This notebook checks cases where no 4  
      or 3 appears 
anywhere .   We also check for any positive coefficients less than 653.   Cases covered are given by first row being

222 222 220 000
222 222 211 000
222 222 111 100
222 221 111 110
222 211 111 111

a  =  {a1, a2, a3, a4, a5, a6, a7, a8, a9,  a10, a11, 
        a12}

b  =  {b1, b2, b3, b4, b5, b6, b7, b8, b9,  b10, b11, 
        b12}

c  =  {c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12}

L[X_ ,  i_ ,  1]  :=  
      D[X,  a[[i]],  a[[i]],  a[[i]],  a[[i]]]/24;  
    L[X_ ,  i_ ,  2]  :=  
      D[X,  a[[i]],  a[[i]],  a[[i]],  b[[i]]]/6; 
    L[X_ ,  i_ ,  3]  :=  
      D[X,  a[[i]],  a[[i]],  a[[i]],  c[[i]]]/6; 
    L[X_ ,  i_ ,  4]  :=  
      D[X,  a[[i]],  a[[i]],  b[[i]],  b[[i]]]/4; 
    L[X_ ,  i_ ,  5]  :=  
      D[X,  a[[i]],  a[[i]],  b[[i]],  c[[i]]]/2; 
    L[X_ ,  i_ ,  6]  :=  
      D[X,  a[[i]],  a[[i]],  c[[i]],  c[[i]]]/4; 
    L[X_ ,  i_ ,  7]  :=  
      D[X,  a[[i]],  b[[i]],  b[[i]],  b[[i]]]/6; 
    L[X_ ,  i_ ,  8]  :=  
      D[X,  a[[i]],  b[[i]],  b[[i]],  c[[i]]]/2; 
    L[X_ ,  i_ ,  9]  :=  
      D[X,  a[[i]],  b[[i]],  c[[i]],  c[[i]]]/2; 
    L[X_ ,  i_ ,  10]  :=  
      D[X,  a[[i]],  c[[i]],  c[[i]],  c[[i]]]/6; 
    L[X_ ,  i_ ,  14]  :=  
      D[X,  b[[i]],  b[[i]],  b[[i]],  b[[i]]]/24; 
    L[X_ ,  i_ ,  12]  :=  
      D[X,  b[[i]],  b[[i]],  b[[i]],  c[[i]]]/6; 
    L[X_ ,  i_ ,  11]  :=  
      D[X,  b[[i]],  b[[i]],  c[[i]],  c[[i]]]/4; 
    L[X_ ,  i_ ,  13]  :=  
      D[X,  b[[i]],  c[[i]],  c[[i]],  c[[i]]]/6; 
    L[X_ ,  i_ ,  15]  :=  
      D[X,  c[[i]],  c[[i]],  c[[i]],  c[[i]]]/
        24;

