2^42 Sedenion Variations (Seds)




This page is a discussion of the standard, 64-bit C program ReverseGamma.c, used to enumerate the overlay pattern of signs on a 16 dimensional sedenion solution.

Identifying free variables was initially done interatively, choosing a new free variable and determining new relationships until either a conflict was found, or all terms were determined.

In this fashion, I determined that 42 free variables were needed to span this space. Several good choices exist for choosing independent cells versus dependent locations. The real criterion is to not overspecify four cells as free, when only three of the four are truly free, with the fourth cell specified by the other three.

The Gamma pattern, shown below, is used by the GiNaC solver SedsViaMult.cp, as the default signs determined correspond to a traditional complexified algebra, with commutative scalar multiplication. However, the sub-algebras, corresponding to octonions, quaternions and complex numbers are in the upper right corner of the subspaces, and the algebraic formulas for the calculated terms do not easily show the sub-space reality.

              Gamma Pattern

        a b c d e f g h i j k l m n o p
        q . . . . . . . r . . . s . t .
        u . . . . . . . v . . . w . . .
        x . . . . . . . A . . . B . . .
        C . . . . . . . D . . . . . . .
        E . . . . . . . F . . . . . . .
        G . . . . . . . H . . . . . . .
        I . . . . . . . J . . . . . . .
        K . . . . . . . . . . . . . . .
        L . . . . . . . . . . . . . . .
        M . . . . . . . . . . . . . . .
        N . . . . . . . . . . . . . . .
        O . . . . . . . . . . . . . . .
        P . . . . . . . . . . . . . . .
        Q . . . . . . . . . . . . . . .
        R . . . . . . . . . . . . . . .


By contrast, the reverse gamma pattern, shown below, clearly demonstrates the subspace nature of comps in quads, quads in octs, octs in seds.
       Reverse Gamma Pattern

    a b d e i j k l t u v w x A B C
    . c . f . . . m . . . . . . . D
    . . . g . . . n . . . . . . . E
    . . . h . . . o . . . . . . . F
    . . . . . . . p . . . . . . . G
    . . . . . . . q . . . . . . . H
    . . . . . . . r . . . . . . . I
    . . . . . . . s . . . . . . . J
    . . . . . . . . . . . . . . . K
    . . . . . . . . . . . . . . . L
    . . . . . . . . . . . . . . . M
    . . . . . . . . . . . . . . . N
    . . . . . . . . . . . . . . . O
    . . . . . . . . . . . . . . . P
    . . . . . . . . . . . . . . . Q
    . . . . . . . . . . . . . . . R


The four-square relationships used in the solver mean that any sign pattern can be flipped, rotated 90, 180 or 270 degrees, and still be valid. The reverse gamma pattern above is useful, as running the program with dimensionality 2, then 4, then 8 and finally 16, present the same cell formulas as seen in the subspaces.

Each cell in the sign matrix corresponds to a 64-bit integer (long long) in the state[row][col] matrix. Bit zero corresponds to a default polarity, (0 means positive, 1 means negative). Bit one is the first free variable, bit two the second free variable, and so on. Given three cells, at coordinates (row, col), (r, col) and (row, c) determine a fourth cell, at coordinate (r,c). The coordinate row, col, r and c are four way dependent - three are free, but the fourth set. The four way dependency for the coordinates is

r^row^c^col = 0
, or
c = r^row^col
, etc. The *value* at these cells, is also four way dependent. Including default sign, we have
1^state[row][col]^state[r][col]^state[r][c]^state[row][c] = 0
, or
state[r][c] = 1^state[row][col]^state[r][col]^state[row][c];

As products are formed, negative factors square to positive, and repeated basis factors null out.


We now present the results of this program for two, four, eight and sixteen dimensions.


Comps

Reverse Gamma Pattern

        a b 
        . c

  
Free Variables

state[0][0] =  2    a
state[0][1] =  4    b
state[1][1] =  8    c

Derived Expressions

state[1][0] = f    SetKnowns used  (1, 0) = 1^[(1, 1),   (0, 1),   (0, 0)] 

Default Sign (T matrix)  

+ + 
- + 

Sign Equations  

S[0][0] = +a;
S[0][1] = +b;

S[1][0] = -a*b*c;
S[1][1] = +c;



Default Component Equations  (different a-b, A-B than above)

(a,b)*(A,B) = (p.t,p,x)


p.t = +s[0][0]*a*A+s[1][1]*b*B ;
p.x = +s[0][1]*a*B+s[1][0]*b*A ;



Quads


 Reverse Gamma Pattern

    a b d e
    . c . f 
    . . . g 
    . . . h 
  
Free Variables

state[0][0] =   2    a
state[0][1] =   4    b
state[1][1] =   8    c
state[0][2] =  10    d
state[0][3] =  20    e
state[1][3] =  40    f
state[2][3] =  80    g
state[3][3] = 100    h

Derived Expressions

state[1][0] =   f    SetKnowns used  (1, 0) = 1^[(1, 1),   (0, 1),   (0, 0)] 
state[1][2] =  71    SetKnowns used  (1, 2) = 1^[(1, 3),   (0, 3),   (0, 2)] 
state[2][0] =  ce    SetKnowns used  (2, 0) = 1^[(2, 3),   (1, 3),   (1, 0)] 
state[2][1] =  a5    SetKnowns used  (2, 1) = 1^[(2, 3),   (0, 3),   (0, 1)] 
state[2][2] =  dd    SetKnowns used  (2, 2) = 1^[(2, 0),   (0, 0),   (0, 2)] 
state[3][0] = 123    SetKnowns used  (3, 0) = 1^[(3, 3),   (0, 3),   (0, 0)] 
state[3][1] = 149    SetKnowns used  (3, 1) = 1^[(3, 3),   (1, 3),   (1, 1)] 
state[3][2] = 15c    SetKnowns used  (3, 2) = 1^[(3, 1),   (0, 1),   (0, 2)] 


Default Sign (T matrix)  

+ + + + 
- + - + 
+ - - + 
- - + + 


Sign Equations  

S[0][0] = +a;
S[0][1] = +b;
S[0][2] = +d;
S[0][3] = +e;

S[1][0] = -a*b*c;
S[1][1] = +c;
S[1][2] = -d*e*f;
S[1][3] = +f;

S[2][0] = +a*b*c*f*g;
S[2][1] = -b*e*g;
S[2][2] = -b*d*c*f*g;
S[2][3] = +g;

S[3][0] = -a*e*h;
S[3][1] = -c*f*h;
S[3][2] = +b*d*c*f*h;
S[3][3] = +h;



Default Component Equations  (different a-d, A-D than above)

(a,b,c,d)*(A,B,C,D) = (p.t,p,x,p.y,p,z)

p.t = +s[0][0]*a*A+s[1][1]*b*B+s[2][2]*c*C+s[3][3]*d*D ;
p.x = +s[0][1]*a*B+s[1][0]*b*A+s[2][3]*c*D+s[3][2]*d*C ;
p.y = +s[0][2]*a*C+s[1][3]*b*D+s[2][0]*c*A+s[3][1]*d*B ;
p.z = +s[0][3]*a*D+s[1][2]*b*C+s[2][1]*c*B+s[3][0]*d*A ;



Octs


 Reverse Gamma Pattern

    a b d e i j k l
    . c . f . . . m
    . . . g . . . n
    . . . h . . . o
    . . . . . . . p
    . . . . . . . q
    . . . . . . . r
    . . . . . . . s
 
Free Variables
  
state[0][0] =     2    a
state[0][1] =     4    b
state[1][1] =     8    c
state[0][2] =    10    d
state[0][3] =    20    e
state[1][3] =    40    f
state[2][3] =    80    g
state[3][3] =   100    h
state[0][4] =   200    i
state[0][5] =   400    j
state[0][6] =   800    k
state[0][7] =  1000    l
state[1][7] =  2000    m
state[2][7] =  4000    n
state[3][7] =  8000    o
state[4][7] = 10000    p
state[5][7] = 20000    q
state[6][7] = 40000    r
state[7][7] = 80000    s

