Two-Variable Rationalization and Sharp Stabilization of Cubic Threefolds

A three-parameter family of smooth cubic threefolds has exact stabilization level two over every extension of its characteristic-zero ground field. The independent construction proves two-variable rationality for every stably rational smooth quartic del Pezzo surface via a saturated rank-three torus quotient of its projective Cox model. In an explicit rational pencil, five elliptic isogeny factors give a local toric-rank formula separating a positive-density squarefree integer family. The companion irrationality and Hodge-conservation theorems give exact levels and infinitely many pairwise nonbirational irrational fourfolds, all rational after one further projective-line factor. Exact computations accompany the conventional proof; no end-to-end Lean formalization is claimed.