p1  = 11;  p2  =  11;  p3  = 11;  L1  = L[T, 1, 11];  
    L2  =  L[L1, 2, 11];  L3  =  L[L2, 3, 11];  
    M = 0; 
For[p4  = 11,  p4 ≤  11,  p4++, 
      L4  =  L[L3,  4, p4]; 
      Print[p4]; 
For[p5  = 4,  
        p5 ≤  6,  p5++, L5  = L[L4, 5, p5]; 
        Print[M]; 
For[p6  = p5,  
          p6  ≤  6, p6++, 
          L6  = L[L5, 6, p6]; 
For[p7 = p6,  
            p7 ≤  6,  p7++, 
            L7  = L[L6, 7, p7]; 
For[p8  =  p7,  
              p8  ≤  6,  p8++, 
              L8  =  L[L7, 8, p8]; 
For[p9  = p8,  
                p9  ≤  6,  p9++, L9  = L[L8, 9, p9];  
                For[p10 = p9,  p10  ≤  6,  p10++, 
                  L10  =  L[L9,  10, p10 ];  
For[
                    p11  = p10,  p11  ≤  6,  p11++,  
                    L11  =  L[L10, 11, p11]; 
                    For[p12  =  p11,  p12 ≤  6,  p12++,  
                      L12 = L[L11, 12, p12]; 
If[L12 < 0,  
                        Print[p1, v, p2, v, p3, v, p4, v, p5, v, p6, v, p7, 
                          v, p8, v, p9,  v , p10,  v, p11, v, p12]];  
                      If[L12 > 0,  M  =  M + 1];  
                      If[ 0  <  L12  <  653,  
                        Print[L12,  v, v,  
p1, v, p2, v, 
                          p3, v, p4, v, p5, v, p6, v, p7, v, p8, v, p9,  
                          v , p10,  v, p11, v, 
                          p12]];
]]]]]]]]];  Print[M]; 
    Print[FINISHED];  M = 0;  
For[p4  = 8,  p4 ≤  9,  p4++, 
      L4  =  L[L3,  4, p4]; 
      Print[p4]; 
For[p5  = p4,  
        p5 ≤  9,  p5++, L5  = L[L4, 5, p5]; 
        Print[M]; 
For[p6  = 4,  
          p6  ≤  6, p6++, 
          L6  = L[L5, 6, p6]; 
For[p7 = p6,  
            p7 ≤  6,  p7++, 
            L7  = L[L6, 7, p7]; 
For[p8  =  p7,  
              p8  ≤  6,  p8++, 
              L8  =  L[L7, 8, p8]; 
For[p9  = p8,  
                p9  ≤  6,  p9++, L9  = L[L8, 9, p9];  
                For[p10  = p9,  p10  ≤  6,  p10++, 
                  L10  =  L[L9,  10, p10 ];  
For[
                    p11  = p10,  p11  ≤  6,  p11++,  
                    L11  =  L[L10, 11, p11]; 
                    For[p12  =  p11,  p12 ≤  6,  p12++,  
                      L12 = L[L11, 12, p12]; 
If[L12 < 0,  
                        Print[p1, v, p2, v, p3, v, p4, v, p5, v, p6, v, p7, 
                          v, p8, v, p9,  v , p10,  v, p11, v, p12]];  
                      If[L12 > 0,  M  =  M + 1];  
                      If[ 0  <  L12  <  653,  
                        Print[L12,  v, v,  
p1, v, p2, v, 
                          p3, v, p4, v, p5, v, p6, v, p7, v, p8, v, p9,  
                          v , p10,  v, p11, v, 
                          p12]];
]]]]]]]]];  Print[M]; 
    Print[FINISHED];
M = 0; 
For[p3  = 8,  p3  ≤  9,  p3++, 
      L3  =  L[L2,  3, p3];  
      Print[p3]; 
For[p4  = p3,  
        p4 ≤  9,  p4++, L4  =  L[L3,  4, p4]; 
        Print[M]; 
For[p5  = p4,  
          p5 ≤  9,  p5++, 
          L5  = L[L4, 5, p5]; 
For[p6  = p5,  
            p6  ≤  9, p6++, 
            L6  = L[L5, 6, p6]; 
For[p7 = 4,  
              p7 ≤  6,  p7++, 
              L7  = L[L6, 7, p7]; 
For[p8  =  p7,  
                p8  ≤  6,  p8++, 
                L8  =  L[L7, 8, p8]; 
For[p9  = p8,  
                  p9  ≤  6,  p9++, L9  = L[L8, 9, p9];  
                  For[p10  = p9,  p10  ≤  6,  p10++, 
                    L10  =  L[L9,  10, p10 ];  
For[
                      p11  = p10,  p11  ≤  6,  p11++,  
                      L11  =  L[L10, 11, p11]; 
                      For[p12  =  p11,  
                        p12 ≤  6,  p12++,  
                        L12 = L[L11, 12, p12]; 
If[
                          L12 < 0,  
                          Print[p1, v, p2, v, p3, v, p4, v, p5, v, p6, v, p7, 
                            v, p8, v, p9,  v , p10,  v, p11, v, p12]];  
                        If[L12 > 0,  M  =  M + 1];  
                        If[ 0  <  L12  <  653,  
                          Print[L12,  v, v,  