Derived Expressions

state[1][0] =                f    SetKnowns used  (1, 0) = 1^[(1, 1),   (0, 1),   (0, 0)] 
state[1][2] =               71    SetKnowns used  (1, 2) = 1^[(1, 3),   (0, 3),   (0, 2)] 
state[1][6] =             3801    SetKnowns used  (1, 6) = 1^[(1, 7),   (0, 7),   (0, 6)] 
state[2][0] =               ce    SetKnowns used  (2, 0) = 1^[(2, 3),   (1, 3),   (1, 0)] 
state[2][1] =               a5    SetKnowns used  (2, 1) = 1^[(2, 3),   (0, 3),   (0, 1)] 
state[2][2] =               dd    SetKnowns used  (2, 2) = 1^[(2, 0),   (0, 0),   (0, 2)] 
state[2][5] =             5401    SetKnowns used  (2, 5) = 1^[(2, 7),   (0, 7),   (0, 5)] 
state[3][0] =              123    SetKnowns used  (3, 0) = 1^[(3, 3),   (0, 3),   (0, 0)] 
state[3][1] =              149    SetKnowns used  (3, 1) = 1^[(3, 3),   (1, 3),   (1, 1)] 
state[3][2] =              15c    SetKnowns used  (3, 2) = 1^[(3, 1),   (0, 1),   (0, 2)] 
state[3][4] =             9201    SetKnowns used  (3, 4) = 1^[(3, 7),   (0, 7),   (0, 4)] 
state[4][0] =            18122    SetKnowns used  (4, 0) = 1^[(4, 7),   (3, 7),   (3, 0)] 
state[4][1] =            140a4    SetKnowns used  (4, 1) = 1^[(4, 7),   (2, 7),   (2, 1)] 
state[4][2] =            12070    SetKnowns used  (4, 2) = 1^[(4, 7),   (1, 7),   (1, 2)] 
state[4][3] =            11021    SetKnowns used  (4, 3) = 1^[(4, 7),   (0, 7),   (0, 3)] 
state[4][4] =            18321    SetKnowns used  (4, 4) = 1^[(4, 0),   (0, 0),   (0, 4)] 
state[4][5] =            144a1    SetKnowns used  (4, 5) = 1^[(4, 1),   (0, 1),   (0, 5)] 
state[4][6] =            12861    SetKnowns used  (4, 6) = 1^[(4, 2),   (0, 2),   (0, 6)] 
state[5][0] =            240cf    SetKnowns used  (5, 0) = 1^[(5, 7),   (2, 7),   (2, 0)] 
state[5][1] =            28148    SetKnowns used  (5, 1) = 1^[(5, 7),   (3, 7),   (3, 1)] 
state[5][2] =            21011    SetKnowns used  (5, 2) = 1^[(5, 7),   (0, 7),   (0, 2)] 
state[5][3] =            22041    SetKnowns used  (5, 3) = 1^[(5, 7),   (1, 7),   (1, 3)] 
state[5][4] =            2834d    SetKnowns used  (5, 4) = 1^[(5, 1),   (0, 1),   (0, 4)] 
state[5][5] =            244cc    SetKnowns used  (5, 5) = 1^[(5, 0),   (0, 0),   (0, 5)] 
state[5][6] =            22860    SetKnowns used  (5, 6) = 1^[(5, 3),   (0, 3),   (0, 6)] 
state[6][0] =            4200e    SetKnowns used  (6, 0) = 1^[(6, 7),   (1, 7),   (1, 0)] 
state[6][1] =            41005    SetKnowns used  (6, 1) = 1^[(6, 7),   (0, 7),   (0, 1)] 
state[6][2] =            4815d    SetKnowns used  (6, 2) = 1^[(6, 7),   (3, 7),   (3, 2)] 
state[6][3] =            44081    SetKnowns used  (6, 3) = 1^[(6, 7),   (2, 7),   (2, 3)] 
state[6][4] =            4834c    SetKnowns used  (6, 4) = 1^[(6, 2),   (0, 2),   (0, 4)] 
state[6][5] =            444a0    SetKnowns used  (6, 5) = 1^[(6, 3),   (0, 3),   (0, 5)] 
state[6][6] =            4280d    SetKnowns used  (6, 6) = 1^[(6, 0),   (0, 0),   (0, 6)] 
state[7][0] =            81003    SetKnowns used  (7, 0) = 1^[(7, 7),   (0, 7),   (0, 0)] 
state[7][1] =            82009    SetKnowns used  (7, 1) = 1^[(7, 7),   (1, 7),   (1, 1)] 
state[7][2] =            840dc    SetKnowns used  (7, 2) = 1^[(7, 7),   (2, 7),   (2, 2)] 
state[7][3] =            88101    SetKnowns used  (7, 3) = 1^[(7, 7),   (3, 7),   (3, 3)] 
state[7][4] =            88320    SetKnowns used  (7, 4) = 1^[(7, 3),   (0, 3),   (0, 4)] 
state[7][5] =            844cd    SetKnowns used  (7, 5) = 1^[(7, 2),   (0, 2),   (0, 5)] 
state[7][6] =            8280c    SetKnowns used  (7, 6) = 1^[(7, 1),   (0, 1),   (0, 6)] 
state[1][4] =             c38c    SetKnowns used  (1, 4) = 1^[(1, 1),   (4, 1),   (4, 4)] 
state[1][5] =             c58d    SetKnowns used  (1, 5) = 1^[(1, 4),   (0, 4),   (0, 5)] 
state[2][4] =             a38d    SetKnowns used  (2, 4) = 1^[(2, 7),   (1, 7),   (1, 4)] 
state[2][6] =             a98c    SetKnowns used  (2, 6) = 1^[(2, 4),   (0, 4),   (0, 6)] 
state[3][5] =             658c    SetKnowns used  (3, 5) = 1^[(3, 7),   (1, 7),   (1, 5)] 
state[3][6] =             698d    SetKnowns used  (3, 6) = 1^[(3, 5),   (0, 5),   (0, 6)] 


Default Sign (T matrix)  

+ + + + + + + + 
- + - + + - - + 
+ - - + - - + + 
- - + + - + - + 
+ + + - - - - + 
- + - - - + + + 
+ - - - + + - + 
- - + - + - + + 



Sign Equations  

S[0][0] = +a;
S[0][1] = +b;
S[0][2] = +d;
S[0][3] = +e;
S[0][4] = +i;
S[0][5] = +j;
S[0][6] = +k;
S[0][7] = +l;

S[1][0] = -a*b*c;
S[1][1] = +c;
S[1][2] = -d*e*f;
S[1][3] = +f;
S[1][4] = +b*i*c*g*n*h*o;
S[1][5] = -b*j*c*g*n*h*o;
S[1][6] = -k*l*m;
S[1][7] = +m;

S[2][0] = +a*b*c*f*g;
S[2][1] = -b*e*g;
S[2][2] = -b*d*c*f*g;
S[2][3] = +g;
S[2][4] = -b*i*c*m*g*h*o;
S[2][5] = -j*l*n;
S[2][6] = +b*k*c*m*g*h*o;
S[2][7] = +n;

S[3][0] = -a*e*h;
S[3][1] = -c*f*h;
S[3][2] = +b*d*c*f*h;
S[3][3] = +h;
S[3][4] = -i*l*o;
S[3][5] = +b*j*c*m*g*n*h;
S[3][6] = -b*k*c*m*g*n*h;
S[3][7] = +o;

S[4][0] = +a*e*h*o*p;
S[4][1] = +b*e*g*n*p;
S[4][2] = +d*e*f*m*p;
S[4][3] = -e*l*p;
S[4][4] = -e*i*h*o*p;
S[4][5] = -e*j*g*n*p;
S[4][6] = -e*k*f*m*p;
S[4][7] = +p;

S[5][0] = -a*b*c*f*g*n*q;
S[5][1] = +c*f*h*o*q;
S[5][2] = -d*l*q;
S[5][3] = -f*m*q;
S[5][4] = -b*i*c*f*h*o*q;
S[5][5] = +b*j*c*f*g*n*q;
S[5][6] = +e*k*f*m*q;
S[5][7] = +q;

S[6][0] = +a*b*c*m*r;
S[6][1] = -b*l*r;
S[6][2] = -b*d*c*f*h*o*r;
S[6][3] = -g*n*r;
S[6][4] = +b*i*c*f*h*o*r;
S[6][5] = +e*j*g*n*r;
S[6][6] = -b*k*c*m*r;
S[6][7] = +r;

S[7][0] = -a*l*s;
S[7][1] = -c*m*s;
S[7][2] = +b*d*c*f*g*n*s;
S[7][3] = -h*o*s;
S[7][4] = +e*i*h*o*s;
S[7][5] = -b*j*c*f*g*n*s;
S[7][6] = +b*k*c*m*s;
S[7][7] = +s;


Default Component Equations  (different a-h, A-H than above)

(a,b,c,d,e,f,g,h)*(A,B,C,D,E,F,G,H) = (p.t,p,x,p.y,p,z,p.E,p,X,p.Y,p,Z)


p.t = +s[0][0]*a*A+s[1][1]*b*B+s[2][2]*c*C+s[3][3]*d*D+s[4][4]*e*E+s[5][5]*f*F+s[6][6]*g*G+s[7][7]*h*H ;
p.x = +s[0][1]*a*B+s[1][0]*b*A+s[2][3]*c*D+s[3][2]*d*C+s[4][5]*e*F+s[5][4]*f*E+s[6][7]*g*H+s[7][6]*h*G ;
p.y = +s[0][2]*a*C+s[1][3]*b*D+s[2][0]*c*A+s[3][1]*d*B+s[4][6]*e*G+s[5][7]*f*H+s[6][4]*g*E+s[7][5]*h*F ;
p.z = +s[0][3]*a*D+s[1][2]*b*C+s[2][1]*c*B+s[3][0]*d*A+s[4][7]*e*H+s[5][6]*f*G+s[6][5]*g*F+s[7][4]*h*E ;
p.E = +s[0][4]*a*E+s[1][5]*b*F+s[2][6]*c*G+s[3][7]*d*H+s[4][0]*e*A+s[5][1]*f*B+s[6][2]*g*C+s[7][3]*h*D ;
p.X = +s[0][5]*a*F+s[1][4]*b*E+s[2][7]*c*H+s[3][6]*d*G+s[4][1]*e*B+s[5][0]*f*A+s[6][3]*g*D+s[7][2]*h*C ;
p.Y = +s[0][6]*a*G+s[1][7]*b*H+s[2][4]*c*E+s[3][5]*d*F+s[4][2]*e*C+s[5][3]*f*D+s[6][0]*g*A+s[7][1]*h*B ;
p.Z = +s[0][7]*a*H+s[1][6]*b*G+s[2][5]*c*F+s[3][4]*d*E+s[4][3]*e*D+s[5][2]*f*C+s[6][1]*g*B+s[7][0]*h*A ;


Seds

Now we get messy. The Seds will have conflicting sign definitions. The verification formula agree with the default 11 out of 15 times, with a conflicting minority report 4 out of 15 times. However, only the sign is in dispute. The polarity contribution of the 42 independent variables is totally consistent. To accommodate this, in the sign equations, I absorb the conflicted sign into a T[row][col] term, to be resolved later, and have the products of the independent variables explicitly shown.

