4
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This question involves the same subject matter as my previous one (How can one achieve the most accurate estimates of certain six-dimensional integrals under specific constraints?), but with another (numerical there, as opposed to symbolic here) perspective emphasized.

I have three positivity constraints (f1, f2, f3):

f1 = 1 - z[2, 3]^2 + z[1, 4]^2 (-1 + z[2, 3]^2) - z[2, 4]^2 + 
  z[1, 3]^2 (-1 + z[2, 4]^2) - 2 z[2, 3] z[2, 4] z[3, 4] - z[3, 4]^2 
  - 2 z[1, 3] z[1, 4] (z[2, 3] z[2, 4] + z[3, 4]) + 
  z[1, 2]^2 (-1 + z[3, 4]^2) + 
  2 z[1, 2] (z[1, 4] (z[2, 4] + z[2, 3] z[3, 4]) + 
  z[1, 3] (z[2, 3] + z[2, 4] z[3, 4]))

and

f2 = 1 - z[1, 2]^2 - z[1, 3]^2 + 2 z[1, 2] z[1, 3] z[2, 3] - z[2, 3]^2

and

f3 = 1 - z[1, 2]^2

The positivity of these (leading principal minor) constraints ensures the positive definiteness of the $4 \times 4$ correlation matrix with 1's on the diagonal entry and ij- and ji-entries equal to z[i,j].

This six-dimensional constrained integration command

Integrate[Boole[f1 > 0 && f2 > 0 && f3 > 0], {z[1, 2], -1, 1}, {z[1, 3], -1, 1}, 
 {z[1, 4], -1, 1}, {z[2, 3], -1, 1}, {z[2, 4], -1, 1}, {z[3, 4], -1, 1}]

yields the volume $\frac{32 \pi^2}{27}$.

Alternatively, I can obtain this result with the two commands:

J6 = GenericCylindricalDecomposition[
   f1 > 0 && f2 > 0 && f3 > 0, {z[1, 2], z[1, 2], z[1, 3], z[1, 4], 
z[2, 3], z[2, 4], z[3, 4]}];

Integrate[
 Boole[J6], {y[1, 2], -1, 1}, {z[1, 2], -1, 1}, {z[1, 3], -1,  1},
 {z[1, 4], -1, 1}, {z[2, 3], -1, 1}, {z[2, 4], -1, 1}, {z[3, 4], -1, 1}]

Now, the challenging part involves the additional implementation of further positivity ("separability") constraints (f4, f5,....) pertaining to another related $4 \times 4$ ("partial transpose") matrix, having an additional parameter $u>0$.

If one could further fully implement the "determinantal" positivity (quartic in $u$) constraint

f4 = -u^4 z[1, 4]^2 - z[2, 3]^2 + 
  2 u z[2, 3] (z[1, 2] z[2, 4] - z[1, 3] z[3, 4]) 
+ 2 u^3 z[1, 4] (z[1, 2] z[1, 3] - z[2, 4] z[3, 4]) 
- u^2 (-1 - z[1, 4]^2 z[2, 3]^2 + 2 z[1, 3] z[1, 4] z[2, 3] z[2, 4] + 
    z[2, 4]^2 - z[1, 3]^2 (-1 + z[2, 4]^2) - 
    2 z[1, 2] (z[1, 4] z[2, 3] + z[1, 3] z[2, 4]) z[3, 4] + 
    z[3, 4]^2 - z[1, 2]^2 (-1 + z[3, 4]^2))

obtaining some unknown function of $u$, the problem would be solved.

