Independent verification run (independent_run_2), check of OWR-12697711-015.  Python 3.13.5, random seed 20260930
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[A] Named matroids (non-realizable over C unless marked 'ctrl'), full analysis
  F7                                 n= 7 d=3 conn=1 circuits=  14 maxflags=    21 layerparts=   21 r'(E)= 5 min(n,2d-1)= 5 dim(S+S)= 5 val(R_Popt)= 5 braidmin=5 strictP=609 minRtested=5 Popt=[7] circ-agree=0 [0.2s]
  F7*                                n= 7 d=4 conn=1 circuits=   7 maxflags=   126 layerparts=   49 r'(E)= 7 min(n,2d-1)= 7 dim(S+S)= 7 val(R_Popt)= 7 braidmin=7 strictP=315 minRtested=7 Popt=[7] circ-agree=22 [0.4s]
  AG(3,2)                            n= 8 d=4 conn=1 circuits=  14 maxflags=   168 layerparts=   84 r'(E)= 7 min(n,2d-1)= 7 dim(S+S)= 7 val(R_Popt)= 7 braidmin=7 strictP=2618 minRtested=7 Popt=[8] circ-agree=4 [1.4s]
  V8                                 n= 8 d=4 conn=1 circuits=  41 maxflags=   276 layerparts=   66 r'(E)= 7 min(n,2d-1)= 7 dim(S+S)= 7 val(R_Popt)= 7 braidmin=7 strictP=2672 minRtested=7 Popt=[8] circ-agree=13 [1.1s]
  V8*                                n= 8 d=4 conn=1 circuits=  41 maxflags=   276 layerparts=   66 r'(E)= 7 min(n,2d-1)= 7 dim(S+S)= 7 val(R_Popt)= 7 braidmin=7 strictP=2672 minRtested=7 Popt=[8] circ-agree=10 [1.0s]
  NP                                 n= 9 d=3 conn=1 circuits=  86 maxflags=    48 layerparts=   36 r'(E)= 5 min(n,2d-1)= 5 dim(S+S)= 5 val(R_Popt)= 5 braidmin=- strictP=- minRtested=5 Popt=[9] circ-agree=0 [0.4s]
  NP*                                n= 9 d=6 conn=1 circuits=  20 maxflags= 12240 layerparts=  198 r'(E)= 9 min(n,2d-1)= 9 dim(S+S)= 9 val(R_Popt)= 9 braidmin=- strictP=- minRtested=9 Popt=[1, 1, 1, 1, 1, 1, 1, 1, 1] circ-agree=45 [2.2s]
  ND                                 n=10 d=3 conn=1 circuits= 156 maxflags=    63 layerparts=   45 r'(E)= 5 min(n,2d-1)= 5 dim(S+S)= 5 val(R_Popt)= 5 braidmin=- strictP=- minRtested=5 Popt=[10] circ-agree=0 [0.4s]
  F7-                                n= 7 d=3 conn=1 circuits=  17 maxflags=    24 layerparts=   21 r'(E)= 5 min(n,2d-1)= 5 dim(S+S)= 5 val(R_Popt)= 5 braidmin=5 strictP=615 minRtested=5 Popt=[7] circ-agree=4 [0.2s]
  Pappus                             n= 9 d=3 conn=1 circuits=  81 maxflags=    45 layerparts=   36 r'(E)= 5 min(n,2d-1)= 5 dim(S+S)= 5 val(R_Popt)= 5 braidmin=- strictP=- minRtested=5 Popt=[9] circ-agree=1 [0.5s]
  (F7-, Pappus are realizable over C: controls)
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[B] Free extensions / sums with nontrivial optimal partitions (the note's Table 1 families + more)
  FE(F7+U2,2)                        n=10 d=5 conn=1 circuits=  42 maxflags=  1176 layerparts=  133 r'(E)= 8 min(n,2d-1)= 9 dim(S+S)= 8 val(R_Popt)= 8 braidmin=- strictP=- minRtested=8 Popt=[1, 1, 1, 7] circ-agree=2 [0.7s]
  FE(AG(3,2)+U2,2)                   n=11 d=6 conn=1 circuits=  70 maxflags= 15960 layerparts=  532 r'(E)=10 min(n,2d-1)=11 dim(S+S)=10 val(R_Popt)=10 braidmin=- strictP=- minRtested=11 Popt=[1, 1, 1, 8] circ-agree=2 [3.4s]
