Approximation of four-variable Boolean costs at treewidth two: Exact regimes and sharp constructions

A finite network of Boolean variables with nonnegative unary and pair costs defines a boundary function by minimizing over its hidden variables. We study uniform approximation of rational tables on four boundary variables by networks of ordinary treewidth at most two with at most eighteen nonseparable pair interactions, retaining absolute costs. A target parity gap survives every approximation of error less than half that gap. Under stated structural and ternary synthesis premises, this gives a finite exact procedure returning the optimum below that threshold, a rational attaining network and a matching lower certificate, or excluding every error below the threshold. Beyond that regime, an elementary four-input, two-output cell comparison gives lower bounds at every width and resource budget. A full-dimensional exact region admits a one-hidden-variable constructor with at most seven interactions and a closed capacity criterion. Under the ternary synthesis and strict-face resource premises, a different two-parameter family has an exact approximation error with ten interactions and three hidden variables jointly necessary and sufficient. A conditional separation shows that the assignment inequalities do not characterize width-two approximation. Rational attainment holds for arbitrary rational targets, but practical general optimization remains open. Targets need not have exact network realizations. The results retain their stated conditional candidate status; the supporting archive provides sources, exact certificates, verification and reproduction instructions.