Robust Amplitude Elimination and Exact Lattice-Phase Selection

A complete exact selection law for a translating common lattice reduces finite robust designs to residue cells and affine envelopes. It gives rational selection probabilities, expected optima and boundary decisions. A four-design example separates the continuous winner from the phase-selection probabilities. A second theorem proves persistent reversal under arbitrarily fine lattice refinement. The principal contributions are the common-lattice phase-selection law, its exact probability and expectation formula, and the persistent-refinement reversal theorem. Scalar monotonicity, interval elimination, homogeneity and line-envelope geometry are established foundations, not separate originality claims. The record includes the mathematical manuscript with full proofs, exact verifier and certificates, a native Blender scene and its source-derived 3840 × 2705 figure.