  FE(V8+U2,2)                        n=11 d=6 conn=1 circuits= 106 maxflags= 24060 layerparts=  424 r'(E)=10 min(n,2d-1)=11 dim(S+S)=10 val(R_Popt)=10 braidmin=- strictP=- minRtested=11 Popt=[1, 1, 1, 8] circ-agree=4 [2.6s]
  FE(NP+U2,2)                        n=12 d=5 conn=1 circuits= 162 maxflags=  2520 layerparts=  225 r'(E)= 8 min(n,2d-1)= 9 dim(S+S)= 8 val(R_Popt)= 8 braidmin=- strictP=- minRtested=- Popt=[1, 1, 1, 9] circ-agree=0 [0.2s]
  FE(ND+U2,2)                        n=13 d=5 conn=1 circuits= 267 maxflags=  3264 layerparts=  280 r'(E)= 8 min(n,2d-1)= 9 dim(S+S)= 8 val(R_Popt)= 8 braidmin=- strictP=- minRtested=- Popt=[1, 1, 1, 10] circ-agree=0 [0.4s]
  FE(F7+U3,3)                        n=11 d=6 conn=1 circuits=  42 maxflags=  8400 layerparts=  217 r'(E)= 9 min(n,2d-1)=11 dim(S+S)= 9 val(R_Popt)= 9 braidmin=- strictP=- minRtested=10 Popt=[1, 1, 1, 1, 7] circ-agree=5 [1.3s]
  FE(AG(3,2)+U3,3)                   n=12 d=7 conn=1 circuits=  70 maxflags=131040 layerparts=  868 r'(E)=11 min(n,2d-1)=12 dim(S+S)=11 val(R_Popt)=11 braidmin=- strictP=- minRtested=- Popt=[1, 1, 1, 1, 8] circ-agree=2 [1.6s]
  FE(V8+U3,3)                        n=12 d=7 conn=1 circuits= 106 maxflags=202320 layerparts=  688 r'(E)=11 min(n,2d-1)=12 dim(S+S)=11 val(R_Popt)=11 braidmin=- strictP=- minRtested=- Popt=[1, 1, 1, 1, 8] circ-agree=10 [1.3s]
  FE(NP+U3,3)                        n=13 d=6 conn=1 circuits= 162 maxflags= 18360 layerparts=  369 r'(E)= 9 min(n,2d-1)=11 dim(S+S)= 9 val(R_Popt)= 9 braidmin=- strictP=- minRtested=- Popt=[1, 1, 1, 1, 9] circ-agree=0 [0.5s]
  FE(F7+U1,2+U1,1)                   n=11 d=5 conn=1 circuits=  71 maxflags=  1176 layerparts=  133 r'(E)= 8 min(n,2d-1)= 9 dim(S+S)= 8 val(R_Popt)= 8 braidmin=- strictP=- minRtested=9 Popt=[1, 1, 2, 7] circ-agree=1 [0.8s]
  FE(AG(3,2)+U1,2+U1,1)              n=12 d=6 conn=1 circuits= 127 maxflags= 15960 layerparts=  532 r'(E)=10 min(n,2d-1)=11 dim(S+S)=10 val(R_Popt)=10 braidmin=- strictP=- minRtested=- Popt=[1, 1, 2, 8] circ-agree=0 [0.6s]
  FE(V8+U1,2+U1,1)                   n=12 d=6 conn=1 circuits= 172 maxflags= 24060 layerparts=  424 r'(E)=10 min(n,2d-1)=11 dim(S+S)=10 val(R_Popt)=10 braidmin=- strictP=- minRtested=- Popt=[1, 1, 2, 8] circ-agree=1 [0.4s]
  FE(NP+U1,2+U1,1)                   n=13 d=5 conn=1 circuits= 239 maxflags=  2520 layerparts=  225 r'(E)= 8 min(n,2d-1)= 9 dim(S+S)= 8 val(R_Popt)= 8 braidmin=- strictP=- minRtested=- Popt=[1, 1, 2, 9] circ-agree=0 [0.4s]
  FE(F7+U1,3+U1,1)                   n=12 d=5 conn=1 circuits= 101 maxflags=  1176 layerparts=  133 r'(E)= 8 min(n,2d-1)= 9 dim(S+S)= 8 val(R_Popt)= 8 braidmin=- strictP=- minRtested=- Popt=[1, 1, 3, 7] circ-agree=0 [0.2s]
  FE(AG(3,2)+U1,3+U1,1)              n=13 d=6 conn=1 circuits= 185 maxflags= 15960 layerparts=  532 r'(E)=10 min(n,2d-1)=11 dim(S+S)=10 val(R_Popt)=10 braidmin=- strictP=- minRtested=- Popt=[1, 1, 3, 8] circ-agree=0 [0.7s]
  FE(V8+U1,3+U1,1)                   n=13 d=6 conn=1 circuits= 239 maxflags= 24060 layerparts=  424 r'(E)=10 min(n,2d-1)=11 dim(S+S)=10 val(R_Popt)=10 braidmin=- strictP=- minRtested=- Popt=[1, 1, 3, 8] circ-agree=0 [0.7s]