        Reverse Gamma Pattern

    a b d e i j k l t u v w x A B C
    . c . f . . . m . . . . . . . D
    . . . g . . . n . . . . . . . E
    . . . h . . . o . . . . . . . F
    . . . . . . . p . . . . . . . G
    . . . . . . . q . . . . . . . H
    . . . . . . . r . . . . . . . I
    . . . . . . . s . . . . . . . J
    . . . . . . . . . . . . . . . K
    . . . . . . . . . . . . . . . L
    . . . . . . . . . . . . . . . M
    . . . . . . . . . . . . . . . N
    . . . . . . . . . . . . . . . O
    . . . . . . . . . . . . . . . P
    . . . . . . . . . . . . . . . Q
    . . . . . . . . . . . . . . . R

 
Free Variables
  
state[0][0] =           2    a
state[0][1] =           4    b
state[1][1] =           8    c

state[0][2] =          10    d
state[0][3] =          20    e
state[1][3] =          40    f
state[2][3] =          80    g
state[3][3] =         100    h

state[0][4] =         200    i
state[0][5] =         400    j
state[0][6] =         800    k
state[0][7] =        1000    l
state[1][7] =        2000    m
state[2][7] =        4000    n
state[3][7] =        8000    o
state[4][7] =       10000    p
state[5][7] =       20000    q
state[6][7] =       40000    r
state[7][7] =       80000    s

state[0][8] =      100000    t
state[0][9] =      200000    u
state[0][a] =      400000    v
state[0][b] =      800000    w
state[0][c] =     1000000    x
state[0][d] =     2000000    A
state[0][e] =     4000000    B
state[0][f] =     8000000    C
state[1][f] =    10000000    D
state[2][f] =    20000000    E
state[3][f] =    40000000    F
state[4][f] =    80000000    G
state[5][f] =   100000000    H
state[6][f] =   200000000    I
state[7][f] =   400000000    J
state[8][f] =   800000000    K
state[9][f] =  1000000000    L
state[a][f] =  2000000000    M
state[b][f] =  4000000000    N
state[c][f] =  8000000000    O
state[d][f] = 10000000000    P
state[e][f] = 20000000000    Q
state[f][f] = 40000000000    R

