Open-ended intelligence is the capacity to adapt to novel problems and environments that are substantially different from those in training. We formalize open-ended intelligence as the 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. A mathematics of open-ended intelligence requires two pillars: a minimal set of representational primitives (e.g., states, actions) and algorithmic primitives (e.g., nearest neighbor), together with composition motifs (e.g., recursion, sequencing) that reflect an acquired compositional grammar. The closure of these two pillars enables the generation of infinite adaptive responses across a wide range of settings. The mathematics supports complementary research agendas, including evaluation metrics for explanation and interpretability, as well as building architectures where compositional generalization is native. We propose next primitive prediction as a novel architectural objective, where the training objective encourages the acquisition of reusable algorithmic primitives and their compositional grammar, such that new solutions are generated through recombination. Curriculum learning and self-play enable lifelong learning and expansion of the closure by discovering reusable primitives and transition motifs across families of tasks and worlds. We ground the framework through case studies in physics, evolution, and neuroscience.

翻译：开放型智能是指适应与训练环境显著不同的新颖问题和环境的能力。本文形式化地将开放型智能定义为由有限原始集 \(P\) 和一组组合算子 \(C\) 生成的闭包。我们刻画了该诱导闭包 \(\mathcal{L}(P,C)\) 的特性，这些特性支撑跨任务族与跨世界族进行无界组合生成。开放型智能的数学基础需要两大支柱：一组最小化的表征原始要素（如状态、动作）与算法原始要素（如最近邻），以及反映习得组合语法的组合模式（如递归、序列化）。这两大支柱的闭包能够在大范围场景中生成无限多的自适应响应。该数学框架支持互补的研究方向，包括用于解释性与可解释性的评估指标，以及构建将组合泛化作为本能的架构。我们提出“下一原始预测”作为新型架构目标——训练目标鼓励习得可复用算法原始要素及其组合语法，从而通过重组生成新解。通过课程学习与自博弈，模型可在跨任务族与跨世界族中发现可复用原始要素与转移模式，实现终身学习与闭包扩展。我们通过物理学、进化论与神经科学中的案例研究对该框架进行实例化验证。