p-adic Manifold Learning and Benchmark Tasks from Impartial Games

We introduce Adelic operation-preserved embeddings (AOE), a training-free representation that captures both a number's real value and its modular (p-adic) signatures. This construction preserves additive and multiplicative structure by design, turning numerical input into embeddings that "speak in the language of mathematics." Unlike prior approaches that rely on task-specific retraining, AOE is plug-and-play and drops seamlessly into existing architectures. On algebraic combinatorics benchmarks, it delivers consistent gains including the first-ever perfect accuracy on the Weaving Pattern task-while suggesting a principled path forward for overcoming the long-standing "number problem" in AI.

Discovery---formulating novel generalizations---is a central part of the scientific process. Despite its importance, there is a gap in the current AI benchmark landscape, with few benchmarks directly probing the capacity for discovering new knowledge with experimentation in controlled environments where the objective is unknown. To address this gap, we release a new benchmark: DiG-bench (Discovery in Games). DiG-bench consists of a set of 70 independent games. Each game is encoded as a short string and has unique transformation rules that must be discovered through interaction and experimentation. The levels of the game present a series of challenges to test whether the rules have been discovered, where the win conditions for each level are also unknown. We provide games at seven tiers of difficulty for AI agents. The lowest tier is routinely solvable by multiple models, while the highest tier challenges the best models in agentic harnesses. All 70 games were solved by at least one human on first attempt. A subset of 21 games is released publicly, and the remainder is held private for secure evaluation.

Multi-task benchmarks have become a central pillar of machine learning research, yet their growing influence has incentivised benchmark gaming -- strategic actions taken to improve the leaderboard rank of a specific model. Treating datasets as voters and models as candidates, we consider benchmark-specific training -- the inclusion of benchmark data in training -- as a form of election manipulation. For any ordinal benchmark, the problem of choosing datasets to train on so that a target model becomes top-ranked corresponds to shift bribery, a class of manipulation problems from computational social choice. Leveraging this identification, we show that the benchmark-specific training problem is NP-hard under Borda count and mean win rate. Complementing this worst-case perspective, we introduce the instance-level robustness, the minimum number of datasets a model developer must include in training to top a given leaderboard, and derive expressions for it under arithmetic mean, median, mean win rate and pairwise majority. We evaluate these expressions on MMLU under HELM and on BIG-Bench Hard (BBH) under the Open LLM Leaderboard. Across both suites, mean win rate is hardest to manipulate: this gap is clear on BBH (24 tasks, 4507 models), where its median robustness is 22 tasks (92%), compared with 13 (54%) under arithmetic mean and 12 (50%) under median and pairwise majority.