Sharp resource tradeoffs and stability in binary Sun representations

A binary pair-cost network on the three-hidden Sun graph defines a ternary response by minimizing over its hidden variables. Fix six positive alternating face margins of that response. We describe all presentations and the exact tradeoffs among nine interaction magnitudes and twenty-eight competing-state energy gaps. An elementary exchange argument forces a unique optimizer at each even row. Twelve nonnegative parameters, with explicit complementarity conditions, describe every presentation modulo observable affine terms and redistribution between edge tables. At a fixed optimizer center the Pareto frontier is a point or a triangle: three groups of five resources share a single unavoidable excess. We obtain every nonnegative weighted optimum and every minimizing presentation, including zero weights and tied centers. We give exact nearest-optimum distances and attained all-budget stability bounds in the essential parameters. The proofs are analytical, and explicit nonnegative tables realize the bounds. The results are presented as candidate theorems supported by written proofs and exact diagnostic checks. The accompanying material supplies the complete sources, software, assurance record, verification and reproduction instructions.