Sparse quadratic compression from bounded min-plus observations

A finite nonnegative cost on three Boolean variables is observed by adding unary penalties on at most two coordinates and taking a minimum. We seek replacements that preserve these absolute observations, use no hidden variables and are sums of nonnegative unary and pair tables. For bounded integer penalty strengths, twelve measurements determine the observation profile. An explicit procedure gives the least possible peak response at each budget of zero, one, two or three genuine pair scopes. The two-pair case reduces to at most twelve tables; the three-pair case to at most fifteen supports and a cube transportation problem. Under the stated common-grid hypotheses, factors can be chosen on that grid. Examples separate bounded-observation matching from full-response and full-min-marginal matching. For Boolean-network sources with finite nonnegative factors of arity at most two and source peak M>0, the optimal boundary-only full-margin peak is strictly below 3M, while the uniform ratio has sharp nonattained supremum three. If all effective response values lie on a grid of spacing h, the sharp bound is 3M-h; no primitive-factor or context-grid condition is imposed for this source-specific bound. The proofs specialize residuation, covering, flow and majority-preservation arguments. The results are presented as candidates, and historical novelty remains unresolved.