Cost–Support Frontiers of Internally Stable Continuations on Trees and Cacti: Realizability, Obstructions, and Representation Size

This research preprint investigates the realizability and representation size of cost–support frontiers arising from internally stable continuations in binary polymatrix games on rooted trees and cacti. The results establish a rooted-tree correspondence with zero–one knapsack value functions, an explicit frontier that no binary polymatrix cactus can realize, and a family requiring a quadratic number of interior players. The analysis also identifies structural constraints imposed by cactus interfaces and demonstrates information loss under scalar tropical evaluation. The results concern fixed-reference, at-most-budget continuations with the root’s Nash inequality deferred. They do not establish dynamic-update complexity bounds or a complete characterization of cactus frontiers. A separate reproducibility archive provides the LaTeX manuscript source, executable verification programs, expected outputs, and reproducibility instructions. Version 1.0 — Research preprint. External peer review and exhaustive historical-priority assessment have not been completed.