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 a replacement that preserves these absolute observations, uses no hidden variables, and is a sum of nonnegative unary and pair tables. For bounded integer penalty strengths, twelve measurements determine the observation profile. We give an explicit procedure for 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. The construction preserves a common coefficient grid, with an explicit rounding rule in the three-pair case. Examples distinguish observation matching from full-response and full-min-marginal matching. The proofs specialize classical residuation, covering and flow arguments. Historical novelty remains unresolved.