Finite exact 3D Delaunay and regular lower-hull cavity repair

Input is a finite indexed list of dyadic sites in ${\mathbb{R}}^3$, each with an optional dyadic weight. Collapse only exact duplicate coordinate-weight pairs, retaining the least identifier. If the projected set has affine dimension less than three, return lower-dimensional, that dimension, and the exact deduplicated list; do not assert tetrahedra.

For affine dimension three, fix the finite lexicographic universe before insertion. Process sites in stable-id order (any stated permutation is allowed if it is fixed before processing). At each prefix, enumerate every four-site subset. If its projected base is affinely dependent, record that tuple with status degenerate-base and omit it from tetrahedral candidates. Otherwise solve the unique affine height plane through its four lifted sites. Retain the tetrahedron exactly when every other lifted site in that prefix lies on or above this plane in the ordered formal field of Stable-id lexicographic lift perturbation for cocircular and power ties.

After adding a site, delete old tetrahedra whose plane the new lift violates, retain old tetrahedra still supporting, and add every newly supporting tetrahedron containing the inserted site. This finite set difference is the cavity repair; no geometric connectivity or running-time bound is asserted. After the final prefix, report the retained lower tetrahedra, the finite degenerate-base tuple list, and every weighted site absent from the lower tetrahedra as hidden, meaning formal-hidden in the fixed formal lift, without asserting original-hidden status.

For ordinary orientation use Certified predicate for exact dyadic geometric inputs. For weighted lower tests, form the exact dyadic matrix with lifted column $\mathrm{\parallel }p{\mathrm{\parallel }}^2-w$ and evaluate the finite formal perturbation coefficients using exact integer determinant signs from Exact dyadic determinant sign by integer clearance. The raw exact predicate keeps zero; the formal lift is applied only by this algorithm's declared tie convention.