### SolvRun

```                         [2, 3, 2, 3], [4, 4, 1, 1]
```

$\stackrel{˜}{\pi }$ = [1, 1, 1, 1]
$\delta$ = [2, 2, 2, 2]

POSSIBLE RANKS

1 x 4
2 x 2

BASE DETERMINANT 117/512, .2285156250

NullSpace of Δ

{1, 2, 3, 4}

Nullspace of A

[{1, 4},{2, 3}]

STRATIFIED CYCLE COVERS

1

0

v[2] v[3] + v[1] v[4]

v[2] v[4] v[3] + v[1] v[4] v[3] + v[1] v[2] v[4] + v[1] v[2] v[3]

2 v[1] v[2] v[4] v[3]

============================================================================

1, [1, 1, 1, 1]

"Coloring", {}

"R", [2, 3, 2, 3]
"B", [4, 4, 1, 1]

NOT SYNC'D

Kernel has RANK, 2,

N= $\left(\begin{array}{cccc}0& \frac{1}{2}& \frac{1}{2}& 1\\ \frac{1}{2}& 0& 1& \frac{1}{2}\\ \frac{1}{2}& 1& 0& \frac{1}{2}\\ 1& \frac{1}{2}& \frac{1}{2}& 0\end{array}\right)$

"R CYCLES", 1 + v[2] v[3]
"B CYCLES", 1 + v[1] v[4]

"Char Poly V" $\frac{-1}{5}$ ( $-5+{t}^{2}$ )

CHECKS

"OM-R" $\left(\begin{array}{cccc}0& \frac{1}{2}& \frac{1}{2}& 0\\ 0& \frac{1}{2}& \frac{1}{2}& 0\\ 0& \frac{1}{2}& \frac{1}{2}& 0\\ 0& \frac{1}{2}& \frac{1}{2}& 0\end{array}\right)$ "CESARO CHECK", true

"OM-B" $\left(\begin{array}{cccc}\frac{1}{2}& 0& 0& \frac{1}{2}\\ \frac{1}{2}& 0& 0& \frac{1}{2}\\ \frac{1}{2}& 0& 0& \frac{1}{2}\\ \frac{1}{2}& 0& 0& \frac{1}{2}\end{array}\right)$ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"Kernel Check OMEGA-R", true $\left(\begin{array}{cccc}\frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right)$

"RANKING B from RC" 1, "vs", 2

"RANKING B-NAT from RC" 1, "vs", 2

"RANKING B from FIRST ROW of RC" $\left(\begin{array}{cccc}\frac{1}{2}& 0& 0& \frac{1}{2}\\ \frac{1}{2}& 0& 0& \frac{1}{2}\end{array}\right)$ 1, "vs", 2

"CHECKING OMEGA-B", true

"Kernel Check OMEGA-B", true $\left(\begin{array}{cccc}\frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\end{array}\right)$

"RANKING R from BC" 1, "vs", 2

"RANKING R-NAT from BC" 1, "vs", 2

"RANKING R from FIRST ROW of BC" $\left(\begin{array}{cccc}0& \frac{1}{2}& \frac{1}{2}& 0\\ 0& \frac{1}{2}& \frac{1}{2}& 0\end{array}\right)$ 1, "vs", 2

NOT SYNC'D

"RANK of R is ", 2
"R ranking is ", 1, "vs", 2
${\text{R}}^{♮}$ = [3, 2, 3, 2]
"RBAR ranking", 1, "vs", 2

"RANK of B is ", 2
"B ranking is ", 1, "vs", 2
${\text{B}}^{♮}$ = [1, 1, 4, 4]
"BBAR ranking", 1, "vs", 2

${\text{R}}^{♮}{\text{B}}^{♮}$ = [4, 1, 4, 1]
${\text{B}}^{♮}{\text{R}}^{♮}$ = [3, 3, 2, 2]

"RNATBNAT ranking", 1, "vs", 2

"BNATRNAT ranking", 1, "vs", 2

"Centralizer " [4, 3, 2, 1] [1, 2, 3, 4]

"Char Poly Commutator" $-2{x}^{2}+{x}^{4}$

"Min Poly Commutator" $-2{x}^{2}+{x}^{4}$

${\text{R}}^{♮}{\text{B}}^{♮}$ CYCLES , 1 + v[1] v[4]
${\text{B}}^{♮}{\text{R}}^{♮}$ CYCLES , 1 + v[2] v[3]

