--- F7*: n=7, rank=4, r'(E)=7
  optimal partition P = {001,010,011,100,101,110,111} ; f(R_P) = 7 (dim R_P=1)
  basis I of M' (|I|=7): {001,010,011,100,101,110,111}
  I1 (basis of M|I) = {011,101,110,111}, I2 = {001,010,011,100}, roots = ['011']
  placement steps: [('root', '011'), ('N-side', '001'), ('N-side', '010'), ('M-side', '101'), ('N-side', '100'), ('M-side', '110'), ('M-side', '111')]
  flag F = {111} < {110,111} < {100,101,110,111} < {001,010,011,100,101,110,111}
  flag G = {100} < {010,100} < {001,010,100,111} < {001,010,011,100,101,110,111}
  layers F: ['{111}', '{110}', '{100,101}', '{001,010,011}']
  layers G: ['{100}', '{010}', '{001,111}', '{011,101,110}']
  H_(F,G) edges on I: ['001:F4-G3', '010:F4-G2', '011:F4-G4', '100:F3-G1', '101:F3-G4', '110:F2-G4', '111:F1-G3']
  forest on I: True ; exact dim(L_F+L_G) over Q = 7 (>= |I| = 7) ; so r'(E) = 7 <= dim(Sigma+Sigma) <= adim <= f(R_P) = 7
--- Vamos: n=8, rank=4, r'(E)=7
  optimal partition P = {a,a',b,b',c,c',d,d'} ; f(R_P) = 7 (dim R_P=1)
  basis I of M' (|I|=7): {a,a',b,b',c,c',d}
  I1 (basis of M|I) = {b',c,c',d}, I2 = {a,a',b,c}, roots = ['c']
  placement steps: [('root', 'c'), ('N-side', 'a'), ('N-side', "a'"), ('N-side', 'b'), ('M-side', "b'"), ('M-side', "c'"), ('M-side', 'd')]
  flag F = {d} < {c',d} < {b',c',d} < {a,a',b,b',c,c',d,d'}
  flag G = {b} < {a',b} < {a,a',b,b'} < {a,a',b,b',c,c',d,d'}
  layers F: ['{d}', "{c'}", "{b'}", "{a,a',b,c,d'}"]
  layers G: ['{b}', "{a'}", "{a,b'}", "{c,c',d,d'}"]
  H_(F,G) edges on I: ['a:F4-G3', "a':F4-G2", 'b:F4-G1', "b':F3-G3", 'c:F4-G4', "c':F2-G4", 'd:F1-G4']
  forest on I: True ; exact dim(L_F+L_G) over Q = 7 (>= |I| = 7) ; so r'(E) = 7 <= dim(Sigma+Sigma) <= adim <= f(R_P) = 7
--- nonPappus*: n=9, rank=6, r'(E)=9
  optimal partition P = {a1} | {a2} | {a3} | {b1} | {b2} | {b3} | {c1} | {c2} | {c3} ; f(R_P) = 9 (dim R_P=9)
  basis I of M' (|I|=9): {a1,a2,a3,b1,b2,b3,c1,c2,c3}
  I1 (basis of M|I) = {a3,b2,b3,c1,c2,c3}, I2 = {a1,a2,a3,b1,b2,b3}, roots = ['a3', 'b2', 'b3']
  placement steps: [('root', 'a3'), ('N-side', 'a1'), ('N-side', 'a2'), ('M-side', 'c1'), ('N-side', 'b1'), ('M-side', 'c2'), ('M-side', 'c3'), ('root', 'b2'), ('root', 'b3')]
  flag F = {b3} < {b2,b3} < {b2,b3,c3} < {b2,b3,c2,c3} < {b1,b2,b3,c1,c2,c3} < {a1,a2,a3,b1,b2,b3,c1,c2,c3}
  flag G = {b3} < {b2,b3} < {b1,b2,b3} < {a2,b1,b2,b3} < {a1,a2,b1,b2,b3} < {a1,a2,a3,b1,b2,b3,c1,c2,c3}
  layers F: ['{b3}', '{b2}', '{c3}', '{c2}', '{b1,c1}', '{a1,a2,a3}']
  layers G: ['{b3}', '{b2}', '{b1}', '{a2}', '{a1}', '{a3,c1,c2,c3}']
  H_(F,G) edges on I: ['a1:F6-G5', 'a2:F6-G4', 'a3:F6-G6', 'b1:F5-G3', 'b2:F2-G2', 'b3:F1-G1', 'c1:F5-G6', 'c2:F4-G6', 'c3:F3-G6']
  forest on I: True ; exact dim(L_F+L_G) over Q = 9 (>= |I| = 9) ; so r'(E) = 9 <= dim(Sigma+Sigma) <= adim <= f(R_P) = 9
--- AG(3,2): n=8, rank=4, r'(E)=7
  optimal partition P = {0,1,2,3,4,5,6,7} ; f(R_P) = 7 (dim R_P=1)
  basis I of M' (|I|=7): {0,1,2,3,4,5,6}
  I1 (basis of M|I) = {3,4,5,6}, I2 = {0,1,2,4}, roots = ['4']
  placement steps: [('root', '4'), ('N-side', '1'), ('N-side', '2'), ('M-side', '3'), ('N-side', '0'), ('M-side', '5'), ('M-side', '6')]
  flag F = {6} < {5,6} < {0,3,5,6} < {0,1,2,3,4,5,6,7}
  flag G = {0} < {0,2} < {0,1,2,3} < {0,1,2,3,4,5,6,7}
  layers F: ['{6}', '{5}', '{0,3}', '{1,2,4,7}']
  layers G: ['{0}', '{2}', '{1,3}', '{4,5,6,7}']
  H_(F,G) edges on I: ['0:F3-G1', '1:F4-G3', '2:F4-G2', '3:F3-G3', '4:F4-G4', '5:F2-G4', '6:F1-G4']
  forest on I: True ; exact dim(L_F+L_G) over Q = 7 (>= |I| = 7) ; so r'(E) = 7 <= dim(Sigma+Sigma) <= adim <= f(R_P) = 7
--- F7: n=7, rank=3, r'(E)=5
  optimal partition P = {001,010,011,100,101,110,111} ; f(R_P) = 5 (dim R_P=1)
  basis I of M' (|I|=5): {001,010,011,100,101}
  I1 (basis of M|I) = {011,100,101}, I2 = {001,010,100}, roots = ['100']
  placement steps: [('root', '100'), ('N-side', '001'), ('N-side', '010'), ('M-side', '011'), ('M-side', '101')]
  flag F = {101} < {011,101,110} < {001,010,011,100,101,110,111}
  flag G = {010} < {001,010,011} < {001,010,011,100,101,110,111}
  layers F: ['{101}', '{011,110}', '{001,010,100,111}']
  layers G: ['{010}', '{001,011}', '{100,101,110,111}']
  H_(F,G) edges on I: ['001:F3-G2', '010:F3-G1', '011:F2-G2', '100:F3-G3', '101:F1-G3']
  forest on I: True ; exact dim(L_F+L_G) over Q = 5 (>= |I| = 5) ; so r'(E) = 5 <= dim(Sigma+Sigma) <= adim <= f(R_P) = 5