Derived Expressions

state[1][0] =                f    SetKnowns used  (1, 0) = 1^[(1, 1),   (0, 1),   (0, 0)] 
state[1][2] =               71    SetKnowns used  (1, 2) = 1^[(1, 3),   (0, 3),   (0, 2)] 
state[1][6] =             3801    SetKnowns used  (1, 6) = 1^[(1, 7),   (0, 7),   (0, 6)] 
state[1][e] =         1c000001    SetKnowns used  (1, e) = 1^[(1, f),   (0, f),   (0, e)] 
state[2][0] =               ce    SetKnowns used  (2, 0) = 1^[(2, 3),   (1, 3),   (1, 0)] 
state[2][1] =               a5    SetKnowns used  (2, 1) = 1^[(2, 3),   (0, 3),   (0, 1)] 
state[2][2] =               dd    SetKnowns used  (2, 2) = 1^[(2, 0),   (0, 0),   (0, 2)] 
state[2][5] =             5401    SetKnowns used  (2, 5) = 1^[(2, 7),   (0, 7),   (0, 5)] 
state[2][d] =         2a000001    SetKnowns used  (2, d) = 1^[(2, f),   (0, f),   (0, d)] 
state[3][0] =              123    SetKnowns used  (3, 0) = 1^[(3, 3),   (0, 3),   (0, 0)] 
state[3][1] =              149    SetKnowns used  (3, 1) = 1^[(3, 3),   (1, 3),   (1, 1)] 
state[3][2] =              15c    SetKnowns used  (3, 2) = 1^[(3, 1),   (0, 1),   (0, 2)] 
state[3][4] =             9201    SetKnowns used  (3, 4) = 1^[(3, 7),   (0, 7),   (0, 4)] 
state[3][c] =         49000001    SetKnowns used  (3, c) = 1^[(3, f),   (0, f),   (0, c)] 
state[4][0] =            18122    SetKnowns used  (4, 0) = 1^[(4, 7),   (3, 7),   (3, 0)] 
state[4][1] =            140a4    SetKnowns used  (4, 1) = 1^[(4, 7),   (2, 7),   (2, 1)] 
state[4][2] =            12070    SetKnowns used  (4, 2) = 1^[(4, 7),   (1, 7),   (1, 2)] 
state[4][3] =            11021    SetKnowns used  (4, 3) = 1^[(4, 7),   (0, 7),   (0, 3)] 
state[4][4] =            18321    SetKnowns used  (4, 4) = 1^[(4, 0),   (0, 0),   (0, 4)] 
state[4][5] =            144a1    SetKnowns used  (4, 5) = 1^[(4, 1),   (0, 1),   (0, 5)] 
state[4][6] =            12861    SetKnowns used  (4, 6) = 1^[(4, 2),   (0, 2),   (0, 6)] 
state[4][b] =         88800001    SetKnowns used  (4, b) = 1^[(4, f),   (0, f),   (0, b)] 
state[5][0] =            240cf    SetKnowns used  (5, 0) = 1^[(5, 7),   (2, 7),   (2, 0)] 
state[5][1] =            28148    SetKnowns used  (5, 1) = 1^[(5, 7),   (3, 7),   (3, 1)] 
state[5][2] =            21011    SetKnowns used  (5, 2) = 1^[(5, 7),   (0, 7),   (0, 2)] 
state[5][3] =            22041    SetKnowns used  (5, 3) = 1^[(5, 7),   (1, 7),   (1, 3)] 
state[5][4] =            2834d    SetKnowns used  (5, 4) = 1^[(5, 1),   (0, 1),   (0, 4)] 
state[5][5] =            244cc    SetKnowns used  (5, 5) = 1^[(5, 0),   (0, 0),   (0, 5)] 
state[5][6] =            22860    SetKnowns used  (5, 6) = 1^[(5, 3),   (0, 3),   (0, 6)] 
state[5][a] =        108400001    SetKnowns used  (5, a) = 1^[(5, f),   (0, f),   (0, a)] 
state[6][0] =            4200e    SetKnowns used  (6, 0) = 1^[(6, 7),   (1, 7),   (1, 0)] 
state[6][1] =            41005    SetKnowns used  (6, 1) = 1^[(6, 7),   (0, 7),   (0, 1)] 
state[6][2] =            4815d    SetKnowns used  (6, 2) = 1^[(6, 7),   (3, 7),   (3, 2)] 
state[6][3] =            44081    SetKnowns used  (6, 3) = 1^[(6, 7),   (2, 7),   (2, 3)] 
state[6][4] =            4834c    SetKnowns used  (6, 4) = 1^[(6, 2),   (0, 2),   (0, 4)] 
state[6][5] =            444a0    SetKnowns used  (6, 5) = 1^[(6, 3),   (0, 3),   (0, 5)] 
state[6][6] =            4280d    SetKnowns used  (6, 6) = 1^[(6, 0),   (0, 0),   (0, 6)] 
state[6][9] =        208200001    SetKnowns used  (6, 9) = 1^[(6, f),   (0, f),   (0, 9)] 
state[7][0] =            81003    SetKnowns used  (7, 0) = 1^[(7, 7),   (0, 7),   (0, 0)] 
state[7][1] =            82009    SetKnowns used  (7, 1) = 1^[(7, 7),   (1, 7),   (1, 1)] 
state[7][2] =            840dc    SetKnowns used  (7, 2) = 1^[(7, 7),   (2, 7),   (2, 2)] 
state[7][3] =            88101    SetKnowns used  (7, 3) = 1^[(7, 7),   (3, 7),   (3, 3)] 
state[7][4] =            88320    SetKnowns used  (7, 4) = 1^[(7, 3),   (0, 3),   (0, 4)] 
state[7][5] =            844cd    SetKnowns used  (7, 5) = 1^[(7, 2),   (0, 2),   (0, 5)] 
state[7][6] =            8280c    SetKnowns used  (7, 6) = 1^[(7, 1),   (0, 1),   (0, 6)] 
state[7][8] =        408100001    SetKnowns used  (7, 8) = 1^[(7, f),   (0, f),   (0, 8)] 
state[8][0] =        c00081002    SetKnowns used  (8, 0) = 1^[(8, f),   (7, f),   (7, 0)] 
state[8][1] =        a00041004    SetKnowns used  (8, 1) = 1^[(8, f),   (6, f),   (6, 1)] 
state[8][2] =        900021010    SetKnowns used  (8, 2) = 1^[(8, f),   (5, f),   (5, 2)] 
state[8][3] =        880011020    SetKnowns used  (8, 3) = 1^[(8, f),   (4, f),   (4, 3)] 
state[8][4] =        840009200    SetKnowns used  (8, 4) = 1^[(8, f),   (3, f),   (3, 4)] 
state[8][5] =        820005400    SetKnowns used  (8, 5) = 1^[(8, f),   (2, f),   (2, 5)] 
state[8][6] =        810003800    SetKnowns used  (8, 6) = 1^[(8, f),   (1, f),   (1, 6)] 
state[8][7] =        808001001    SetKnowns used  (8, 7) = 1^[(8, f),   (0, f),   (0, 7)] 
state[8][8] =        c00181001    SetKnowns used  (8, 8) = 1^[(8, 0),   (0, 0),   (0, 8)] 
state[8][9] =        a00241001    SetKnowns used  (8, 9) = 1^[(8, 1),   (0, 1),   (0, 9)] 
state[8][a] =        900421001    SetKnowns used  (8, a) = 1^[(8, 2),   (0, 2),   (0, a)] 
state[8][b] =        880811001    SetKnowns used  (8, b) = 1^[(8, 3),   (0, 3),   (0, b)] 
state[8][c] =        841009001    SetKnowns used  (8, c) = 1^[(8, 4),   (0, 4),   (0, c)] 
state[8][d] =        822005001    SetKnowns used  (8, d) = 1^[(8, 5),   (0, 5),   (0, d)] 
state[8][e] =        814003001    SetKnowns used  (8, e) = 1^[(8, 6),   (0, 6),   (0, e)] 
state[9][0] =       120004200f    SetKnowns used  (9, 0) = 1^[(9, f),   (6, f),   (6, 0)] 
state[9][1] =       1400082008    SetKnowns used  (9, 1) = 1^[(9, f),   (7, f),   (7, 1)] 
state[9][2] =       1080012071    SetKnowns used  (9, 2) = 1^[(9, f),   (4, f),   (4, 2)] 
state[9][3] =       1100022040    SetKnowns used  (9, 3) = 1^[(9, f),   (5, f),   (5, 3)] 
state[9][6] =       1008000801    SetKnowns used  (9, 6) = 1^[(9, f),   (0, f),   (0, 6)] 
state[9][7] =       1010002001    SetKnowns used  (9, 7) = 1^[(9, f),   (1, f),   (1, 7)] 
state[9][8] =       140018200d    SetKnowns used  (9, 8) = 1^[(9, 1),   (0, 1),   (0, 8)] 
state[9][9] =       120024200c    SetKnowns used  (9, 9) = 1^[(9, 0),   (0, 0),   (0, 9)] 
state[9][a] =       1100422061    SetKnowns used  (9, a) = 1^[(9, 3),   (0, 3),   (0, a)] 
state[9][b] =       1080812060    SetKnowns used  (9, b) = 1^[(9, 2),   (0, 2),   (0, b)] 
state[9][e] =       1014003000    SetKnowns used  (9, e) = 1^[(9, 7),   (0, 7),   (0, e)] 
state[a][0] =       21000240ce    SetKnowns used  (a, 0) = 1^[(a, f),   (5, f),   (5, 0)] 
state[a][1] =       20800140a5    SetKnowns used  (a, 1) = 1^[(a, f),   (4, f),   (4, 1)] 
state[a][2] =       24000840dd    SetKnowns used  (a, 2) = 1^[(a, f),   (7, f),   (7, 2)] 
state[a][3] =       2200044080    SetKnowns used  (a, 3) = 1^[(a, f),   (6, f),   (6, 3)] 
state[a][5] =       2008000401    SetKnowns used  (a, 5) = 1^[(a, f),   (0, f),   (0, 5)] 
state[a][7] =       2020004001    SetKnowns used  (a, 7) = 1^[(a, f),   (2, f),   (2, 7)] 
state[a][8] =       24001840cc    SetKnowns used  (a, 8) = 1^[(a, 2),   (0, 2),   (0, 8)] 
state[a][9] =       22002440a1    SetKnowns used  (a, 9) = 1^[(a, 3),   (0, 3),   (0, 9)] 
state[a][a] =       21004240cd    SetKnowns used  (a, a) = 1^[(a, 0),   (0, 0),   (0, a)] 
state[a][b] =       20808140a0    SetKnowns used  (a, b) = 1^[(a, 1),   (0, 1),   (0, b)] 
state[a][d] =       2022005000    SetKnowns used  (a, d) = 1^[(a, 7),   (0, 7),   (0, d)] 
state[b][0] =       4080018123    SetKnowns used  (b, 0) = 1^[(b, f),   (4, f),   (4, 0)] 
state[b][1] =       4100028149    SetKnowns used  (b, 1) = 1^[(b, f),   (5, f),   (5, 1)] 
state[b][2] =       420004815c    SetKnowns used  (b, 2) = 1^[(b, f),   (6, f),   (6, 2)] 
state[b][3] =       4400088100    SetKnowns used  (b, 3) = 1^[(b, f),   (7, f),   (7, 3)] 
state[b][4] =       4008000201    SetKnowns used  (b, 4) = 1^[(b, f),   (0, f),   (0, 4)] 
state[b][7] =       4040008001    SetKnowns used  (b, 7) = 1^[(b, f),   (3, f),   (3, 7)] 
state[b][8] =       4400188121    SetKnowns used  (b, 8) = 1^[(b, 3),   (0, 3),   (0, 8)] 
state[b][9] =       420024814d    SetKnowns used  (b, 9) = 1^[(b, 2),   (0, 2),   (0, 9)] 
state[b][a] =       410042814c    SetKnowns used  (b, a) = 1^[(b, 1),   (0, 1),   (0, a)] 
state[b][b] =       4080818120    SetKnowns used  (b, b) = 1^[(b, 0),   (0, 0),   (0, b)] 
state[b][c] =       4041009000    SetKnowns used  (b, c) = 1^[(b, 7),   (0, 7),   (0, c)] 
state[c][0] =       8040000122    SetKnowns used  (c, 0) = 1^[(c, f),   (3, f),   (3, 0)] 
state[c][1] =       80200000a4    SetKnowns used  (c, 1) = 1^[(c, f),   (2, f),   (2, 1)] 
state[c][2] =       8010000070    SetKnowns used  (c, 2) = 1^[(c, f),   (1, f),   (1, 2)] 
state[c][3] =       8008000021    SetKnowns used  (c, 3) = 1^[(c, f),   (0, f),   (0, 3)] 
state[c][4] =       8400088321    SetKnowns used  (c, 4) = 1^[(c, f),   (7, f),   (7, 4)] 
state[c][5] =       82000444a1    SetKnowns used  (c, 5) = 1^[(c, f),   (6, f),   (6, 5)] 
state[c][6] =       8100022861    SetKnowns used  (c, 6) = 1^[(c, f),   (5, f),   (5, 6)] 
state[c][7] =       8080010001    SetKnowns used  (c, 7) = 1^[(c, f),   (4, f),   (4, 7)] 
state[c][8] =       8400188120    SetKnowns used  (c, 8) = 1^[(c, 4),   (0, 4),   (0, 8)] 
state[c][9] =       82002440a0    SetKnowns used  (c, 9) = 1^[(c, 5),   (0, 5),   (0, 9)] 
state[c][a] =       8100422060    SetKnowns used  (c, a) = 1^[(c, 6),   (0, 6),   (0, a)] 
state[c][b] =       8080811000    SetKnowns used  (c, b) = 1^[(c, 7),   (0, 7),   (0, b)] 
state[c][c] =       8041000121    SetKnowns used  (c, c) = 1^[(c, 0),   (0, 0),   (0, c)] 