${\text{R}}^{♮}\text{B}$ = [1, 4, 1, 4], (v[1] + 1) (v[4] + 1)
${\text{B}}^{♮}\text{R}$ = [2, 2, 3, 3], (v[2] + 1) (v[3] + 1)

"RNATB NAT RANK", 2
"BNATR NAT RANK", 2

"IDEMSOLVRANK", 2, "LOCAL TRACE", 0
"IDEMSOLVRANK", 2, "LOCAL TRACE", 0

"IDEMSOLVER", [1, 4, 1, 4]
"IDEMSOLVER", [2, 2, 3, 3]

"IDEMSOLVABLE?", false

"ABELIAN? " false

"CMM = " $\left(\begin{array}{cccc}0& 0& -1& 1\\ 1& 0& -1& 0\\ 0& -1& 0& 1\\ 1& -1& 0& 0\end{array}\right)$

"ISIDEM?", false

RBSOLVRANK, 2, with index , 2

NOT SOLVABLE

"COLORING IS CC"

============================================================================

2, [1, -1, 1, 1]

"Coloring", {2}

"R", [2, 4, 2, 3]
"B", [4, 3, 1, 1]

SYNC'D

"R CYCLES", 1 + v[2] v[4] v[3]
"B CYCLES", 1 + v[1] v[4]

"Char Poly V" $\frac{-1}{10}$ ( $-10-3t+{t}^{3}$ )

CHECKS

"OM-R" $\left(\begin{array}{cccc}0& \frac{1}{3}& \frac{1}{3}& \frac{1}{3}\\ 0& \frac{1}{3}& \frac{1}{3}& \frac{1}{3}\\ 0& \frac{1}{3}& \frac{1}{3}& \frac{1}{3}\\ 0& \frac{1}{3}& \frac{1}{3}& \frac{1}{3}\end{array}\right)$ "CESARO CHECK", true

"OM-B" $\left(\begin{array}{cccc}\frac{1}{2}& 0& 0& \frac{1}{2}\\ \frac{1}{2}& 0& 0& \frac{1}{2}\\ \frac{1}{2}& 0& 0& \frac{1}{2}\\ \frac{1}{2}& 0& 0& \frac{1}{2}\end{array}\right)$ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"RANKING B from RC" 3, "vs", 3

"RANKING B-NAT from RC" 1, "vs", 2

"RANKING B from FIRST ROW of RC" $\left(\begin{array}{cccc}\frac{2}{3}& 0& \frac{1}{3}& 0\\ \frac{1}{3}& 0& 0& \frac{2}{3}\\ \frac{2}{3}& 0& 0& \frac{1}{3}\end{array}\right)$ 3, "vs", 3

"CHECKING OMEGA-B", true

"RANKING R from BC" 3, "vs", 3

"RANKING R-NAT from BC" 1, "vs", 3

"RANKING R from FIRST ROW of BC" $\left(\begin{array}{cccc}0& \frac{1}{2}& \frac{1}{2}& 0\\ 0& \frac{1}{2}& 0& \frac{1}{2}\\ 0& 0& \frac{1}{2}& \frac{1}{2}\end{array}\right)$ 3, "vs", 3

SYNC'D

"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
${\text{R}}^{♮}$ = [3, 2, 3, 4]
"RBAR ranking", 3, "vs", 3

"RANK of B is ", 3
"B ranking is ", 2, "vs", 3
${\text{B}}^{♮}$ = [1, 1, 4, 4]
"BBAR ranking", 1, "vs", 2

${\text{R}}^{♮}{\text{B}}^{♮}$ = [4, 1, 4, 4]
${\text{B}}^{♮}{\text{R}}^{♮}$ = [3, 3, 4, 4]

"RNATBNAT ranking", 2, "vs", 2

"BNATRNAT ranking", 2, "vs", 2

"Centralizer "

"Char Poly Commutator" ${x}^{4}$

"Min Poly Commutator" ${x}^{3}$

${\text{R}}^{♮}{\text{B}}^{♮}$ CYCLES , v[4] + 1
${\text{B}}^{♮}{\text{R}}^{♮}$ CYCLES , v[4] + 1

${\text{R}}^{♮}\text{B}$ = [1, 3, 1, 1], v[1] + 1
${\text{B}}^{♮}\text{R}$ = [2, 2, 3, 3], (v[2] + 1) (v[3] + 1)

"RNATB NAT RANK", 1
"BNATR NAT RANK", 2

"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1

"IDEMSOLVER", [4, 4, 4, 4]
"IDEMSOLVER", [4, 4, 4, 4]

"IDEMSOLVABLE?", true

"ABELIAN? " false

"CMM = " $\left(\begin{array}{cccc}0& 0& -1& 1\\ 1& 0& -1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

"ISIDEM?", false

RB-RANKED. KERNEL IS RANK ONE.

