Terminal Symmetry as a Carrier of Asymmetric Process Knowledge: Statewise Refinement for Anytime Verified Construction

Many sequential construction tasks have exact terminal symmetries even though execution is directed and depends on history. Our insight is that terminal symmetry can carry independently grounded process knowledge across equivalent outcomes even when execution dynamics are asymmetric. The resulting carrier framework transports reusable structure through terminal correspondence, refines its relevance from the realized state, and certifies actions with a fixed verifier. SymBuild combines transported process and state residual ranks through ordinal rank meet. Its threshold sets are unions of the corresponding input sets, yielding a tight worst-case verifier query bound under prefix information. In computer-aided design (CAD) assembly, Mini-Programs, and exact-fill packing, SymBuild improves the area under the anytime verified success curve over Static by up to 6.77, 21.75, and 8.68 percentage points, respectively. Refresh gains extend across alternative aggregation, planning, and learned scoring methods. In Geometric Reasoning Network (GRN) target removal, direct Combined refresh achieves the lowest mean verifier query score at all three scales. It reduces distinct scored states by factors of 6.48 to 12.22 relative to refreshed population search. These results demonstrate how terminal symmetry supports knowledge reuse and statewise refinement improves verified construction.