state[c][d] =       80220000a1    SetKnowns used  (c, d) = 1^[(c, 1),   (0, 1),   (0, d)] 
state[c][e] =       8014000061    SetKnowns used  (c, e) = 1^[(c, 2),   (0, 2),   (0, e)] 
state[d][0] =      100200000cf    SetKnowns used  (d, 0) = 1^[(d, f),   (2, f),   (2, 0)] 
state[d][1] =      10040000148    SetKnowns used  (d, 1) = 1^[(d, f),   (3, f),   (3, 1)] 
state[d][2] =      10008000011    SetKnowns used  (d, 2) = 1^[(d, f),   (0, f),   (0, 2)] 
state[d][3] =      10010000041    SetKnowns used  (d, 3) = 1^[(d, f),   (1, f),   (1, 3)] 
state[d][4] =      1020004834d    SetKnowns used  (d, 4) = 1^[(d, f),   (6, f),   (6, 4)] 
state[d][5] =      104000844cc    SetKnowns used  (d, 5) = 1^[(d, f),   (7, f),   (7, 5)] 
state[d][6] =      10080012860    SetKnowns used  (d, 6) = 1^[(d, f),   (4, f),   (4, 6)] 
state[d][7] =      10100020001    SetKnowns used  (d, 7) = 1^[(d, f),   (5, f),   (5, 7)] 
state[d][8] =      104001840cd    SetKnowns used  (d, 8) = 1^[(d, 5),   (0, 5),   (0, 8)] 
state[d][9] =      1020024814c    SetKnowns used  (d, 9) = 1^[(d, 4),   (0, 4),   (0, 9)] 
state[d][a] =      10100421000    SetKnowns used  (d, a) = 1^[(d, 7),   (0, 7),   (0, a)] 
state[d][b] =      10080812061    SetKnowns used  (d, b) = 1^[(d, 6),   (0, 6),   (0, b)] 
state[d][c] =      1004100014d    SetKnowns used  (d, c) = 1^[(d, 1),   (0, 1),   (0, c)] 
state[d][d] =      100220000cc    SetKnowns used  (d, d) = 1^[(d, 0),   (0, 0),   (0, d)] 
state[d][e] =      10014000060    SetKnowns used  (d, e) = 1^[(d, 3),   (0, 3),   (0, e)] 
state[e][0] =      2001000000e    SetKnowns used  (e, 0) = 1^[(e, f),   (1, f),   (1, 0)] 
state[e][1] =      20008000005    SetKnowns used  (e, 1) = 1^[(e, f),   (0, f),   (0, 1)] 
state[e][2] =      2004000015d    SetKnowns used  (e, 2) = 1^[(e, f),   (3, f),   (3, 2)] 
state[e][3] =      20020000081    SetKnowns used  (e, 3) = 1^[(e, f),   (2, f),   (2, 3)] 
state[e][4] =      2010002834c    SetKnowns used  (e, 4) = 1^[(e, f),   (5, f),   (5, 4)] 
state[e][5] =      200800144a0    SetKnowns used  (e, 5) = 1^[(e, f),   (4, f),   (4, 5)] 
state[e][6] =      2040008280d    SetKnowns used  (e, 6) = 1^[(e, f),   (7, f),   (7, 6)] 
state[e][7] =      20200040001    SetKnowns used  (e, 7) = 1^[(e, f),   (6, f),   (6, 7)] 
state[e][8] =      2040018200c    SetKnowns used  (e, 8) = 1^[(e, 6),   (0, 6),   (0, 8)] 
state[e][9] =      20200241000    SetKnowns used  (e, 9) = 1^[(e, 7),   (0, 7),   (0, 9)] 
state[e][a] =      2010042814d    SetKnowns used  (e, a) = 1^[(e, 4),   (0, 4),   (0, a)] 
state[e][b] =      200808140a1    SetKnowns used  (e, b) = 1^[(e, 5),   (0, 5),   (0, b)] 
state[e][c] =      2004100014c    SetKnowns used  (e, c) = 1^[(e, 2),   (0, 2),   (0, c)] 
state[e][d] =      200220000a0    SetKnowns used  (e, d) = 1^[(e, 3),   (0, 3),   (0, d)] 
state[e][e] =      2001400000d    SetKnowns used  (e, e) = 1^[(e, 0),   (0, 0),   (0, e)] 
state[f][0] =      40008000003    SetKnowns used  (f, 0) = 1^[(f, f),   (0, f),   (0, 0)] 
state[f][1] =      40010000009    SetKnowns used  (f, 1) = 1^[(f, f),   (1, f),   (1, 1)] 
state[f][2] =      400200000dc    SetKnowns used  (f, 2) = 1^[(f, f),   (2, f),   (2, 2)] 
state[f][3] =      40040000101    SetKnowns used  (f, 3) = 1^[(f, f),   (3, f),   (3, 3)] 
state[f][4] =      40080018320    SetKnowns used  (f, 4) = 1^[(f, f),   (4, f),   (4, 4)] 
state[f][5] =      401000244cd    SetKnowns used  (f, 5) = 1^[(f, f),   (5, f),   (5, 5)] 
state[f][6] =      4020004280c    SetKnowns used  (f, 6) = 1^[(f, f),   (6, f),   (6, 6)] 
state[f][7] =      40400080001    SetKnowns used  (f, 7) = 1^[(f, f),   (7, f),   (7, 7)] 
state[f][8] =      40400181000    SetKnowns used  (f, 8) = 1^[(f, 7),   (0, 7),   (0, 8)] 
state[f][9] =      4020024200d    SetKnowns used  (f, 9) = 1^[(f, 6),   (0, 6),   (0, 9)] 
state[f][a] =      401004240cc    SetKnowns used  (f, a) = 1^[(f, 5),   (0, 5),   (0, a)] 
state[f][b] =      40080818121    SetKnowns used  (f, b) = 1^[(f, 4),   (0, 4),   (0, b)] 
state[f][c] =      40041000120    SetKnowns used  (f, c) = 1^[(f, 3),   (0, 3),   (0, c)] 
state[f][d] =      400220000cd    SetKnowns used  (f, d) = 1^[(f, 2),   (0, 2),   (0, d)] 
state[f][e] =      4001400000c    SetKnowns used  (f, e) = 1^[(f, 1),   (0, 1),   (0, e)] 
state[1][4] =             c38c    SetKnowns used  (1, 4) = 1^[(1, 1),   (4, 1),   (4, 4)] 
state[1][5] =             c58d    SetKnowns used  (1, 5) = 1^[(1, 4),   (0, 4),   (0, 5)] 
state[1][8] =        6001c000c    SetKnowns used  (1, 8) = 1^[(1, 1),   (8, 1),   (8, 8)] 
state[1][9] =        6002c000d    SetKnowns used  (1, 9) = 1^[(1, 8),   (0, 8),   (0, 9)] 
state[1][a] =        180430060    SetKnowns used  (1, a) = 1^[(1, 3),   (8, 3),   (8, a)] 
state[1][b] =        180830061    SetKnowns used  (1, b) = 1^[(1, a),   (0, a),   (0, b)] 
state[1][c] =         6100018d    SetKnowns used  (1, c) = 1^[(1, 5),   (8, 5),   (8, c)] 
state[1][d] =         6200018c    SetKnowns used  (1, d) = 1^[(1, c),   (0, c),   (0, d)] 
state[2][4] =             a38d    SetKnowns used  (2, 4) = 1^[(2, 7),   (1, 7),   (1, 4)] 
state[2][6] =             a98c    SetKnowns used  (2, 6) = 1^[(2, 4),   (0, 4),   (0, 6)] 
state[2][8] =        5001a00cd    SetKnowns used  (2, 8) = 1^[(2, 2),   (8, 2),   (8, 8)] 
state[2][9] =        2802500a0    SetKnowns used  (2, 9) = 1^[(2, 3),   (8, 3),   (8, 9)] 
state[2][a] =        5004a00cc    SetKnowns used  (2, a) = 1^[(2, 8),   (0, 8),   (0, a)] 
state[2][b] =        2808500a1    SetKnowns used  (2, b) = 1^[(2, 9),   (0, 9),   (0, b)] 
state[2][c] =         5100018c    SetKnowns used  (2, c) = 1^[(2, f),   (1, f),   (1, c)] 
state[2][e] =         5400018d    SetKnowns used  (2, e) = 1^[(2, c),   (0, c),   (0, e)] 
state[3][5] =             658c    SetKnowns used  (3, 5) = 1^[(3, 7),   (1, 7),   (1, 5)] 
state[3][6] =             698d    SetKnowns used  (3, 6) = 1^[(3, 5),   (0, 5),   (0, 6)] 
state[3][8] =        480190120    SetKnowns used  (3, 8) = 1^[(3, 3),   (8, 3),   (8, 8)] 
state[3][9] =        30026014c    SetKnowns used  (3, 9) = 1^[(3, 8),   (2, 8),   (2, 9)] 
state[3][a] =        30046014d    SetKnowns used  (3, a) = 1^[(3, 9),   (0, 9),   (0, a)] 
state[3][b] =        480890121    SetKnowns used  (3, b) = 1^[(3, 8),   (0, 8),   (0, b)] 
state[3][d] =         3200018d    SetKnowns used  (3, d) = 1^[(3, f),   (1, f),   (1, d)] 
state[3][e] =         3400018c    SetKnowns used  (3, e) = 1^[(3, d),   (0, d),   (0, e)] 
state[4][8] =        440190121    SetKnowns used  (4, 8) = 1^[(4, f),   (3, f),   (3, 8)] 
state[4][9] =        2202500a1    SetKnowns used  (4, 9) = 1^[(4, f),   (2, f),   (2, 9)] 
state[4][a] =        110430061    SetKnowns used  (4, a) = 1^[(4, f),   (1, f),   (1, a)] 
state[4][c] =        441090120    SetKnowns used  (4, c) = 1^[(4, 8),   (0, 8),   (0, c)] 
state[4][d] =        2220500a0    SetKnowns used  (4, d) = 1^[(4, 9),   (0, 9),   (0, d)] 
state[4][e] =        114030060    SetKnowns used  (4, e) = 1^[(4, a),   (0, a),   (0, e)] 
state[5][8] =        4201a00cc    SetKnowns used  (5, 8) = 1^[(5, f),   (2, f),   (2, 8)] 
state[5][9] =        24026014d    SetKnowns used  (5, 9) = 1^[(5, f),   (3, f),   (3, 9)] 
state[5][b] =         90830060    SetKnowns used  (5, b) = 1^[(5, f),   (1, f),   (1, b)] 
state[5][c] =        24106014c    SetKnowns used  (5, c) = 1^[(5, 9),   (0, 9),   (0, c)] 
state[5][d] =        4220a00cd    SetKnowns used  (5, d) = 1^[(5, 8),   (0, 8),   (0, d)] 
state[5][e] =         94030061    SetKnowns used  (5, e) = 1^[(5, b),   (0, b),   (0, e)] 
state[6][8] =        4101c000d    SetKnowns used  (6, 8) = 1^[(6, f),   (1, f),   (1, 8)] 
state[6][a] =        14046014c    SetKnowns used  (6, a) = 1^[(6, f),   (3, f),   (3, a)] 
state[6][b] =         a08500a0    SetKnowns used  (6, b) = 1^[(6, f),   (2, f),   (2, b)] 
state[6][c] =        14106014d    SetKnowns used  (6, c) = 1^[(6, a),   (0, a),   (0, c)] 
state[6][d] =         a20500a1    SetKnowns used  (6, d) = 1^[(6, b),   (0, b),   (0, d)] 
state[6][e] =        4140c000c    SetKnowns used  (6, e) = 1^[(6, 8),   (0, 8),   (0, e)] 
state[7][9] =        2102c000c    SetKnowns used  (7, 9) = 1^[(7, f),   (1, f),   (1, 9)] 
state[7][a] =        1204a00cd    SetKnowns used  (7, a) = 1^[(7, f),   (2, f),   (2, a)] 
state[7][b] =         c0890120    SetKnowns used  (7, b) = 1^[(7, f),   (3, f),   (3, b)] 
state[7][c] =         c1090121    SetKnowns used  (7, c) = 1^[(7, b),   (0, b),   (0, c)] 
state[7][d] =        1220a00cc    SetKnowns used  (7, d) = 1^[(7, a),   (0, a),   (0, d)] 
state[7][e] =        2140c000d    SetKnowns used  (7, e) = 1^[(7, 9),   (0, 9),   (0, e)] 
state[9][4] =       102000a38c    SetKnowns used  (9, 4) = 1^[(9, f),   (2, f),   (2, 4)] 
state[9][5] =       104000658d    SetKnowns used  (9, 5) = 1^[(9, e),   (2, e),   (2, 5)] 
state[9][c] =       104100618c    SetKnowns used  (9, c) = 1^[(9, 5),   (0, 5),   (0, c)] 
state[9][d] =       102200a18d    SetKnowns used  (9, d) = 1^[(9, 4),   (0, 4),   (0, d)] 
state[a][4] =       201000c38d    SetKnowns used  (a, 4) = 1^[(a, f),   (1, f),   (1, 4)] 
state[a][6] =       204000698c    SetKnowns used  (a, 6) = 1^[(a, d),   (1, d),   (1, 6)] 
state[a][c] =       204100618d    SetKnowns used  (a, c) = 1^[(a, 6),   (0, 6),   (0, c)] 
state[a][e] =       201400c18c    SetKnowns used  (a, e) = 1^[(a, 4),   (0, 4),   (0, e)] 
state[b][5] =       401000c58c    SetKnowns used  (b, 5) = 1^[(b, f),   (1, f),   (1, 5)] 
state[b][6] =       402000a98d    SetKnowns used  (b, 6) = 1^[(b, c),   (1, c),   (1, 6)] 
state[b][d] =       402200a18c    SetKnowns used  (b, d) = 1^[(b, 6),   (0, 6),   (0, d)] 
state[b][e] =       401400c18d    SetKnowns used  (b, e) = 1^[(b, 5),   (0, 5),   (0, e)] 