  FE(F7+U2,2+U1,2)                   n=12 d=6 conn=1 circuits=  71 maxflags=  8400 layerparts=  217 r'(E)= 9 min(n,2d-1)=11 dim(S+S)= 9 val(R_Popt)= 9 braidmin=- strictP=- minRtested=- Popt=[1, 1, 1, 2, 7] circ-agree=0 [0.3s]
  FE(AG(3,2)+U2,2+U1,2)              n=13 d=7 conn=1 circuits= 127 maxflags=131040 layerparts=  868 r'(E)=11 min(n,2d-1)=13 dim(S+S)=11 val(R_Popt)=11 braidmin=- strictP=- minRtested=- Popt=[1, 1, 1, 2, 8] circ-agree=4 [2.0s]
  FE(V8+U2,2+U1,2)                   n=13 d=7 conn=1 circuits= 172 maxflags=202320 layerparts=  688 r'(E)=11 min(n,2d-1)=13 dim(S+S)=11 val(R_Popt)=11 braidmin=- strictP=- minRtested=- Popt=[1, 1, 1, 2, 8] circ-agree=3 [1.3s]
  FE^2(F7+U3,3)                      n=12 d=6 conn=1 circuits= 175 maxflags= 21120 layerparts=  456 r'(E)=10 min(n,2d-1)=11 dim(S+S)=10 val(R_Popt)=10 braidmin=- strictP=- minRtested=- Popt=[1, 1, 1, 1, 1, 7] circ-agree=1 [0.5s]
  FE^2(AG(3,2)+U3,3)                 n=13 d=7 conn=1 circuits= 350 maxflags=358560 layerparts= 1828 r'(E)=12 min(n,2d-1)=13 dim(S+S)=12 val(R_Popt)=12 braidmin=- strictP=- minRtested=- Popt=[1, 1, 1, 1, 1, 8] circ-agree=1 [6.2s]
  FE^2(V8+U3,3)                      n=13 d=7 conn=1 circuits= 422 maxflags=520560 layerparts= 1468 r'(E)=12 min(n,2d-1)=13 dim(S+S)=12 val(R_Popt)=12 braidmin=- strictP=- minRtested=- Popt=[1, 1, 1, 1, 1, 8] circ-agree=5 [4.8s]
  Table reproduction (n,d) r'(E) < min(n,2d-1), connected?:
    FE(N+U2,2)         (10,5) 8<9 c=1 | (11,6) 10<11 c=1 | (11,6) 10<11 c=1 | (12,5) 8<9 c=1 | (13,5) 8<9 c=1
    FE(N+U3,3)         (11,6) 9<11 c=1 | (12,7) 11<12 c=1 | (12,7) 11<12 c=1 | (13,6) 9<11 c=1 | --
    FE(N+U1,2+U1,1)    (11,5) 8<9 c=1 | (12,6) 10<11 c=1 | (12,6) 10<11 c=1 | (13,5) 8<9 c=1 | --
    FE(N+U1,3+U1,1)    (12,5) 8<9 c=1 | (13,6) 10<11 c=1 | (13,6) 10<11 c=1 | -- | --
    FE(N+U2,2+U1,2)    (12,6) 9<11 c=1 | (13,7) 11<13 c=1 | (13,7) 11<13 c=1 | -- | --
    FE^2(N+U3,3)       (12,6) 10<11 c=1 | (13,7) 12<13 c=1 | (13,7) 12<13 c=1 | -- | --
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[C] Further non-realizable constructions
  FE(F7+U1,2+U1,2)                   n=12 d=5 conn=1 circuits= 128 maxflags=  1176 layerparts=  133 r'(E)= 8 min(n,2d-1)= 9 dim(S+S)= 8 val(R_Popt)= 8 braidmin=- strictP=- minRtested=- Popt=[1, 2, 2, 7] circ-agree=0 [0.2s]
  T(F7+U2,2)                         n= 9 d=4 conn=1 circuits=  42 maxflags=   189 layerparts=   70 r'(E)= 7 min(n,2d-1)= 7 dim(S+S)= 7 val(R_Popt)= 7 braidmin=- strictP=- minRtested=7 Popt=[9] circ-agree=1 [0.4s]
  T(V8+U1,1)                         n= 9 d=4 conn=1 circuits= 106 maxflags=   444 layerparts=   94 r'(E)= 7 min(n,2d-1)= 7 dim(S+S)= 7 val(R_Popt)= 7 braidmin=- strictP=- minRtested=7 Popt=[9] circ-agree=4 [0.5s]
  FE(F7+U2,3)                        n=11 d=5 conn=1 circuits=  99 maxflags=  1722 layerparts=  231 r'(E)= 9 min(n,2d-1)= 9 dim(S+S)= 9 val(R_Popt)= 9 braidmin=- strictP=- minRtested=- Popt=[11] circ-agree=1 [0.2s]
  FE^2(V8)                           n=10 d=4 conn=1 circuits= 227 maxflags=   660 layerparts=  130 r'(E)= 7 min(n,2d-1)= 7 dim(S+S)= 7 val(R_Popt)= 7 braidmin=- strictP=- minRtested=7 Popt=[10] circ-agree=1 [0.7s]
