Open-ended intelligence is the capacity to adapt to novel problems and environments that are substantially different from those in training. A mathematics of open-ended intelligence requires two pillars: first, a minimal set of representational primitives (e.g., states, actions) and algorithmic primitives (e.g., nearest neighbor); and second, an acquired compositional grammar for selection, recursion, and branching that produces sequences of operations and recurring motifs. We formalize open-ended intelligence in terms of the compositional closure induced by a finite primitive set $P$ and a set of composition operators $C$. We characterize properties of the induced closure $\mathcal{L}(P,C)$ that support unbounded compositional generation across families of tasks and worlds. The closure of the two pillars yields infinite adaptive responses across a wide range of settings. The mathematics supports complementary research agendas, including evaluation metrics for explanation and interpretability, and novel architectures where compositional generalization is native. We propose next primitive prediction (NPP) as a novel architectural objective, where training encourages the acquisition of reusable algorithmic primitives and their compositional grammar, such that new solutions are generated through recombination. Given such an objective, curriculum learning and self-play can enable lifelong learning, expanding the closure by discovering reusable primitives and transition motifs across settings. We ground the framework through case studies in physics, evolution, and neuroscience.

翻译：开放式智能是指适应与训练环境存在本质差异的新问题和新环境的能力。此类智能的数学化需要两大支柱：首先是一组最小表示基元（如状态、动作）和算法基元（如最近邻算法）；其次是经过习得的组合语法体系，包含选择、递归和分支操作，用以生成操作序列与重复性模式。我们通过有限基元集合$P$与组合算子集合$C$诱导的组合闭包来形式化描述开放式智能，并刻画了该诱导闭包$\mathcal{L}(P,C)$支持跨任务族和世界族进行无界组合生成的性质。这两大支柱的闭包可在广泛场景中产生无限的自适应响应。该数学框架支持多项互补性研究议程，包括用于解释性和可解释性的评估指标，以及原生支持组合泛化的新型架构。我们提出下一基元预测（NPP）作为新型架构目标——通过训练促使模型习得可复用算法基元及其组合语法，从而通过重组生成新解决方案。在此目标下，课程学习与自我博弈可通过跨场景发现可复用基元与转换模式来扩展闭包，最终实现终身学习。我们通过物理学、进化论和神经科学领域的案例研究对该框架进行实证验证。