RBSOLVRANK, 1, with index , 2

RBSOLVABLE

============================================================================

3, [1, 1, -1, 1]

"Coloring", {3}

"R", [2, 3, 1, 3]
"B", [4, 4, 2, 1]

SYNC'D

"R CYCLES", 1 + v[1] v[2] v[3]
"B CYCLES", 1 + v[1] v[4]

"Char Poly V" $\frac{-1}{10}$ ( $-10-3t+{t}^{3}$ )

CHECKS

"OM-R" $\left(\begin{array}{cccc}\frac{1}{3}& \frac{1}{3}& \frac{1}{3}& 0\\ \frac{1}{3}& \frac{1}{3}& \frac{1}{3}& 0\\ \frac{1}{3}& \frac{1}{3}& \frac{1}{3}& 0\\ \frac{1}{3}& \frac{1}{3}& \frac{1}{3}& 0\end{array}\right)$ "CESARO CHECK", true

"OM-B" $\left(\begin{array}{cccc}\frac{1}{2}& 0& 0& \frac{1}{2}\\ \frac{1}{2}& 0& 0& \frac{1}{2}\\ \frac{1}{2}& 0& 0& \frac{1}{2}\\ \frac{1}{2}& 0& 0& \frac{1}{2}\end{array}\right)$ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"RANKING B from RC" 3, "vs", 3

"RANKING B-NAT from RC" 1, "vs", 2

"RANKING B from FIRST ROW of RC" $\left(\begin{array}{cccc}0& \frac{1}{3}& 0& \frac{2}{3}\\ \frac{2}{3}& 0& 0& \frac{1}{3}\\ \frac{1}{3}& 0& 0& \frac{2}{3}\end{array}\right)$ 3, "vs", 3

"CHECKING OMEGA-B", true

"RANKING R from BC" 3, "vs", 3

"RANKING R-NAT from BC" 1, "vs", 3

"RANKING R from FIRST ROW of BC" $\left(\begin{array}{cccc}0& \frac{1}{2}& \frac{1}{2}& 0\\ \frac{1}{2}& 0& \frac{1}{2}& 0\\ \frac{1}{2}& \frac{1}{2}& 0& 0\end{array}\right)$ 3, "vs", 3

SYNC'D

"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
${\text{R}}^{♮}$ = [1, 2, 3, 2]
"RBAR ranking", 3, "vs", 3

"RANK of B is ", 3
"B ranking is ", 2, "vs", 3
${\text{B}}^{♮}$ = [1, 1, 4, 4]
"BBAR ranking", 1, "vs", 2

${\text{R}}^{♮}{\text{B}}^{♮}$ = [1, 1, 4, 1]
${\text{B}}^{♮}{\text{R}}^{♮}$ = [1, 1, 2, 2]

"RNATBNAT ranking", 2, "vs", 2

"BNATRNAT ranking", 2, "vs", 2

"Centralizer "

"Char Poly Commutator" ${x}^{4}$

"Min Poly Commutator" ${x}^{3}$

${\text{R}}^{♮}{\text{B}}^{♮}$ CYCLES , v[1] + 1
${\text{B}}^{♮}{\text{R}}^{♮}$ CYCLES , v[1] + 1

${\text{R}}^{♮}\text{B}$ = [4, 4, 2, 4], v[4] + 1
${\text{B}}^{♮}\text{R}$ = [2, 2, 3, 3], (v[2] + 1) (v[3] + 1)

"RNATB NAT RANK", 1
"BNATR NAT RANK", 2

"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1

"IDEMSOLVER", [1, 1, 1, 1]
"IDEMSOLVER", [1, 1, 1, 1]

"IDEMSOLVABLE?", true

"ABELIAN? " false

"CMM = " $\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& -1& 0& 1\\ 1& -1& 0& 0\end{array}\right)$

"ISIDEM?", false

RB-RANKED. KERNEL IS RANK ONE.