Default Sign (T matrix)  

+ + + + + + + + + + + + + + + + 
- + - + + - - + + - + - - + - + 
+ - - + - - + + - + + - + - - + 
- - + + - + - + + + - - - - + + 
+ + + - - - - + - - - - + + + + 
- + - - - + + + + - - + + - - + 
+ - - - + + - + - - + + - - + + 
- - + - + - + + - + - + - + - + 
+ + + + + + + - - - - - - - - + 
- + - + + - - - - + - + + - + + 
+ - - + - - + - + - - + - + + + 
- - + + - + - - - - + + + + - + 
+ + + - - - - - + + + + - - - + 
- + - - - + + - - + + - - + + + 
+ - - - + + - - + + - - + + - + 
- - + - + - + - + - + - + - + + 

Pair Sign Equations  - Note sign conflict hidden in T[row][col]

S[0][0] = a*T[0][0]
S[0][1] = b*T[0][1]
S[0][2] = d*T[0][2]
S[0][3] = e*T[0][3]
S[0][4] = i*T[0][4]
S[0][5] = j*T[0][5]
S[0][6] = k*T[0][6]
S[0][7] = l*T[0][7]
S[0][8] = t*T[0][8]
S[0][9] = u*T[0][9]
S[0][10] = v*T[0][10]
S[0][11] = w*T[0][11]
S[0][12] = x*T[0][12]
S[0][13] = A*T[0][13]
S[0][14] = B*T[0][14]
S[0][15] = C*T[0][15]

S[1][0] = a*b*c*T[1][0]
S[1][1] = c*T[1][1]
S[1][2] = d*e*f*T[1][2]
S[1][3] = f*T[1][3]
S[1][4] = b*i*c*g*n*h*o*T[1][4]
S[1][5] = b*j*c*g*n*h*o*T[1][5]
S[1][6] = k*l*m*T[1][6]
S[1][7] = m*T[1][7]
S[1][8] = b*t*c*r*I*s*J*T[1][8]
S[1][9] = b*u*c*r*I*s*J*T[1][9]
S[1][10] = e*v*f*p*G*q*H*T[1][10]
S[1][11] = e*w*f*p*G*q*H*T[1][11]
S[1][12] = b*x*c*g*E*h*F*T[1][12]
S[1][13] = b*A*c*g*E*h*F*T[1][13]
S[1][14] = B*C*D*T[1][14]
S[1][15] = D*T[1][15]

S[2][0] = a*b*c*f*g*T[2][0]
S[2][1] = b*e*g*T[2][1]
S[2][2] = b*d*c*f*g*T[2][2]
S[2][3] = g*T[2][3]
S[2][4] = b*i*c*m*g*h*o*T[2][4]
S[2][5] = j*l*n*T[2][5]
S[2][6] = b*k*c*m*g*h*o*T[2][6]
S[2][7] = n*T[2][7]
S[2][8] = b*t*c*f*g*q*H*s*J*T[2][8]
S[2][9] = e*u*g*p*G*r*I*T[2][9]
S[2][10] = b*v*c*f*g*q*H*s*J*T[2][10]
S[2][11] = e*w*g*p*G*r*I*T[2][11]
S[2][12] = b*x*c*D*g*h*F*T[2][12]
S[2][13] = A*C*E*T[2][13]
S[2][14] = b*B*c*D*g*h*F*T[2][14]
S[2][15] = E*T[2][15]

S[3][0] = a*e*h*T[3][0]
S[3][1] = c*f*h*T[3][1]
S[3][2] = b*d*c*f*h*T[3][2]
S[3][3] = h*T[3][3]
S[3][4] = i*l*o*T[3][4]
S[3][5] = b*j*c*m*g*n*h*T[3][5]
S[3][6] = b*k*c*m*g*n*h*T[3][6]
S[3][7] = o*T[3][7]
S[3][8] = e*t*h*p*G*s*J*T[3][8]
S[3][9] = b*u*c*f*h*q*H*r*I*T[3][9]
S[3][10] = b*v*c*f*h*q*H*r*I*T[3][10]
S[3][11] = e*w*h*p*G*s*J*T[3][11]
S[3][12] = x*C*F*T[3][12]
S[3][13] = b*A*c*D*g*E*h*T[3][13]
S[3][14] = b*B*c*D*g*E*h*T[3][14]
S[3][15] = F*T[3][15]

S[4][0] = a*e*h*o*p*T[4][0]
S[4][1] = b*e*g*n*p*T[4][1]
S[4][2] = d*e*f*m*p*T[4][2]
S[4][3] = e*l*p*T[4][3]
S[4][4] = e*i*h*o*p*T[4][4]
S[4][5] = e*j*g*n*p*T[4][5]
S[4][6] = e*k*f*m*p*T[4][6]
S[4][7] = p*T[4][7]
S[4][8] = e*t*h*F*p*s*J*T[4][8]
S[4][9] = e*u*g*E*p*r*I*T[4][9]
S[4][10] = e*v*f*D*p*q*H*T[4][10]
S[4][11] = w*C*G*T[4][11]
S[4][12] = e*x*h*F*p*s*J*T[4][12]
S[4][13] = e*A*g*E*p*r*I*T[4][13]
S[4][14] = e*B*f*D*p*q*H*T[4][14]
S[4][15] = G*T[4][15]

