Sharp sparse quadratization of weighted quartic Boolean costs

A quadratization represents a Boolean cost by minimizing a quadratic cost over auxiliary variables. For a four-variable family with one positive quartic coefficient and six sufficiently negative quadratic coefficients, this work determines the exact uniform error with at most six genuine pair interactions, allowing any finite auxiliary set. The optimum is the minimum of an explicit quadratic fallback and thirty-nine scalar capacities, explained by three response mechanisms. Interval endpoints and rational max flow construct attaining nonnegative rational networks with at most two auxiliary variables, without optimization linear programs. A structural lower bound covers all finite auxiliary sets. The paper also gives the separate one-auxiliary seven-, eight- and nine-interaction formulas, the one-auxiliary six-interaction procedure with 57 rational linear programs, and an exact interaction frontier. Absolute costs are retained. The results are proof-supported candidate theorems for this structured family; no claim about arbitrary quartic targets or hardware performance is made.