RBSOLVRANK, 1, with index , 2

RBSOLVABLE

============================================================================

4, [1, 1, 1, -1]

"Coloring", {4}

"R", [2, 3, 2, 1]
"B", [4, 4, 1, 3]

SYNC'D

"R CYCLES", 1 + v[2] v[3]
"B CYCLES", 1 + v[1] v[4] v[3]

"Char Poly V" $\frac{1}{10}$ ( $10-3t+{t}^{3}$ )

CHECKS

"OM-R" $\left(\begin{array}{cccc}0& \frac{1}{2}& \frac{1}{2}& 0\\ 0& \frac{1}{2}& \frac{1}{2}& 0\\ 0& \frac{1}{2}& \frac{1}{2}& 0\\ 0& \frac{1}{2}& \frac{1}{2}& 0\end{array}\right)$ "CESARO CHECK", true

"OM-B" $\left(\begin{array}{cccc}\frac{1}{3}& 0& \frac{1}{3}& \frac{1}{3}\\ \frac{1}{3}& 0& \frac{1}{3}& \frac{1}{3}\\ \frac{1}{3}& 0& \frac{1}{3}& \frac{1}{3}\\ \frac{1}{3}& 0& \frac{1}{3}& \frac{1}{3}\end{array}\right)$ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"RANKING B from RC" 3, "vs", 3

"RANKING B-NAT from RC" 1, "vs", 3

"RANKING B from FIRST ROW of RC" $\left(\begin{array}{cccc}\frac{1}{2}& 0& 0& \frac{1}{2}\\ 0& 0& \frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& 0& \frac{1}{2}& 0\end{array}\right)$ 3, "vs", 3

"CHECKING OMEGA-B", true

"RANKING R from BC" 3, "vs", 3

"RANKING R-NAT from BC" 1, "vs", 2

"RANKING R from FIRST ROW of BC" $\left(\begin{array}{cccc}\frac{1}{3}& \frac{2}{3}& 0& 0\\ 0& \frac{1}{3}& \frac{2}{3}& 0\\ 0& \frac{2}{3}& \frac{1}{3}& 0\end{array}\right)$ 3, "vs", 3

SYNC'D

"RANK of R is ", 3
"R ranking is ", 2, "vs", 3
${\text{R}}^{♮}$ = [3, 2, 3, 2]
"RBAR ranking", 1, "vs", 2

"RANK of B is ", 3
"B ranking is ", 3, "vs", 3
${\text{B}}^{♮}$ = [1, 1, 3, 4]
"BBAR ranking", 3, "vs", 3

${\text{R}}^{♮}{\text{B}}^{♮}$ = [3, 1, 3, 1]
${\text{B}}^{♮}{\text{R}}^{♮}$ = [3, 3, 3, 2]

"RNATBNAT ranking", 2, "vs", 2

"BNATRNAT ranking", 2, "vs", 2

"Centralizer "

"Char Poly Commutator" ${x}^{4}$

"Min Poly Commutator" ${x}^{3}$

${\text{R}}^{♮}{\text{B}}^{♮}$ CYCLES , v[3] + 1
${\text{B}}^{♮}{\text{R}}^{♮}$ CYCLES , v[3] + 1

${\text{R}}^{♮}\text{B}$ = [1, 4, 1, 4], (v[1] + 1) (v[4] + 1)
${\text{B}}^{♮}\text{R}$ = [2, 2, 2, 1], v[2] + 1

"RNATB NAT RANK", 2
"BNATR NAT RANK", 1

"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1

"IDEMSOLVER", [3, 3, 3, 3]
"IDEMSOLVER", [3, 3, 3, 3]

"IDEMSOLVABLE?", true

"ABELIAN? " false

"CMM = " $\left(\begin{array}{cccc}0& 0& 0& 0\\ 1& 0& -1& 0\\ 0& 0& 0& 0\\ 1& -1& 0& 0\end{array}\right)$

"ISIDEM?", false

RB-RANKED. KERNEL IS RANK ONE.

RBSOLVRANK, 1, with index , 2

RBSOLVABLE

============================================================================

5, [1, -1, -1, 1]

"Coloring", {2, 3}

"R", [2, 4, 1, 3]
"B", [4, 3, 2, 1]

NOT SYNC'D

Kernel has RANK, 4,

N= $\left(\begin{array}{cccc}0& 1& 1& 1\\ 1& 0& 1& 1\\ 1& 1& 0& 1\\ 1& 1& 1& 0\end{array}\right)$

"R CYCLES", 1 + v[1] v[2] v[4] v[3]
"B CYCLES", (1 + v[2] v[3]) (1 + v[1] v[4])

"Char Poly V" $\frac{1}{5}$ ( $5-2t+{t}^{2}$ ) ( $1+t$ )

"CYCLE CHECK B"

