             9 bern= 9 forest=True roots=1 f(R_P)= 9 [P={0,1,2,3,4,5,6,7,8,9}] minf(randomR)= 9 lemmaA=True random sparse paving OK (10.6s)
SP(4,10;20)                  n=10 r=4 #layerparts=  160 r'= 7 dSS=             7 bern= 7 forest=True roots=1 f(R_P)= 7 [P={0,1,2,3,4,5,6,7,8,9}] minf(randomR)= 7 lemmaA=True random sparse paving OK (6.6s)
SP(5,10;24)                  n=10 r=5 #layerparts=  330 r'= 9 dSS=             9 bern= 9 forest=True roots=1 f(R_P)= 9 [P={0,1,2,3,4,5,6,7,8,9}] minf(randomR)= 9 lemmaA=True random sparse paving OK (15.1s)
=== duals of random rank-3 matroids with many 3-point lines ===
SP(3,8;8)*                   n= 8 r=5 #layerparts=  110 r'= 8 dSS=             8 bern= 8 forest=True roots=2 f(R_P)= 8 [P={0} {1} {2} {3} {4} {5} {6} {7}] minf(randomR)= 8 lemmaA=True dual of random rank-3 OK (2.8s)
SP(3,9;10)*                  n= 9 r=6 #layerparts=  216 r'= 9 dSS=             9 bern= 9 forest=True roots=3 f(R_P)= 9 [P={0} {1} {2} {3} {4} {5} {6} {7} {8}] minf(randomR)= 9 lemmaA=True dual of random rank-3 OK (10.2s)
SP(3,10;11)*                 n=10 r=7 #layerparts=  364 r'=10 dSS=            10 bern=10 forest=True roots=4 f(R_P)=10 [P={0} {1} {2} {3} {4} {5} {6} {7} {8} {9}] minf(randomR)=10 lemmaA=True dual of random rank-3 OK (23.1s)
=== random realizable (Q) matroids with dependencies ===
Lin(3,7)                     n= 7 r=3 #layerparts=   15 r'= 5 dSS=             5 bern= 5 forest=True roots=1 f(R_P)= 5 [P={0,1,2,3,4,5,6}] minf(randomR)= 5 lemmaA=True realizable over Q OK (0.2s)
Lin(4,8)                     n= 8 r=4 #layerparts=   58 r'= 7 dSS=             7 bern= 7 forest=True roots=1 f(R_P)= 7 [P={0,1,2,3,4,5,6,7}] minf(randomR)= 7 lemmaA=True realizable over Q OK (1.2s)
Lin(4,9)                     n= 9 r=4 #layerparts=   96 r'= 7 dSS=             7 bern= 7 forest=True roots=1 f(R_P)= 7 [P={0,1,2,3,4,5,6,7,8}] minf(randomR)= 7 lemmaA=True realizable over Q OK (3.3s)
Lin(5,9)                     n= 9 r=5 #layerparts=  163 r'= 9 dSS=             9 bern= 9 forest=True roots=1 f(R_P)= 9 [P={0,1,2,3,4,5,6,7,8}] minf(randomR)= 9 lemmaA=True realizable over Q OK (5.2s)
=== two-matroid form (Bernstein Thm 3.5) on non-realizable pairs ===
  pair F7                     & F7-                    n=7 rank M(rM+rN-1)= 5 max_FG=   5 bern= 5 forest=True F7 vs non-Fano OK
  pair F7                     & F7*                    n=7 rank M(rM+rN-1)= 6 max_FG=   6 bern= 6 forest=True F7 vs F7* OK
  pair Vamos                  & SP(4,8;9)              n=8 rank M(rM+rN-1)= 7 max_FG=   7 bern= 7 forest=True Vamos vs random SP OK
  pair Vamos                  & AG(3,2)                n=8 rank M(rM+rN-1)= 7 max_FG=   7 bern= 7 forest=True Vamos vs AG(3,2) OK
  pair nonPappus              & nonPappus*             n=9 rank M(rM+rN-1)= 8 max_FG=   8 bern= 8 forest=True nonPappus vs dual OK
  pair U(2,8)                 & Vamos                  n=8 rank M(rM+rN-1)= 5 max_FG=   5 bern= 5 forest=True U(2,8) vs Vamos OK
  pair SP(3,8;7)              & SP(4,8;9)              n=8 rank M(rM+rN-1)= 6 max_FG=   6 bern= 6 forest=True random SP pair OK
  pair SP(4,8;8)              & SP(4,8;14)             n=8 rank M(rM+rN-1)= 7 max_FG=   7 bern= 7 forest=True random SP pair OK
  pair SP(3,9;8)              & SP(5,9;15)             n=9 rank M(rM+rN-1)= 7 max_FG=   7 bern= 7 forest=True random SP pair OK
  pair SP(4,9;13)             & SP(5,9;15)             n=9 rank M(rM+rN-1)= 8 max_FG=   8 bern= 8 forest=True random SP pair OK
ALL OK