  FE(F7*)                            n= 8 d=4 conn=1 circuits=  35 maxflags=   252 layerparts=   70 r'(E)= 7 min(n,2d-1)= 7 dim(S+S)= 7 val(R_Popt)= 7 braidmin=7 strictP=2660 minRtested=7 Popt=[8] circ-agree=7 [0.9s]
  T(FE(F7+U3,3))                     n=11 d=5 conn=1 circuits= 147 maxflags=  2544 layerparts=  239 r'(E)= 9 min(n,2d-1)= 9 dim(S+S)= 9 val(R_Popt)= 9 braidmin=- strictP=- minRtested=- Popt=[11] circ-agree=1 [0.2s]
  (F7+U2,3)                          n=10 d=5 conn=0 circuits=  15 maxflags=   630 layerparts=   63 r'(E)= 8 min(n,2d-1)= 9 dim(S+S)= 8 val(R_Popt)= 8 braidmin=- strictP=- minRtested=8 Popt=[3, 7] circ-agree=2 [0.4s]
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[D] Random sparse paving matroids (most are non-realizable); full braid check for n<=8
  115 random sparse paving matroids: dim(S+S) = r'(E) = value at R_Popt in all cases; 0 with r'(E) < min(n,2d-1)
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[E] Realizable controls over Q (DEPRY Thm 1.3.1 applies): Nisse's example with random stars, random matrices
  Nisse(4x7)                         n= 7 d=4 conn=1 circuits=  10 maxflags=   114 layerparts=   25 r'(E)= 6 min(n,2d-1)= 7 dim(S+S)= 6 val(R_Popt)= 6 braidmin=6 strictP=295 minRtested=6 Popt=[1, 1, 1, 4] circ-agree=18 [0.2s]
  randQ(3x7)                         n= 7 d=3 conn=1 circuits=  10 maxflags=    12 layerparts=    6 r'(E)= 4 min(n,2d-1)= 5 dim(S+S)= 4 val(R_Popt)= 4 braidmin=4 strictP=396 minRtested=5 Popt=[1, 1, 2, 3] circ-agree=3 [0.1s]
  randQ(4x8)                         n= 8 d=4 conn=1 circuits=  26 maxflags=   174 layerparts=   41 r'(E)= 7 min(n,2d-1)= 7 dim(S+S)= 7 val(R_Popt)= 7 braidmin=7 strictP=2339 minRtested=7 Popt=[8] circ-agree=7 [0.5s]
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[F] Pairs (M,N): Prop 4.2  dim(S(M)+S(N)) = min_P sum(rM+rN-1) = value of first min at R_P; Lemma 3.6 greedy on pairs
  F7             F7-*           n=7 dim(S(M)+S(N))=6 min_P sum(rM+rN-1)=6 value at R_Popt=6 min over tested R=6 ok=True
  F7             U3,7           n=7 dim(S(M)+S(N))=5 min_P sum(rM+rN-1)=5 value at R_Popt=5 min over tested R=5 ok=True
  V8             AG(3,2)        n=8 dim(S(M)+S(N))=7 min_P sum(rM+rN-1)=7 value at R_Popt=7 min over tested R=7 ok=True
  AG(3,2)        AG*            n=8 dim(S(M)+S(N))=7 min_P sum(rM+rN-1)=7 value at R_Popt=7 min over tested R=7 ok=True
  V8             U2,8           n=8 dim(S(M)+S(N))=5 min_P sum(rM+rN-1)=5 value at R_Popt=5 min over tested R=5 ok=True
  F7             F7-            n=7 dim(S(M)+S(N))=5 min_P sum(rM+rN-1)=5 value at R_Popt=5 min over tested R=5 ok=True
  SP(7,4,5)      SP(7,5,3)      n=7 dim(S(M)+S(N))=7 min_P sum(rM+rN-1)=7 value at R_Popt=7 min over tested R=7 ok=True
  SP(8,3,7)      SP(8,5,7)      n=8 dim(S(M)+S(N))=7 min_P sum(rM+rN-1)=7 value at R_Popt=7 min over tested R=7 ok=True
  SP(8,3,6)      SP(8,3,6)      n=8 dim(S(M)+S(N))=5 min_P sum(rM+rN-1)=5 value at R_Popt=5 min over tested R=5 ok=True
  SP(8,4,8)      SP(8,4,7)      n=8 dim(S(M)+S(N))=7 min_P sum(rM+rN-1)=7 value at R_Popt=7 min over tested R=7 ok=True
