Certified geometric predicate for interval-valued coordinates

Input: finite closed dyadic intervals for all coordinates in one orientation, incircle, or insphere determinant. Output: positive, negative, or indeterminate. Construct the orientation or lifted matrix as in Dyadic orientation and lifted sphere determinants. For a lifted matrix, form each squared-norm entry by squaring its coordinate intervals and adding the results from coordinate index $1$ through $d$ from left to right, using the outward interval operations of Outward-rounded interval, rectangle and ball operations. If matrix construction returns a nonfinite status, return indeterminate.

Enumerate the permutations in lexicographic tuple order. For each permutation, start with $[1,1]$ and multiply the selected row-entry intervals from row $0$ through row $s-1$, from left to right, using outward interval multiplication. For an odd permutation, replace the product $T$ by $[0,0]-T$ using outward interval subtraction. Initialize the determinant interval $J$ with the first signed term, then add each later signed term from left to right using outward interval addition. If any operation returns a nonfinite status, return indeterminate. For a finite $J=[a,b]$, return positive when $a>0$, negative when $b<0$, and indeterminate when $0\in J$. This procedure reports a uniform raw determinant sign over the represented coordinate box; it does not choose a center point or invoke exact point fallback.