CHECKS

"OM-R" $\left(\begin{array}{cccc}\frac{1}{4}& \frac{1}{4}& \frac{1}{4}& \frac{1}{4}\\ \frac{1}{4}& \frac{1}{4}& \frac{1}{4}& \frac{1}{4}\\ \frac{1}{4}& \frac{1}{4}& \frac{1}{4}& \frac{1}{4}\\ \frac{1}{4}& \frac{1}{4}& \frac{1}{4}& \frac{1}{4}\end{array}\right)$ "CESARO CHECK", true

"OM-B" $\left(\begin{array}{cccc}\frac{1}{2}& 0& 0& \frac{1}{2}\\ 0& \frac{1}{2}& \frac{1}{2}& 0\\ 0& \frac{1}{2}& \frac{1}{2}& 0\\ \frac{1}{2}& 0& 0& \frac{1}{2}\end{array}\right)$ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"Kernel Check OMEGA-R", true $\left(\begin{array}{cccc}\frac{1}{4}& \frac{1}{4}& \frac{1}{4}& \frac{1}{4}\\ \frac{1}{4}& \frac{1}{4}& \frac{1}{4}& \frac{1}{4}\\ \frac{1}{4}& \frac{1}{4}& \frac{1}{4}& \frac{1}{4}\\ \frac{1}{4}& \frac{1}{4}& \frac{1}{4}& \frac{1}{4}\end{array}\right)$

"RANKING B from RC" 1, "vs", 4

"RANKING B-NAT from RC" 1, "vs", 4

"RANKING B from FIRST ROW of RC" $\left(\begin{array}{cccc}\frac{1}{4}& \frac{1}{4}& \frac{1}{4}& \frac{1}{4}\\ \frac{1}{4}& \frac{1}{4}& \frac{1}{4}& \frac{1}{4}\\ \frac{1}{4}& \frac{1}{4}& \frac{1}{4}& \frac{1}{4}\\ \frac{1}{4}& \frac{1}{4}& \frac{1}{4}& \frac{1}{4}\end{array}\right)$ 1, "vs", 4

"CHECKING OMEGA-B", true

"Kernel Check OMEGA-B", false $\left(\begin{array}{cccc}\frac{1}{2}& 0& 0& \frac{1}{2}\\ 0& \frac{1}{2}& \frac{1}{2}& 0\\ 0& \frac{1}{2}& \frac{1}{2}& 0\\ \frac{1}{2}& 0& 0& \frac{1}{2}\end{array}\right)$

"RANKING R from BC" 2, "vs", 4

"RANKING R-NAT from BC" 2, "vs", 4

"RANKING R from FIRST ROW of BC" $\left(\begin{array}{cccc}0& \frac{1}{2}& \frac{1}{2}& 0\\ \frac{1}{2}& 0& 0& \frac{1}{2}\\ 0& \frac{1}{2}& \frac{1}{2}& 0\\ \frac{1}{2}& 0& 0& \frac{1}{2}\end{array}\right)$ 2, "vs", 4

NOT SYNC'D

"RANK of R is ", 4
"R ranking is ", 1, "vs", 4
${\text{R}}^{♮}$ = [1, 2, 3, 4]
"RBAR ranking", 1, "vs", 4

"RANK of B is ", 4
"B ranking is ", 1, "vs", 4
${\text{B}}^{♮}$ = [1, 2, 3, 4]
"BBAR ranking", 1, "vs", 4

${\text{R}}^{♮}{\text{B}}^{♮}$ = [1, 2, 3, 4]
${\text{B}}^{♮}{\text{R}}^{♮}$ = [1, 2, 3, 4]

"RNATBNAT ranking", 1, "vs", 4

"BNATRNAT ranking", 1, "vs", 4

"Centralizer " [1, 2, 3, 4] [4, 3, 2, 1] [3, 1, 4, 2] [2, 4, 1, 3]

"Char Poly Commutator" ${x}^{4}$

"Min Poly Commutator" $x$

${\text{R}}^{♮}{\text{B}}^{♮}$ CYCLES , (v[1] + 1) (v[2] + 1) (v[3] + 1) (v[4] + 1)
${\text{B}}^{♮}{\text{R}}^{♮}$ CYCLES , (v[1] + 1) (v[2] + 1) (v[3] + 1) (v[4] + 1)

${\text{R}}^{♮}\text{B}$ = [4, 3, 2, 1], (1 + v[2] v[3]) (1 + v[1] v[4])
${\text{B}}^{♮}\text{R}$ = [2, 4, 1, 3], 1 + v[1] v[2] v[4] v[3]