Sub-optimal (and presumably "easier" to obtain) solutions could be obtained by implementing (in addition, of course, to f1, f2, f3) one or more of the $3 \times 3$ principal minor (quadratic in $u$) constraints:

f5 = 1 - z[1, 2]^2 - z[1, 3]^2 + 2 u z[1, 2] z[1, 3] z[1, 4] - u^2 z[1, 4]^2,

f6 = u^2 - u^2 z[1, 2]^2 - z[2, 3]^2 + 2 u z[1, 2] z[2, 3] z[2, 4] - u^2 z[2, 4]^2

f7 = u^2 - u^2 z[1, 3]^2 - z[2, 3]^2 + 2 u z[1, 3] z[2, 3] z[3, 4] - u^2 z[3, 4]^2

f8 = 1 - u^2 z[1, 4]^2 - z[2, 4]^2 + 2 u z[1, 4] z[2, 4] z[3, 4] - z[3, 4]^2

I have succeeded in so implementing f5, obtaining $\frac{27 \pi ^4 u^2-7 \pi ^4}{192 u^3}$ for $u>1$ and $\frac{\pi ^3 \left(3 \left(27 u^2-7\right) \sin ^{-1}(u)+u \sqrt{1-u^2} \left(2 u^4+37 u^2+21\right)\right)}{288 u^3}$ for $1>u>0$. (The first [polynomial] result is sharper than the second.)

Presumably, implementing f6, f7 or f8 individually would yield solutions of the same nature.

So, the issue in which I am interested is whether one can implement two or more of f5, f6, f7 or f8, or f4 itself (solving the problem). (Also, I think that implementation of any three of f5, f6, f7 and f8 would provide the full solution.)

I have been playing around with this problem, in particular using the GenericCylindricalDecomposition command, but am not sure I have been proceeding most effectively. (Also the relative effectiveness is not clear of first implementing GenericCylindricalDecomposition and employing that in further Boole integrations, or putting the "raw" constraints in the integration [per the two approaches above in obtaining the volume $\frac{32 \pi^2}{27}$].)

Also, I'm not sure of whether it helps or possibly hinders further computations to use Simplify and/or FullSimplify on the results of GenericCylindricalDecomposition. At times, I've wanted to extract from the GenericCylindricalDecomposition result, the specific limits for one of the variables, and was not quite sure how to proceed.

Of course, one could also try reparameterizing the underlying correlation matrix, but my attempts (Cholesky decompostion, partial correlations,...) along these lines have led to bulky, seemingly non-promising constraints.

Different orderings of the six integration variables (z[i,j]) might lead to different performance.

I did obtain a cylindrical decomposition (with $u$ as the first of the now seven variables) using f1, f2, f3 AND f4. The LeafCount of the result (R) was 314,459. (Simplify[PowerExpand[R,u]] reduces this to 10,075 and FullSimplify to 10,051. I'm trying to include the code below--but my first attempt was too long.) How to further managably proceed with it, however, is not quite clear.

One expects that a solution--say $g(u)$--to the full (f1, f2, f3, f4) problem will be such that $g(u)=g(\frac{1}{u})$.

In a manner of speaking, a solution to this problem has recently been constructed (avoiding "brute force" computations) by Lovas and Andai (https://arxiv.org/abs/1610.01410), but not in terms of the variable $u$, but in terms of a ratio ($\epsilon$) of singular values of $2 \times 2$ matrices, the two variables ($u, \epsilon$) only being equal for diagonal matrices. It seems quite challenging to convert one framework into another. The construction of $g(u)$, therefore, remains of strong interest.

I'm now trying to include the code (but not succeeding--too long apparently). So, I'll just insert the first (LeafCount=4374) of the three parts:

0 < u < 1/Sqrt[2] && 

u (-1 + z[1, 2]^2) (-u z[1, 2] z[1, 4] z[2, 3] + u^2 z[1, 4] z[2, 4] + z[1, 3] (z[2, 3] - u z[1, 2] z[2, 4]) - (-1 + z[1, 2]^2) [Sqrt](( 1/((-1 + z[1, 2]^2)^2))(-1 + z[1, 2]^2 + z[1, 3]^2 - 2 u z[1, 2] z[1, 3] z[1, 4] + u^2 z[1, 4]^2) (z[2, 3]^2 - 2 u z[1, 2] z[2, 3] z[2, 4] + u^2 (-1 + z[1, 2]^2 + z[2, 4]^2))) + u (-1 + z[1, 2]^2) z[3, 4]) < 0 && u (-1 + z[1, 2]^2) (-u z[1, 2] z[1, 4] z[2, 3] + u^2 z[1, 4] z[2, 4] + z[1, 3] (z[2, 3] - u z[1, 2] z[2, 4]) + (-1 + z[1, 2]^2) [Sqrt](( 1/((-1 + z[1, 2]^2)^2))(-1 + z[1, 2]^2 + z[1, 3]^2 - 2 u z[1, 2] z[1, 3] z[1, 4] + u^2 z[1, 4]^2) (z[2, 3]^2 - 2 u z[1, 2] z[2, 3] z[2, 4] + u^2 (-1 + z[1, 2]^2 + z[2, 4]^2))) + u (-1 + z[1, 2]^2) z[3, 4]) > 0 && ((Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + z[2, 3] > z[1, 2] z[1, 3] && ((u + z[2, 3] < 0 && ((z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)])/u < z[1, 4] || u (-z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + u z[1, 4]) < 0) && ((Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] > u z[1, 2] + z[1, 3] && ((u z[1, 2] + Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] > 0 && ((1 + z[1, 2] > 0 && Sqrt[1 - u^2] + z[1, 2] < 0) || (Sqrt[1 - u^2] < z[1, 2] && z[1, 2] < 1))) || (Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] > u z[1, 2] && ((u > z[1, 2] && z[1, 2] > 0) || (u < z[1, 2] && Sqrt[1 - u^2] > z[1, 2]))) || (u z[1, 2] + Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] < z[1, 3] && ((u + z[1, 2] < 0 && Sqrt[1 - u^2] + z[1, 2] > 0) || (z[1, 2] < 0 && u + z[1, 2] > 0))))) || (u z[1, 2] + Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] > 0 && ((Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] < u z[1, 2] && ((u > z[1, 2] && z[1, 2] > 0) || (u < z[1, 2] && Sqrt[1 - u^2] > z[1, 2]))) || (u z[1, 2] + Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] > z[1, 3] && ((u + z[1, 2] < 0 && Sqrt[1 - u^2] + z[1, 2] > 0) || (z[1, 2] < 0 && u + z[1, 2] > 0))))))) || (z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] > z[2, 3] && ((1 + z[1, 2] > 0 && Sqrt[1 - u^2] + z[1, 2] < 0 && (((( z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)])/u < z[1, 4] || u (-z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + u z[1, 4]) < 0) && ((1 + z[1, 3] > 0 && Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] < u z[1, 2]) || (Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] < u z[1, 2] + z[1, 3] && z[1, 3] < 1))) || (u z[1, 2] + Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] < z[1, 3] && u z[1, 2] + Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] < 0 && (((( z[1, 2] z[2, 3] + Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)])/u < z[2, 4] || u (-z[1, 2] z[2, 3] + Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)] + u z[2, 4]) < 0) && (( z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)])/u < z[1, 4] || u (-z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + u z[1, 4]) < 0)) || (( z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)])/u > z[1, 4] && u (-z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + u z[1, 4]) > 0 && ( z[1, 2] z[2, 3] + Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)])/u > z[2, 4] && u (-z[1, 2] z[2, 3] + Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)] + u z[2, 4]) > 0))))) || ((( z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)])/u < z[1, 4] || u (-z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + u z[1, 4]) < 0) && ((((1 + z[1, 3] > 0 && u z[1, 2] + Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] < 0) || (u z[1, 2] + Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] < z[1, 3] && z[1, 3] < 1)) && ((u < z[1, 2] && Sqrt[1 - u^2] > z[1, 2]) || (Sqrt[1 - u^2] < z[1, 2] && z[1, 2] < 1))) || (u + z[1, 2] < 0 && Sqrt[1 - u^2] + z[1, 2] > 0 && ((1 + z[1, 3] > 0 && Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] < u z[1, 2]) || (Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] < u z[1, 2] + z[1, 3] && z[1, 3] < 1))))) || ((((( z[1, 2] z[2, 3] + Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)])/u < z[2, 4] || u (-z[1, 2] z[2, 3] + Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)] + u z[2, 4]) < 0) && (( z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)])/u < z[1, 4] || u (-z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + u z[1, 4]) < 0)) || (( z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)])/u > z[1, 4] && u (-z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + u z[1, 4]) > 0 && ( z[1, 2] z[2, 3] + Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)])/u > z[2, 4] && u (-z[1, 2] z[2, 3] + Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)] + u z[2, 4]) > 0)) && ((Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] < u z[1, 2] + z[1, 3] && ((Sqrt[1 - u^2] < z[1, 2] && z[1, 2] < 1 && Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] < u z[1, 2]) || (u + z[1, 2] > 0 && z[1, 2] < 0 && z[1, 3] < 1))) || (1 + z[1, 3] >