p1, v, p2, 
                            v, p3, v, p4, v, p5, v, p6, v, p7, v, p8, v, 
                            p9,  v , p10,  v, p11, v, 
                            p12]];
]]]]]]]]]];  Print[M]; 
    Print[FINISHED]; M = 0; 
For[p2  = 8,  p2  ≤  9,  p2++, 
      L2  =  L[L1,  2,  p2]; 
      Print[p2]; 
For[p3  = p2,  
        p3  ≤  9,  p3++, L3  =  L[L2,  3, p3];  
        Print[p3]; 
For[p4  = p3,  
          p4 ≤  9,  p4++, L4  =  L[L3,  4, p4]; 
          Print[M]; 
For[p5  = p4,  
            p5 ≤  9,  p5++, 
            L5  = L[L4, 5, p5]; 
For[p6  = p5,  
              p6  ≤  9, p6++, 
              L6  = L[L5, 6, p6]; 
For[p7 = p6,  
                p7 ≤  9,  p7++, 
                L7  = L[L6, 7, p7]; 
For[p8  =  4,  
                  p8  ≤  6,  p8++, 
                  L8  =  L[L7, 8, p8]; 
For[p9  = p8,  
                    p9  ≤  6,  p9++, L9  = L[L8, 9, p9];  
                    For[p10  = p9,  p10  ≤  6,  p10++, 
                      L10  =  L[L9,  10, p10 ];  
For[
                        p11  = p10,  p11  ≤  6,  p11++,  
                        L11  =  L[L10, 11, p11]; 
                        For[p12  =  p11,  
                          p12 ≤  6,  p12++,  
                          L12 = L[L11, 12, p12]; 
If[
                            L12 < 0,  
                            Print[p1, v, p2, v, p3, v, p4, v, p5, v, p6, v, 
                              p7, v, p8, v, p9,  v , p10,  v, p11, v, 
                              p12]];  If[L12 > 0,  M  =  M + 1];  
                          If[ 0  <  L12  <  653,  
                            Print[L12,  v, v,  
p1, v, 
                              p2, v, p3, v, p4, v, p5, v, p6, v, p7, v, p8, 
                              v, p9,  v , p10,  v, p11, v, 
                              p12]];
]]]]]]]]]]];  
    Print[M]; Print[FINISHED]; 
    M = 0;  For[p1 = 8,  
      p1  ≤  9,  p1++,  L1  =  
        L[T, 1, p1];
For[p2  = p1,  
        p2  ≤  9,  p2++, L2  =  L[L1,  2,  p2]; 
        Print[p2]; 
For[p3  = p2,  
          p3  ≤  9,  p3++, L3  =  L[L2,  3, p3];  
          Print[p3]; 
For[p4  = p3,  
            p4 ≤  9,  p4++, L4  =  L[L3,  4, p4]; 
            Print[M]; 
For[p5  = p4,  
              p5 ≤  9,  p5++, 
              L5  = L[L4, 5, p5]; 
For[p6  = p5,  
                p6  ≤  9, p6++, 
                L6  = L[L5, 6, p6]; 
For[p7 = p6,  
                  p7 ≤  9,  p7++, 
                  L7  = L[L6, 7, p7]; 
For[p8  =  p7,  
                    p8  ≤  9,  p8++, 
                    L8  =  
                      L[L7, 8, 
                        p850
For[p9  = 4,  
                            p9  ≤  6,  p9++, 
                            L9  = L[L8, 9, p9];  
                            For[p10  = p9,  
                              p10  ≤  6,  p10++, 
                              L10  =  
                                L[L9,  10, p10 ];  
For[
                                p11  = p10,  
                                p11  ≤  6,  p11++,  
                                L11  =  L[L10, 11, p11]; 
                                For[p12  =  p11,  
                                  p12 ≤  6,  p12++,  
                                  L12 = 
                                    L[L11, 12, p12]; 
If[
                                    L12 < 0,  
                                    Print[p1, v, p2, v, p3, v, p4, v, p5, v, 
                                      p6, v, p7, v, p8, v, p9,  v , p10,  
                                      v, p11, v, p12]];  
                                  If[L12 > 0,  M  =  M + 1];  
                                  If[ 0  <  L12  <  653,  
                                    Print[L12,  v, 
                                      v,  
p1, v, p2, v, 
                                      p3, v, p4, v, p5, v, p6, v, p7, v, p8, 
                                      v, p9,  v , p10,  v, p11, v, 
                                      p12]];
]]]]]]]]]]]]; Print[M];Print[FINISHED];