"RNATB NAT RANK", 4
"BNATR NAT RANK", 4

"IDEMSOLVRANK", 4, "LOCAL TRACE", 4
"IDEMSOLVRANK", 4, "LOCAL TRACE", 4

"IDEMSOLVER", [1, 2, 3, 4]
"IDEMSOLVER", [1, 2, 3, 4]

"IDEMSOLVABLE?", true

"ABELIAN? " true

"CMM = " $\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

"ISIDEM?", true

RBSOLVRANK, 4, with index , 4

NOT SOLVABLE

============================================================================

6, [1, -1, 1, -1]

"Coloring", {2, 4}

"R", [2, 4, 2, 1]
"B", [4, 3, 1, 3]

SYNC'D

"R CYCLES", 1 + v[1] v[2] v[4]
"B CYCLES", 1 + v[1] v[4] v[3]

"Char Poly V" $\frac{1}{5}$ ( $5+{t}^{2}$ )

CHECKS

"OM-R" $\left(\begin{array}{cccc}\frac{1}{3}& \frac{1}{3}& 0& \frac{1}{3}\\ \frac{1}{3}& \frac{1}{3}& 0& \frac{1}{3}\\ \frac{1}{3}& \frac{1}{3}& 0& \frac{1}{3}\\ \frac{1}{3}& \frac{1}{3}& 0& \frac{1}{3}\end{array}\right)$ "CESARO CHECK", true

"OM-B" $\left(\begin{array}{cccc}\frac{1}{3}& 0& \frac{1}{3}& \frac{1}{3}\\ \frac{1}{3}& 0& \frac{1}{3}& \frac{1}{3}\\ \frac{1}{3}& 0& \frac{1}{3}& \frac{1}{3}\\ \frac{1}{3}& 0& \frac{1}{3}& \frac{1}{3}\end{array}\right)$ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"RANKING B from RC" 3, "vs", 3

"RANKING B-NAT from RC" 1, "vs", 3

"RANKING B from FIRST ROW of RC" $\left(\begin{array}{cccc}0& 0& \frac{2}{3}& \frac{1}{3}\\ \frac{2}{3}& 0& \frac{1}{3}& 0\\ \frac{1}{3}& 0& 0& \frac{2}{3}\end{array}\right)$ 3, "vs", 3

"CHECKING OMEGA-B", true

"RANKING R from BC" 3, "vs", 3

"RANKING R-NAT from BC" 1, "vs", 3

"RANKING R from FIRST ROW of BC" $\left(\begin{array}{cccc}\frac{1}{3}& \frac{2}{3}& 0& 0\\ 0& \frac{1}{3}& 0& \frac{2}{3}\\ \frac{2}{3}& 0& 0& \frac{1}{3}\end{array}\right)$ 3, "vs", 3

SYNC'D

"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
${\text{R}}^{♮}$ = [1, 2, 1, 4]
"RBAR ranking", 3, "vs", 3

"RANK of B is ", 3
"B ranking is ", 3, "vs", 3
${\text{B}}^{♮}$ = [1, 4, 3, 4]
"BBAR ranking", 3, "vs", 3

${\text{R}}^{♮}{\text{B}}^{♮}$ = [1, 4, 1, 4]
${\text{B}}^{♮}{\text{R}}^{♮}$ = [1, 4, 1, 4]

"RNATBNAT ranking", 1, "vs", 2

"BNATRNAT ranking", 1, "vs", 2

"Centralizer "

"Char Poly Commutator" ${x}^{4}$

"Min Poly Commutator" $x$

${\text{R}}^{♮}{\text{B}}^{♮}$ CYCLES , (v[1] + 1) (v[4] + 1)
${\text{B}}^{♮}{\text{R}}^{♮}$ CYCLES , (v[1] + 1) (v[4] + 1)

${\text{R}}^{♮}\text{B}$ = [4, 3, 4, 3], 1 + v[4] v[3]
${\text{B}}^{♮}\text{R}$ = [2, 1, 2, 1], 1 + v[1] v[2]

"RNATB NAT RANK", 2
"BNATR NAT RANK", 2

"IDEMSOLVRANK", 2, "LOCAL TRACE", 2
"IDEMSOLVRANK", 2, "LOCAL TRACE", 2

"IDEMSOLVER", [1, 4, 1, 4]
"IDEMSOLVER", [1, 4, 1, 4]

"IDEMSOLVABLE?", true

"ABELIAN? " true

"CMM = " $\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

"ISIDEM?", true

RB-RANKED. KERNEL IS RANK ONE.