S[5][0] = a*b*c*f*g*n*q*T[5][0]
S[5][1] = c*f*h*o*q*T[5][1]
S[5][2] = d*l*q*T[5][2]
S[5][3] = f*m*q*T[5][3]
S[5][4] = b*i*c*f*h*o*q*T[5][4]
S[5][5] = b*j*c*f*g*n*q*T[5][5]
S[5][6] = e*k*f*m*q*T[5][6]
S[5][7] = q*T[5][7]
S[5][8] = b*t*c*f*g*E*q*s*J*T[5][8]
S[5][9] = b*u*c*f*h*F*q*r*I*T[5][9]
S[5][10] = v*C*H*T[5][10]
S[5][11] = e*w*f*D*p*G*q*T[5][11]
S[5][12] = b*x*c*f*h*F*q*r*I*T[5][12]
S[5][13] = b*A*c*f*g*E*q*s*J*T[5][13]
S[5][14] = e*B*f*D*p*G*q*T[5][14]
S[5][15] = H*T[5][15]

S[6][0] = a*b*c*m*r*T[6][0]
S[6][1] = b*l*r*T[6][1]
S[6][2] = b*d*c*f*h*o*r*T[6][2]
S[6][3] = g*n*r*T[6][3]
S[6][4] = b*i*c*f*h*o*r*T[6][4]
S[6][5] = e*j*g*n*r*T[6][5]
S[6][6] = b*k*c*m*r*T[6][6]
S[6][7] = r*T[6][7]
S[6][8] = b*t*c*D*r*s*J*T[6][8]
S[6][9] = u*C*I*T[6][9]
S[6][10] = b*v*c*f*h*F*q*H*r*T[6][10]
S[6][11] = e*w*g*E*p*G*r*T[6][11]
S[6][12] = b*x*c*f*h*F*q*H*r*T[6][12]
S[6][13] = e*A*g*E*p*G*r*T[6][13]
S[6][14] = b*B*c*D*r*s*J*T[6][14]
S[6][15] = I*T[6][15]

S[7][0] = a*l*s*T[7][0]
S[7][1] = c*m*s*T[7][1]
S[7][2] = b*d*c*f*g*n*s*T[7][2]
S[7][3] = h*o*s*T[7][3]
S[7][4] = e*i*h*o*s*T[7][4]
S[7][5] = b*j*c*f*g*n*s*T[7][5]
S[7][6] = b*k*c*m*s*T[7][6]
S[7][7] = s*T[7][7]
S[7][8] = t*C*J*T[7][8]
S[7][9] = b*u*c*D*r*I*s*T[7][9]
S[7][10] = b*v*c*f*g*E*q*H*s*T[7][10]
S[7][11] = e*w*h*F*p*G*s*T[7][11]
S[7][12] = e*x*h*F*p*G*s*T[7][12]
S[7][13] = b*A*c*f*g*E*q*H*s*T[7][13]
S[7][14] = b*B*c*D*r*I*s*T[7][14]
S[7][15] = J*T[7][15]

S[8][0] = a*l*s*J*K*T[8][0]
S[8][1] = b*l*r*I*K*T[8][1]
S[8][2] = d*l*q*H*K*T[8][2]
S[8][3] = e*l*p*G*K*T[8][3]
S[8][4] = i*l*o*F*K*T[8][4]
S[8][5] = j*l*n*E*K*T[8][5]
S[8][6] = k*l*m*D*K*T[8][6]
S[8][7] = l*C*K*T[8][7]
S[8][8] = l*t*s*J*K*T[8][8]
S[8][9] = l*u*r*I*K*T[8][9]
S[8][10] = l*v*q*H*K*T[8][10]
S[8][11] = l*w*p*G*K*T[8][11]
S[8][12] = l*x*o*F*K*T[8][12]
S[8][13] = l*A*n*E*K*T[8][13]
S[8][14] = l*B*m*D*K*T[8][14]
S[8][15] = K*T[8][15]

S[9][0] = a*b*c*m*r*I*L*T[9][0]
S[9][1] = c*m*s*J*L*T[9][1]
S[9][2] = d*e*f*m*p*G*L*T[9][2]
S[9][3] = f*m*q*H*L*T[9][3]
S[9][4] = b*i*c*m*g*E*h*o*L*T[9][4]
S[9][5] = b*j*c*m*g*n*h*F*L*T[9][5]
S[9][6] = k*C*L*T[9][6]
S[9][7] = m*D*L*T[9][7]
S[9][8] = b*t*c*m*s*J*L*T[9][8]
S[9][9] = b*u*c*m*r*I*L*T[9][9]
S[9][10] = e*v*f*m*q*H*L*T[9][10]
S[9][11] = e*w*f*m*p*G*L*T[9][11]
S[9][12] = b*x*c*m*g*n*h*F*L*T[9][12]
S[9][13] = b*A*c*m*g*E*h*o*L*T[9][13]
S[9][14] = l*B*m*D*L*T[9][14]
S[9][15] = L*T[9][15]

S[10][0] = a*b*c*f*g*n*q*H*M*T[10][0]
S[10][1] = b*e*g*n*p*G*M*T[10][1]
S[10][2] = b*d*c*f*g*n*s*J*M*T[10][2]
S[10][3] = g*n*r*I*M*T[10][3]
S[10][4] = b*i*c*D*g*n*h*o*M*T[10][4]
S[10][5] = j*C*M*T[10][5]
S[10][6] = b*k*c*m*g*n*h*F*M*T[10][6]
S[10][7] = n*E*M*T[10][7]
S[10][8] = b*t*c*f*g*n*s*J*M*T[10][8]
S[10][9] = e*u*g*n*r*I*M*T[10][9]
S[10][10] = b*v*c*f*g*n*q*H*M*T[10][10]
S[10][11] = e*w*g*n*p*G*M*T[10][11]
S[10][12] = b*x*c*m*g*n*h*F*M*T[10][12]
S[10][13] = l*A*n*E*M*T[10][13]
S[10][14] = b*B*c*D*g*n*h*o*M*T[10][14]
S[10][15] = M*T[10][15]

S[11][0] = a*e*h*o*p*G*N*T[11][0]
S[11][1] = c*f*h*o*q*H*N*T[11][1]
S[11][2] = b*d*c*f*h*o*r*I*N*T[11][2]
S[11][3] = h*o*s*J*N*T[11][3]
S[11][4] = i*C*N*T[11][4]
S[11][5] = b*j*c*D*g*n*h*o*N*T[11][5]
S[11][6] = b*k*c*m*g*E*h*o*N*T[11][6]
S[11][7] = o*F*N*T[11][7]
S[11][8] = e*t*h*o*s*J*N*T[11][8]
S[11][9] = b*u*c*f*h*o*r*I*N*T[11][9]
S[11][10] = b*v*c*f*h*o*q*H*N*T[11][10]
S[11][11] = e*w*h*o*p*G*N*T[11][11]
S[11][12] = l*x*o*F*N*T[11][12]
S[11][13] = b*A*c*m*g*E*h*o*N*T[11][13]
S[11][14] = b*B*c*D*g*n*h*o*N*T[11][14]
S[11][15] = N*T[11][15]

S[12][0] = a*e*h*F*O*T[12][0]
S[12][1] = b*e*g*E*O*T[12][1]
S[12][2] = d*e*f*D*O*T[12][2]
S[12][3] = e*C*O*T[12][3]
S[12][4] = e*i*h*o*s*J*O*T[12][4]
S[12][5] = e*j*g*n*r*I*O*T[12][5]
S[12][6] = e*k*f*m*q*H*O*T[12][6]
S[12][7] = p*G*O*T[12][7]
S[12][8] = e*t*h*o*s*J*O*T[12][8]
S[12][9] = e*u*g*n*r*I*O*T[12][9]
S[12][10] = e*v*f*m*q*H*O*T[12][10]
S[12][11] = l*w*p*G*O*T[12][11]
S[12][12] = e*x*h*F*O*T[12][12]
S[12][13] = e*A*g*E*O*T[12][13]
S[12][14] = e*B*f*D*O*T[12][14]
S[12][15] = O*T[12][15]

S[13][0] = a*b*c*f*g*E*P*T[13][0]
S[13][1] = c*f*h*F*P*T[13][1]
S[13][2] = d*C*P*T[13][2]
S[13][3] = f*D*P*T[13][3]
S[13][4] = b*i*c*f*h*o*r*I*P*T[13][4]
S[13][5] = b*j*c*f*g*n*s*J*P*T[13][5]
S[13][6] = e*k*f*m*p*G*P*T[13][6]
S[13][7] = q*H*P*T[13][7]
S[13][8] = b*t*c*f*g*n*s*J*P*T[13][8]
S[13][9] = b*u*c*f*h*o*r*I*P*T[13][9]
S[13][10] = l*v*q*H*P*T[13][10]
S[13][11] = e*w*f*m*p*G*P*T[13][11]
S[13][12] = b*x*c*f*h*F*P*T[13][12]
S[13][13] = b*A*c*f*g*E*P*T[13][13]
S[13][14] = e*B*f*D*P*T[13][14]
S[13][15] = P*T[13][15]

