Sharp parity-resource bounds and Sun-tree realizations: Nonnegative Boolean pair costs at treewidth two

A Boolean network with nonnegative unary and pair costs defines a boundary response by minimizing its hidden variables. At ordinary treewidth two, we determine the least resources permitting a strict global parity gap on n >= 3 boundary bits: 4n - 9 hidden variables and 9n - 18 genuine pair scopes, minimized separately. Every scope-optimal graph, after hidden isolates are removed, is a tree of n - 2 three-hidden Sun graphs. The proof removes a boundary cherry and its next triangle, saving four hidden variables and nine scopes. We then characterize responses whose even-parity values are all equal on each such tree. Exact minimum identities across its cuts determine compatibility; a sum of budgets determined by four odd residuals per block gives the least absolute baseline. A single nonnegative energy identity supplies the local budget bound, replacing a finite certificate argument. An exact subset dynamic program optimizes the baseline over all labelled trees, retains ties and has a separately checkable full-input certificate. Explicit nonnegative tables attain the bounds. The structural proofs use the stated ternary interface results from the companion compression paper; no graph census is a premise. The results are presented as candidate theorems supported by written proofs and exact checks.