Sharp approximation and capacity of ternary Boolean costs: Two independent stars, nonnegative costs, and unequal tolerances

We study the error incurred when a nonnegative ternary Boolean cost, realizable by a network of treewidth at most two, is approximated by the sum of two independent hidden-variable stars. Costs are retained absolutely, including their constant part. We give an exact weighted maximum-norm distance formula and attained cost bounds. In the uniform norm the sharp bounds are one fifth of the largest entry and one thirty-first of the total; we describe their equality cases and the exact effect of a common lower floor. For eight independently positive tolerance scales, the least total and least peak needed for unit error have explicit finite min-max formulas. The peak formula admits an analytical construction using four aggregate parameters, replacing a previous thirty-vertex interpolation. We also give the total-cost law with eight tolerance scales and a lower floor. When the scales are constant on each cube parity, joint total and peak budgets have at most two Pareto-minimal cost pairs. The arguments use clipping at zero, an elementary weighted-box construction, and the root-capacity criterion from the companion compression paper. The results are presented as candidate theorems supported by written proofs and exact software checks. The accompanying support supplies the complete source, exact code, verification and reproduction records.