              0 && ((u + z[1, 2] > 0 && z[1, 2] < 0 && 
                Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] < 
                u z[1, 2]) || (u > z[1, 2] && z[1, 2] > 0 && 
                u z[1, 2] + Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + 
                z[1, 3] < 0))) || (u z[1, 2] + 
               Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] < z[1, 3] && 
             z[1, 3] < 1 && u > z[1, 2] && 
             z[1, 2] > 0))))) || (u > 
      z[2, 3] && ((((
            z[1, 2] z[2, 3] + 
             Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)])/u < 
            z[2, 4] || 
           u (-z[1, 2] z[2, 3] + 
               Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)] + 
               u z[2, 4]) < 0) && ((
            z[1, 2] z[1, 3] + 
             Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)])/u < 
            z[1, 4] || 
           u (-z[1, 2] z[1, 3] + 
               Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + 
               u z[1, 4]) < 0)) || ((
          z[1, 2] z[1, 3] + 
           Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)])/u > z[1, 4] &&
          u (-z[1, 2] z[1, 3] + 
             Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + 
             u z[1, 4]) > 0 && (
          z[1, 2] z[2, 3] + 
           Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)])/u > 
          z[2, 4] && 
         u (-z[1, 2] z[2, 3] + 
             Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)] + 
             u z[2, 4]) > 
          0)) && ((u z[1, 2] + Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] >
           z[1, 3] && ((Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] < 
              u z[1, 2] + 
               z[1, 3] && ((u < z[1, 2] && 
                Sqrt[1 - u^2] > z[1, 2]) || (u > z[1, 2] && 
                z[1, 2] > 0))) || (Sqrt[(-1 + u^2) (-1 + 
                z[1, 2]^2)] + z[1, 3] > 
              u z[1, 2] && ((Sqrt[1 - u^2] < z[1, 2] && 
                z[1, 2] < 1) || (1 + z[1, 2] > 0 && 
                Sqrt[1 - u^2] + z[1, 2] < 0))))) || (Sqrt[(-1 + 
              u^2) (-1 + z[1, 2]^2)] + z[1, 3] > u z[1, 2] && 
         u z[1, 2] + Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] <
           0 && ((z[1, 2] < 0 && 
             u + z[1, 2] > 0) || (u + z[1, 2] < 0 && 
             Sqrt[1 - u^2] + z[1, 2] > 0))))))) || (z[1, 2] z[1, 
     3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] > 
  z[2, 3] && ((u < 
      z[2, 3] && ((
        z[1, 2] z[1, 3] + 
         Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)])/u < z[1, 4] || 
       u (-z[1, 2] z[1, 3] + 
           Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + u z[1, 4]) < 
        0) && ((Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] > 