  SP(8,4,8)      SP(8,4,10)     n=8 dim(S(M)+S(N))=7 min_P sum(rM+rN-1)=7 value at R_Popt=7 min over tested R=7 ok=True
  SP(8,4,8)      SP(8,3,6)      n=8 dim(S(M)+S(N))=6 min_P sum(rM+rN-1)=6 value at R_Popt=6 min over tested R=6 ok=True
  SP(8,3,5)      SP(8,3,7)      n=8 dim(S(M)+S(N))=5 min_P sum(rM+rN-1)=5 value at R_Popt=5 min over tested R=5 ok=True
  SP(8,3,7)      SP(8,4,8)      n=8 dim(S(M)+S(N))=6 min_P sum(rM+rN-1)=6 value at R_Popt=6 min over tested R=6 ok=True
  SP(8,4,7)      SP(8,3,7)      n=8 dim(S(M)+S(N))=6 min_P sum(rM+rN-1)=6 value at R_Popt=6 min over tested R=6 ok=True
  SP(8,4,8)      SP(8,4,8)      n=8 dim(S(M)+S(N))=7 min_P sum(rM+rN-1)=7 value at R_Popt=7 min over tested R=7 ok=True
  SP(7,3,5)      SP(7,5,3)      n=7 dim(S(M)+S(N))=7 min_P sum(rM+rN-1)=7 value at R_Popt=7 min over tested R=7 ok=True
  SP(7,3,6)      SP(7,5,3)      n=7 dim(S(M)+S(N))=7 min_P sum(rM+rN-1)=7 value at R_Popt=7 min over tested R=7 ok=True
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[G] Lemma 3.6 stress test: random pairs, random I independent in M(rM+rN-1), random tie-breaking
  300 random instances: every assertion of the proof of Lemma 3.6 and of Lemma 3.5 held; forests verified
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[H] Prop 5.1: Phi = union of the four coordinate 3-planes R^{S_v} of K4 (coordinates = 6 edges)
  dim(Phi+Phi) = 5
  min of 2dim(Phi+R)-dim R over 4267 tested subspaces (64 coordinate, 203 braid, 3000 random, 1000 plane-structured) = 6
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[I] DEPRY sec. 1.4 states r~(P) = 2dim(Sigma+R_P) - dim R_P for EVERY partition P; the version of the note checked
    by this run repeated this after Lemma 3.3 ('In fact equality holds in the second claim').  Full braid scans found strict inequality:
    F7                    : 609/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    F7*                   : 315/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    AG(3,2)               : 2618/4140 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6, 7]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    V8                    : 2672/4140 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6, 7]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    V8*                   : 2672/4140 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6, 7]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    F7-                   : 615/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    FE(F7*)               : 2660/4140 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6, 7]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,3,5)             : 620/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,5)             : 620/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,5)             : 620/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,6)             : 615/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,4)             : 626/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,5)             : 620/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,5)             : 