RBSOLVRANK, 2, with index , 1

NOT SOLVABLE

============================================================================

7, [1, 1, -1, -1]

"Coloring", {3, 4}

"R", [2, 3, 1, 1]
"B", [4, 4, 2, 3]

SYNC'D

"R CYCLES", 1 + v[1] v[2] v[3]
"B CYCLES", 1 + v[2] v[4] v[3]

"Char Poly V" $\frac{1}{5}$ ( $5+{t}^{2}$ )

CHECKS

"OM-R" $\left(\begin{array}{cccc}\frac{1}{3}& \frac{1}{3}& \frac{1}{3}& 0\\ \frac{1}{3}& \frac{1}{3}& \frac{1}{3}& 0\\ \frac{1}{3}& \frac{1}{3}& \frac{1}{3}& 0\\ \frac{1}{3}& \frac{1}{3}& \frac{1}{3}& 0\end{array}\right)$ "CESARO CHECK", true

"OM-B" $\left(\begin{array}{cccc}0& \frac{1}{3}& \frac{1}{3}& \frac{1}{3}\\ 0& \frac{1}{3}& \frac{1}{3}& \frac{1}{3}\\ 0& \frac{1}{3}& \frac{1}{3}& \frac{1}{3}\\ 0& \frac{1}{3}& \frac{1}{3}& \frac{1}{3}\end{array}\right)$ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"RANKING B from RC" 3, "vs", 3

"RANKING B-NAT from RC" 1, "vs", 3

"RANKING B from FIRST ROW of RC" $\left(\begin{array}{cccc}0& \frac{1}{3}& 0& \frac{2}{3}\\ 0& 0& \frac{2}{3}& \frac{1}{3}\\ 0& \frac{2}{3}& \frac{1}{3}& 0\end{array}\right)$ 3, "vs", 3

"CHECKING OMEGA-B", true

"RANKING R from BC" 3, "vs", 3

"RANKING R-NAT from BC" 1, "vs", 3

"RANKING R from FIRST ROW of BC" $\left(\begin{array}{cccc}\frac{2}{3}& 0& \frac{1}{3}& 0\\ \frac{1}{3}& \frac{2}{3}& 0& 0\\ 0& \frac{1}{3}& \frac{2}{3}& 0\end{array}\right)$ 3, "vs", 3

SYNC'D

"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
${\text{R}}^{♮}$ = [1, 2, 3, 3]
"RBAR ranking", 3, "vs", 3

"RANK of B is ", 3
"B ranking is ", 3, "vs", 3
${\text{B}}^{♮}$ = [2, 2, 3, 4]
"BBAR ranking", 3, "vs", 3

${\text{R}}^{♮}{\text{B}}^{♮}$ = [2, 2, 3, 3]
${\text{B}}^{♮}{\text{R}}^{♮}$ = [2, 2, 3, 3]

"RNATBNAT ranking", 1, "vs", 2

"BNATRNAT ranking", 1, "vs", 2

"Centralizer "

"Char Poly Commutator" ${x}^{4}$

"Min Poly Commutator" $x$

${\text{R}}^{♮}{\text{B}}^{♮}$ CYCLES , (v[2] + 1) (v[3] + 1)
${\text{B}}^{♮}{\text{R}}^{♮}$ CYCLES , (v[2] + 1) (v[3] + 1)

${\text{R}}^{♮}\text{B}$ = [4, 4, 2, 2], 1 + v[2] v[4]
${\text{B}}^{♮}\text{R}$ = [3, 3, 1, 1], 1 + v[1] v[3]

"RNATB NAT RANK", 2
"BNATR NAT RANK", 2

"IDEMSOLVRANK", 2, "LOCAL TRACE", 2
"IDEMSOLVRANK", 2, "LOCAL TRACE", 2

"IDEMSOLVER", [2, 2, 3, 3]
"IDEMSOLVER", [2, 2, 3, 3]

"IDEMSOLVABLE?", true

"ABELIAN? " true

"CMM = " $\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

"ISIDEM?", true

RB-RANKED. KERNEL IS RANK ONE.