          u z[1, 2] && ((Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] > 
              u z[1, 2] + 
               z[1, 3] && ((u > z[1, 2] && 
                z[1, 2] > 0) || (u < z[1, 2] && 
                Sqrt[1 - u^2] > z[1, 2]))) || (u z[1, 2] + 
               Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] > 
              z[1, 3] && ((1 + z[1, 2] > 0 && 
                Sqrt[1 - u^2] + z[1, 2] < 0) || (Sqrt[1 - u^2] < 
                z[1, 2] && z[1, 2] < 1))) || (u z[1, 2] + 
               Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] < 
              0 && ((u + z[1, 2] < 0 && 
                Sqrt[1 - u^2] + z[1, 2] > 0) || (z[1, 2] < 0 && 
                u + z[1, 2] > 0))))) || (u z[1, 2] + 
           Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] > 
          z[1, 3] && ((u z[1, 2] + 
               Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] > 
              0 && ((u + z[1, 2] < 0 && 
                Sqrt[1 - u^2] + z[1, 2] > 0) || (z[1, 2] < 0 && 
                u + z[1, 2] > 0))) || (Sqrt[(-1 + u^2) (-1 + 
                z[1, 2]^2)] < 
              u z[1, 2] + 
               z[1, 3] && ((u > z[1, 2] && 
                z[1, 2] > 0) || (u < z[1, 2] && 
                Sqrt[1 - u^2] > z[1, 2]))))))) || (u + z[2, 3] > 
      0 && ((((
            z[1, 2] z[2, 3] + 
             Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)])/u < 
            z[2, 4] || 
           u (-z[1, 2] z[2, 3] + 
               Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)] + 
               u z[2, 4]) < 0) && ((
            z[1, 2] z[1, 3] + 
             Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)])/u < 
            z[1, 4] || 
           u (-z[1, 2] z[1, 3] + 
               Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + 
               u z[1, 4]) < 0)) || ((
          z[1, 2] z[1, 3] + 
           Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)])/u > z[1, 4] &&
          u (-z[1, 2] z[1, 3] + 
             Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + 
             u z[1, 4]) > 0 && (
          z[1, 2] z[2, 3] + 
           Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)])/u > 
          z[2, 4] && 
         u (-z[1, 2] z[2, 3] + 
             Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)] + 
             u z[2, 4]) > 
          0)) && ((u z[1, 2] + Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] +
            z[1, 3] > 
          0 && ((Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] < 
              u z[1, 2] && ((u < z[1, 2] && 
                Sqrt[1 - u^2] > z[1, 2]) || (u > z[1, 2] && 
                z[1, 2] > 0))) || (Sqrt[(-1 + u^2) (-1 + 
                z[1, 2]^2)] > 
              u z[1, 2] + 
               z[1, 3] && ((Sqrt[1 - u^2] < z[1, 2] && 
                z[1, 2] < 1) || (1 + z[1, 2] > 0 && 
                Sqrt[1 - u^2] + z[1, 2] < 0))))) || (Sqrt[(-1 + 
             u^2) (-1 + z[1, 2]^2)] > u z[1, 2] + z[1, 3] && 
         u z[1, 2] + Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] < 
          z[1, 3] && ((z[1, 2] < 0 && 
             u + z[1, 2] > 0) || (u + z[1, 2] < 0 && 
             Sqrt[1 - u^2] + z[1, 2] > 0))))))) || (u > z[2, 3] &&
  u + z[2, 3] > 
  0 && ((((
        z[1, 2] z[2, 3] + 
         Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)])/u < z[2, 4] ||
        u (-z[1, 2] z[2, 3] + 
           Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)] + 
           u z[2, 4]) < 0) && ((
        z[1, 2] z[1, 3] + 
         Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)])/u < z[1, 4] || 
       u (-z[1, 2] z[1, 3] + 
           Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + u z[1, 4]) < 
        0)) || ((
      z[1, 2] z[1, 3] + Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)])/
      u > z[1, 4] && 
     u (-z[1, 2] z[1, 3] + 
         Sqrt[(-1 + z[1, 2]^2) (-1 + z[1, 3]^2)] + u z[1, 4]) > 
      0 && (z[1, 2] z[2, 3] + 
       Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)])/u > z[2, 4] && 
     u (-z[1, 2] z[2, 3] + 
         Sqrt[(-1 + z[1, 2]^2) (-u^2 + z[2, 3]^2)] + u z[2, 4]) > 
      0)) && ((Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] > 
      u z[1, 2] + z[1, 3] && 
     Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] > 
      u z[1, 2] && ((u > z[1, 2] && z[1, 2] > 0) || (u < z[1, 2] &&
          Sqrt[1 - u^2] > z[1, 2]))) || (u z[1, 2] + 
       Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] > z[1, 3] && 
     u z[1, 2] + Sqrt[(-1 + u^2) (-1 + z[1, 2]^2)] + z[1, 3] > 
      0 && ((u + z[1, 2] < 0 && 
         Sqrt[1 - u^2] + z[1, 2] > 0) || (z[1, 2] < 0 && 
         u + z[1, 2] > 0))))))
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  • $\begingroup$ Paul if you are trying to share an idea or brainstorm a research w/ help of Mathematica in a free discussion formyou might try also Wolfram Community. $\endgroup$ – Vitaliy Kaurov Oct 25 '18 at 1:20
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This is a limited answer--only advancing to the imposition (in addition to f1, f2, f3) as constraints of two principal $3 \times 3$ minors (f5, f7), rather than just f5, as indicated was accomplished in the question. However, the major caveat now is that this "advancement" was only achieved by setting the correlation parameters $z_{12}$ and $z_{34}$ to 0. (This lower-dimensional case does, however, have particular interest, in that than the parameter $u$ becomes equal to the parameter $\varepsilon$ in the important related study of Lovas and Andai referenced above (https://arxiv.org/pdf/1610.01410.pdf). In the general case, \begin{equation} \varepsilon=\exp \left(-\cosh ^{-1}\left(\frac{-u^2+2 u z_{12} z_{34}-1}{2 u \sqrt{z_{12}^2-1} \sqrt{z_{34}^2-1}}\right)\right). \end{equation} Now, the (Mathematica-generated) result I have obtained for the imposition of f1, f2, f3, f5 and f7 in this specific case, is \begin{equation} \begin{cases} \frac{4 \pi \left(\sqrt{u^2-1}+u^2 \csc ^{-1}(u)\right)}{3 u^2} & u>1 \\ \frac{2}{3} \pi \left(2 \sqrt{1-u^2} u+2 i \log \left(u+i \sqrt{1-u^2}\right)+\pi \right) & 0<u<1 \end{cases}. \end{equation} An interesting observation here is that by replacing $u$ by $\frac{1}{u}$ in either one of these two functions, the resultant pair (original plus transformed) functions become equivalent to one another over the positive real axis. This equality was just explained by Claude Leibovici (https://math.stackexchange.com/questions/2213869/show-that-two-functions-are-identical-over-the-real-positive-axis).

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  • $\begingroup$ Michael Trott has indicated to me that rather than using GenericCylindricalDecomposition[ f1 > 0 && f2 > 0 && f3 > 0 && f4 > 0, {u,z[1, 2], z[1, 3], z[1, 4], z[2, 3], z[2, 4], z[3, 4]}] I should expand the command to GenericCylindricalDecomposition[ f1 > 0 && f2 > 0 && f3 > 0 && f4 > 0&&-1 < z[1, 2] < 1 && -1 < z[1, 3] < 1 && -1 < z[1, 4] < 1 && -1 < z[2, 3] < 1 && -1 < z[2, 4] < 1 && -1 < z[3, 4] < 1, {z[1, 2], z[1, 3], z[1, 4], z[2, 3], z[2, 4], z[3, 4]}] . Maybe the use of f5 >0 in the CAD in addition to f1>0,f2>0,f3>0,f4>0 might also address the same issue. $\endgroup$ – Paul B. Slater Apr 6 '17 at 17:01

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