S[14][0] = a*b*c*D*Q*T[14][0]
S[14][1] = b*C*Q*T[14][1]
S[14][2] = b*d*c*f*h*F*Q*T[14][2]
S[14][3] = g*E*Q*T[14][3]
S[14][4] = b*i*c*f*h*o*q*H*Q*T[14][4]
S[14][5] = e*j*g*n*p*G*Q*T[14][5]
S[14][6] = b*k*c*m*s*J*Q*T[14][6]
S[14][7] = r*I*Q*T[14][7]
S[14][8] = b*t*c*m*s*J*Q*T[14][8]
S[14][9] = l*u*r*I*Q*T[14][9]
S[14][10] = b*v*c*f*h*o*q*H*Q*T[14][10]
S[14][11] = e*w*g*n*p*G*Q*T[14][11]
S[14][12] = b*x*c*f*h*F*Q*T[14][12]
S[14][13] = e*A*g*E*Q*T[14][13]
S[14][14] = b*B*c*D*Q*T[14][14]
S[14][15] = Q*T[14][15]

S[15][0] = a*C*R*T[15][0]
S[15][1] = c*D*R*T[15][1]
S[15][2] = b*d*c*f*g*E*R*T[15][2]
S[15][3] = h*F*R*T[15][3]
S[15][4] = e*i*h*o*p*G*R*T[15][4]
S[15][5] = b*j*c*f*g*n*q*H*R*T[15][5]
S[15][6] = b*k*c*m*r*I*R*T[15][6]
S[15][7] = s*J*R*T[15][7]
S[15][8] = l*t*s*J*R*T[15][8]
S[15][9] = b*u*c*m*r*I*R*T[15][9]
S[15][10] = b*v*c*f*g*n*q*H*R*T[15][10]
S[15][11] = e*w*h*o*p*G*R*T[15][11]
S[15][12] = e*x*h*F*R*T[15][12]
S[15][13] = b*A*c*f*g*E*R*T[15][13]
S[15][14] = b*B*c*D*R*T[15][14]
S[15][15] = R*T[15][15]


Default Component Equations  (a..p)*(A..P) = terms below.

+s[0][0]*a*A+s[1][1]*b*B+s[2][2]*c*C+s[3][3]*d*D+s[4][4]*e*E+s[5][5]*f*F+s[6][6]*g*G+s[7][7]*h*H
      +s[8][8]*i*I+s[9][9]*j*J+s[10][10]*k*K+s[11][11]*l*L+s[12][12]*m*M+s[13][13]*n*N+s[14][14]*o*O+s[15][15]*p*P ;
      
+s[0][1]*a*B+s[1][0]*b*A+s[2][3]*c*D+s[3][2]*d*C+s[4][5]*e*F+s[5][4]*f*E+s[6][7]*g*H+s[7][6]*h*G
      +s[8][9]*i*J+s[9][8]*j*I+s[10][11]*k*L+s[11][10]*l*K+s[12][13]*m*N+s[13][12]*n*M+s[14][15]*o*P+s[15][14]*p*O ;
      
+s[0][2]*a*C+s[1][3]*b*D+s[2][0]*c*A+s[3][1]*d*B+s[4][6]*e*G+s[5][7]*f*H+s[6][4]*g*E+s[7][5]*h*F
      +s[8][10]*i*K+s[9][11]*j*L+s[10][8]*k*I+s[11][9]*l*J+s[12][14]*m*O+s[13][15]*n*P+s[14][12]*o*M+s[15][13]*p*N ;
            
+s[0][3]*a*D+s[1][2]*b*C+s[2][1]*c*B+s[3][0]*d*A+s[4][7]*e*H+s[5][6]*f*G+s[6][5]*g*F+s[7][4]*h*E
      +s[8][11]*i*L+s[9][10]*j*K+s[10][9]*k*J+s[11][8]*l*I+s[12][15]*m*P+s[13][14]*n*O+s[14][13]*o*N+s[15][12]*p*M ;
            
+s[0][4]*a*E+s[1][5]*b*F+s[2][6]*c*G+s[3][7]*d*H+s[4][0]*e*A+s[5][1]*f*B+s[6][2]*g*C+s[7][3]*h*D
      +s[8][12]*i*M+s[9][13]*j*N+s[10][14]*k*O+s[11][15]*l*P+s[12][8]*m*I+s[13][9]*n*J+s[14][10]*o*K+s[15][11]*p*L ;
            
+s[0][5]*a*F+s[1][4]*b*E+s[2][7]*c*H+s[3][6]*d*G+s[4][1]*e*B+s[5][0]*f*A+s[6][3]*g*D+s[7][2]*h*C
      +s[8][13]*i*N+s[9][12]*j*M+s[10][15]*k*P+s[11][14]*l*O+s[12][9]*m*J+s[13][8]*n*I+s[14][11]*o*L+s[15][10]*p*K ;
      
+s[0][6]*a*G+s[1][7]*b*H+s[2][4]*c*E+s[3][5]*d*F+s[4][2]*e*C+s[5][3]*f*D+s[6][0]*g*A+s[7][1]*h*B
      +s[8][14]*i*O+s[9][15]*j*P+s[10][12]*k*M+s[11][13]*l*N+s[12][10]*m*K+s[13][11]*n*L+s[14][8]*o*I+s[15][9]*p*J ;
      
+s[0][7]*a*H+s[1][6]*b*G+s[2][5]*c*F+s[3][4]*d*E+s[4][3]*e*D+s[5][2]*f*C+s[6][1]*g*B+s[7][0]*h*A
      +s[8][15]*i*P+s[9][14]*j*O+s[10][13]*k*N+s[11][12]*l*M+s[12][11]*m*L+s[13][10]*n*K+s[14][9]*o*J+s[15][8]*p*I ;
            
+s[0][8]*a*I+s[1][9]*b*J+s[2][10]*c*K+s[3][11]*d*L+s[4][12]*e*M+s[5][13]*f*N+s[6][14]*g*O+s[7][15]*h*P
      +s[8][0]*i*A+s[9][1]*j*B+s[10][2]*k*C+s[11][3]*l*D+s[12][4]*m*E+s[13][5]*n*F+s[14][6]*o*G+s[15][7]*p*H ;
      
+s[0][9]*a*J+s[1][8]*b*I+s[2][11]*c*L+s[3][10]*d*K+s[4][13]*e*N+s[5][12]*f*M+s[6][15]*g*P+s[7][14]*h*O
      +s[8][1]*i*B+s[9][0]*j*A+s[10][3]*k*D+s[11][2]*l*C+s[12][5]*m*F+s[13][4]*n*E+s[14][7]*o*H+s[15][6]*p*G ;
      
+s[0][10]*a*K+s[1][11]*b*L+s[2][8]*c*I+s[3][9]*d*J+s[4][14]*e*O+s[5][15]*f*P+s[6][12]*g*M+s[7][13]*h*N
      +s[8][2]*i*C+s[9][3]*j*D+s[10][0]*k*A+s[11][1]*l*B+s[12][6]*m*G+s[13][7]*n*H+s[14][4]*o*E+s[15][5]*p*F ;
      
+s[0][11]*a*L+s[1][10]*b*K+s[2][9]*c*J+s[3][8]*d*I+s[4][15]*e*P+s[5][14]*f*O+s[6][13]*g*N+s[7][12]*h*M
      +s[8][3]*i*D+s[9][2]*j*C+s[10][1]*k*B+s[11][0]*l*A+s[12][7]*m*H+s[13][6]*n*G+s[14][5]*o*F+s[15][4]*p*E ;
      
+s[0][12]*a*M+s[1][13]*b*N+s[2][14]*c*O+s[3][15]*d*P+s[4][8]*e*I+s[5][9]*f*J+s[6][10]*g*K+s[7][11]*h*L
      +s[8][4]*i*E+s[9][5]*j*F+s[10][6]*k*G+s[11][7]*l*H+s[12][0]*m*A+s[13][1]*n*B+s[14][2]*o*C+s[15][3]*p*D ;
      
+s[0][13]*a*N+s[1][12]*b*M+s[2][15]*c*P+s[3][14]*d*O+s[4][9]*e*J+s[5][8]*f*I+s[6][11]*g*L+s[7][10]*h*K
      +s[8][5]*i*F+s[9][4]*j*E+s[10][7]*k*H+s[11][6]*l*G+s[12][1]*m*B+s[13][0]*n*A+s[14][3]*o*D+s[15][2]*p*C ;
      
+s[0][14]*a*O+s[1][15]*b*P+s[2][12]*c*M+s[3][13]*d*N+s[4][10]*e*K+s[5][11]*f*L+s[6][8]*g*I+s[7][9]*h*J
      +s[8][6]*i*G+s[9][7]*j*H+s[10][4]*k*E+s[11][5]*l*F+s[12][2]*m*C+s[13][3]*n*D+s[14][0]*o*A+s[15][1]*p*B ;
      
+s[0][15]*a*P+s[1][14]*b*O+s[2][13]*c*N+s[3][12]*d*M+s[4][11]*e*L+s[5][10]*f*K+s[6][9]*g*J+s[7][8]*h*I
      +s[8][7]*i*H+s[9][6]*j*G+s[10][5]*k*F+s[11][4]*l*E+s[12][3]*m*D+s[13][2]*n*C+s[14][1]*o*B+s[15][0]*p*A ;


Conclusions

The 42 free variables for seds will change the polarity of the product terms giving 2^42 variations. The task now remaining is to decide what for a T[row][col] solution will be for a base product.

Due to spam, no e-mail address is listed. However, a person can very likely guess correctly in a few attempts.