Exact realization of nonflat Boolean parity costs on Sun trees: A finite criterion and rational constructor at treewidth two

A nonnegative Boolean cost table has a strict parity gap when every value on one parity class is smaller than every value on the other. We study its exact realization by nonnegative unary and pair costs, minimized over hidden Boolean variables, at ordinary treewidth at most two. Conditional on the published equality-graph and ternary synthesis results, we give a finite necessary-and-sufficient criterion for attaining the minimum 9(n-2) genuine pair scopes, for every n >= 3 and every Sun-tree shape. Preferred local values are recovered by leaf removal; canonical odd values are forced by maxima over boundary contexts. The remaining allocation problem is a finite union of bounded rational polyhedra in 2(n-3) variables. A global condition preserves at least one attaining branch at every odd boundary row. An eight-variable example shows why local feasibility and separate essential-row tests do not suffice. A successful test constructs 4n-9 hidden bits and 9(n-2) genuine pair scopes, with primitive cells between zero and the maximum target value. For a positive integer D, input values on the 1/D grid admit some output common denominator at most 12^(n-3)D. No polynomial-time or sharp-denominator claim is made. The results retain conditional candidate status; proofs, software and verification records accompany the paper.