RBSOLVRANK, 2, with index , 1

NOT SOLVABLE

============================================================================

8, [1, -1, -1, -1]

"Coloring", {2, 3, 4}

"R", [2, 4, 1, 1]
"B", [4, 3, 2, 3]

SYNC'D

"R CYCLES", 1 + v[1] v[2] v[4]
"B CYCLES", 1 + v[2] v[3]

"Char Poly V" $\frac{-1}{10}$ ( $-10-3t+{t}^{3}$ )

CHECKS

"OM-R" $\left(\begin{array}{cccc}\frac{1}{3}& \frac{1}{3}& 0& \frac{1}{3}\\ \frac{1}{3}& \frac{1}{3}& 0& \frac{1}{3}\\ \frac{1}{3}& \frac{1}{3}& 0& \frac{1}{3}\\ \frac{1}{3}& \frac{1}{3}& 0& \frac{1}{3}\end{array}\right)$ "CESARO CHECK", true

"OM-B" $\left(\begin{array}{cccc}0& \frac{1}{2}& \frac{1}{2}& 0\\ 0& \frac{1}{2}& \frac{1}{2}& 0\\ 0& \frac{1}{2}& \frac{1}{2}& 0\\ 0& \frac{1}{2}& \frac{1}{2}& 0\end{array}\right)$ "CESARO CHECK", true

"CHECKING OMEGA-R", true

"RANKING B from RC" 3, "vs", 3

"RANKING B-NAT from RC" 1, "vs", 2

"RANKING B from FIRST ROW of RC" $\left(\begin{array}{cccc}0& 0& \frac{2}{3}& \frac{1}{3}\\ 0& \frac{2}{3}& \frac{1}{3}& 0\\ 0& \frac{1}{3}& \frac{2}{3}& 0\end{array}\right)$ 3, "vs", 3

"CHECKING OMEGA-B", true

"RANKING R from BC" 3, "vs", 3

"RANKING R-NAT from BC" 1, "vs", 3

"RANKING R from FIRST ROW of BC" $\left(\begin{array}{cccc}\frac{1}{2}& 0& 0& \frac{1}{2}\\ \frac{1}{2}& \frac{1}{2}& 0& 0\\ 0& \frac{1}{2}& 0& \frac{1}{2}\end{array}\right)$ 3, "vs", 3

SYNC'D

"RANK of R is ", 3
"R ranking is ", 3, "vs", 3
${\text{R}}^{♮}$ = [1, 2, 4, 4]
"RBAR ranking", 3, "vs", 3

"RANK of B is ", 3
"B ranking is ", 2, "vs", 3
${\text{B}}^{♮}$ = [3, 2, 3, 2]
"BBAR ranking", 1, "vs", 2

${\text{R}}^{♮}{\text{B}}^{♮}$ = [3, 2, 2, 2]
${\text{B}}^{♮}{\text{R}}^{♮}$ = [4, 2, 4, 2]

"RNATBNAT ranking", 2, "vs", 2

"BNATRNAT ranking", 2, "vs", 2

"Centralizer "

"Char Poly Commutator" ${x}^{4}$

"Min Poly Commutator" ${x}^{3}$

${\text{R}}^{♮}{\text{B}}^{♮}$ CYCLES , v[2] + 1
${\text{B}}^{♮}{\text{R}}^{♮}$ CYCLES , v[2] + 1

${\text{R}}^{♮}\text{B}$ = [4, 3, 3, 3], v[3] + 1
${\text{B}}^{♮}\text{R}$ = [1, 4, 1, 4], (v[1] + 1) (v[4] + 1)

"RNATB NAT RANK", 1
"BNATR NAT RANK", 2

"IDEMSOLVRANK", 1, "LOCAL TRACE", 1
"IDEMSOLVRANK", 1, "LOCAL TRACE", 1

"IDEMSOLVER", [2, 2, 2, 2]
"IDEMSOLVER", [2, 2, 2, 2]

"IDEMSOLVABLE?", true

"ABELIAN? " false

"CMM = " $\left(\begin{array}{cccc}0& 0& 1& -1\\ 0& 0& 0& 0\\ 0& 1& 0& -1\\ 0& 0& 0& 0\end{array}\right)$

"ISIDEM?", false

RB-RANKED. KERNEL IS RANK ONE.

RBSOLVRANK, 1, with index , 2

RBSOLVABLE

============================================================================

"SANDWICH SUMMARY", 0

"RG SUMMARY", 0

"SOLVSUMMARY", 4
2 . {2}, rank: 1/1
3 . {3}, rank: 1/1
4 . {4}, rank: 1/1
8 . {2, 3, 4}, rank: 1/1

"IDEMSOLVSUMMARY", 7
2 . {2}, rank: 1/1
3 . {3}, rank: 1/1
4 . {4}, rank: 1/1
5 . {2, 3}, rank: 4/1
6 . {2, 4}, rank: 2/1
7 . {3, 4}, rank: 2/1
8 . {2, 3, 4}, rank: 1/1

"SUMMARY: NON-SYNCED CC", 1
{[1, {}, sw, 2]}