Many sequential construction tasks have exact terminal symmetries even though execution is directed and depends on history. Process evidence supplies order; terminal correspondence transports it between equivalent outcomes; the realized state updates relevance. These roles define a carrier framework: transport what the outcome preserves; refine what history changes. SymBuild combines transported process and state residual ranks by ordinal rank meet; its top-$k$ prefix exactly equals their top-$k$ union, yielding a tight worst-case verifier query bound under prefix information. We evaluate SymBuild in three construction domains: computer-aided design (CAD) assembly, Mini-Programs, and exact-fill packing, and test additional framework instantiations in all four domains. SymBuild improves the area under the anytime verified success curve by up to 6.77, 21.75, and 8.68 points over initial-static in the three construction domains. Refresh gains recur beyond SymBuild under alternative aggregation, planning, and learned scoring methods; on Geometric Reasoning Network (GRN) target removal, direct Combined refresh has the lowest mean capped verifier cost at all three scales and reduces learned state evaluations by factors of 6.57-12.20 relative to refreshed population-guided search. Together, these results support the carrier framework and demonstrate that SymBuild is an effective, analyzable method for anytime verified construction.

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