620/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,7)             : 609/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,5)             : 620/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,5)             : 620/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,5)             : 620/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,5)             : 620/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,5)             : 620/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,6)             : 615/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,5)             : 620/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,5)             : 621/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,4)             : 626/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,6)             : 615/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,5)             : 620/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,7)             : 609/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,6)             : 615/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,6)             : 615/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,7)             : 609/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,7)             : 609/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,3,5)             : 621/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    SP(7,4,7)             : 315/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,4)             : 324/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,7)             : 315/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,6)             : 318/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,7)             : 315/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,6)             : 318/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,6)             : 318/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,7)             : 315/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,7)             : 315/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,4)             : 324/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    SP(7,4,5)             : 321/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    Nisse(4x7)            : 295/877 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    randQ(3x7)            : 396/877 partitions strict; smallest example P=[[0, 1, 2], [3, 4, 5, 6]]: 2dim(S+R_P)-dim R_P=6 < r~(P)=8
    randQ(4x8)            : 2339/4140 partitions strict; smallest example P=[[0, 1], [2, 3, 4, 5, 6, 7]]: 2dim(S+R_P)-dim R_P=8 < r~(P)=10
    exact check F7 (lines 013,124,235,346,450,561,602), P={0,1}|{2,...,6}: max_F dim(L_F+R_P) = 4 (sum r(P_i) = 5); 2*4-2 = 6 < r~(P) = 8
    Reason: 1_E lies in L_F and in R_P, so dim(L_F+R_P) <= d+k-1 < sum r(P_i) here.  Only the inequality <= is true in general;
    equality does hold for OPTIMAL partitions (a consequence of Thm 1.2, asserted above for every optimal P in the scans).
====================================================================================================
ALL CHECKS PASSED  (total 66s)
