Primitive Conflicts and Realizability Hierarchies in Binary Polymatrix Games: Universal Obstructions, Optimal Local Constructions, and Inverse Max-Plus Factorization

This research preprint develops a structural theory of inverse realizability for cost–support frontiers in binary polymatrix games on finite rooted cactus graphs. The central question is which numerical frontiers can arise from internally Nash-stable continuations and how many interior players are required to realize them. Working with a fixed root action and reference profile, at-most-budget semantics, and the root’s best-response condition deferred, the paper establishes several results: Primitive forcing and realizability obstructions. Physical cactus interfaces impose restrictions on attainable cost–support patterns. These yield pair conflicts, higher-order hyperconflicts, primitive support-capacity bounds, and an infinite family of numerically pairwise-compatible frontiers that remain globally unrealizable. Exact local synthesis and terminal-primitive classification. Explicit binary polymatrix constructions realize admissible local patterns through nested activation and OR, neutral, and AND mechanisms. These constructions provide a necessary-and-sufficient classification of complete frontiers whose terminal jump is primitive. Complete-frontier atomization and optimal representation size. A complete normalized frontier is cactus-realizable if and only if it admits a finite at-most max-plus factorization into permitted numerical cactus atoms. For every nonzero realizable complete frontier, the minimum number of interior players equals its first saturation budget. Exact finite-prefix synthesis. For strictly superadditive prescribed prefixes, the minimum cactus realization size is characterized by a weighted partition problem whose admissible blocks are determined by additive resonance triples. The resulting theory establishes sharp representation bounds, optimal interface requirements, and explicit families exhibiting threefold asymptotic compression relative to rooted-tree realizations. Inverse factorization and computational boundaries. The paper establishes nonuniqueness of numerical representations and distinguishes complete-frontier realizability from finite-prefix synthesis. It also provides an NP-membership upper bound for complete-frontier recognition under an explicit rational input representation. The results distinguish numerical root-conditioned frontier expressiveness from the representation of full Nash equilibrium sets. Classical knapsack optimization and max-plus convolution provide the algebraic foundation; the principal contributions concern cactus-specific physical constraints, exact realizability, and optimal synthesis. Scope and verification: The manuscript concerns binary polymatrix games with rational, potentially asymmetric endpoint payoffs. It does not establish a general polynomial-time inverse-realizability algorithm or dynamic equilibrium-maintenance complexity bounds. A separate reproducibility archive contains the manuscript source, exact-rational verification programs, expected outputs, and supporting audit documentation. Computational verification supplies finite falsification evidence and does not replace universal mathematical proof. Version 1.2 — Author research preprint, November 2026. Independent external refereeing and comprehensive historical-priority assessment remain open.