Sharp compression and rigidity of ternary Boolean costs: Exact nonnegative representations at treewidth two

A finite Boolean network with nonnegative unary and pair costs defines a response on three boundary variables by minimizing over its hidden variables. Every such response of ordinary treewidth at most two has an exact representation of the same width with at most three hidden bits and nine pair scopes, preserving all eight absolute values. For rational source cells, the output can use any rational additive grid containing the response values, with every cell between zero and their maximum. Compatible boundary pairs admit a one-hidden-bit, five-scope replacement. Six strict alternating face inequalities force at least three hidden bits and nine scopes, even against signed-real competitors of the same width. At exactly nine genuine scopes, the original graph, after isolated hidden vertices are removed, must be a three-hidden Sun. Its preferred-row optimizers are unique and form a labelled claw, also for finite signed real cells. The arguments use interface replacement, disagreement exchange and integral flow allocation without a graph census or a catalogue of optimization certificates. An exact algorithm combines separator dynamic programming on a supplied width-two decomposition with a constant-size constructor. The results are presented as candidate theorems supported by written proofs and exact software checks. The accompanying material provides the source, software, verification record and reproduction instructions.