We present DualityCert, a symbolic verifier for candidate Seiberg-duality claims in four-dimensional N=1 quiver gauge theories. The verifier evaluates 't Hooft anomaly matching, superpotential R-charge consistency, central-charge matching, and a bounded chiral-ring proxy. A claim that passes receives a consistency certificate, which states that no tested inconsistency was found, not that the duality is proven. We use the verifier as a repair environment for language-model agents, which receive a deliberately broken claim and must edit it until it certifies. On a preregistered benchmark of 145 broken claims, with the analysis fixed before the first confirmatory model call, verifier-gated retry improves final repair success over a single attempt by +8.3 percentage points (pp) on deepseek-chat and +7.1 pp on qwen-plus (Holm-adjusted p<0.002). Under an equal budget of eleven attempts, the stop-first strategy portfolio underperforms independent verifier-filtered resampling by 10.3 percentage points on deepseek-chat but outperforms it by 14.7 points on qwen-plus, reversing the ordering of the two tested verifier-exploitation policies across the two confirmatory models. On qwen-plus, category-level verifier feedback is worth +8.7 pp over content-free retry, and interpretable obligation identities alone are worth +6.4 pp over structurally identical masked feedback. Neither effect is detected on deepseek-chat. Separately, a preregistered MiniMax-M2.5 extension again finds an iteration gain and independent verifier-filtered resampling outperforming the strategy portfolio. Which policy is better thus differs between the two models, while every winning policy uses the same cheap certificate. The verifier, benchmark, protocol, and all per-attempt records are released.
World models enable language-model agents to predict environment dynamics and plan before acting. In text environments, the model must learn symbolic action effects from serialized state descriptions, but the role of serialization structure remains underexplored. We present HyperWorld, a controlled study of state serialization for learned textual world models. We compare raw observations with three symbolic serializations of the same ground-truth state: independent sentences, pairwise triples, and entity-centered hyperedge units that group multiple related facts around entities and relations. All variants use the same training objective: given a state and an action, predict symbolic effects or judge the action infeasible. Across model scales, data budgets, and in-distribution and out-of-distribution test worlds, hyperedge serialization gives the clearest gains for 0.5B--1.5B models and under distribution shift. Larger models reduce the gap, and pairwise triples can match or slightly exceed hyperedges on in-distribution exact match, but hyperedges achieve the strongest out-of-distribution fact F1 and the best small-to-medium scale trade-off between feasibility detection and effect prediction. In downstream greedy planning, the hyperedge world model also attains the highest success rate among the tested representations. These results show that higher-order state organization is a simple but effective inductive bias for learned symbolic world models, especially when model capacity is limited or test environments differ from training.