Real-world phenomena do not generate arbitrary variability: their signals concentrate on compact, low-variability subsets of functional space, enabling rapid generalization from few examples. A small child can recognize a dog after extremely limited exposure because the perceptual manifold of "dog" is compact, structured, and low-dimensional. We formalize this principle through a deterministic functional-topological framework in which the set of valid realizations produced by a physical process forms a compact subset of a Banach space, endowed with stable invariants, a finite Hausdorff radius, and an induced continuous perceptual functional. This geometry provides explicit limits on knowledge, conditions for identifiability, and guarantees for generalization from sparse evidence -- properties fundamental to both natural and artificial intelligence. Across electromechanical, electrochemical, and physiological domains, we show that real-world processes consistently generate compact perceptual manifolds with the same geometric characteristics. Their boundaries can be discovered in a fully self-supervised manner as the empirical radius saturates with increasing sampling, even when the governing equations are unknown. These results demonstrate that deterministic functional topology offers a unified mathematical foundation for perception, representation, and world-model construction. It provides a geometric explanation for why biological learners and self-supervised AI systems can generalize from few observations, and establishes compact perceptual manifolds as a fundamental building block for future AI architectures. Finally, this work unifies biological perception and modern self-supervised models under a single geometric principle: both derive their generalization ability from the compactness and invariants of real-world perceptual manifolds.

翻译：现实世界现象不会产生任意变异性：其信号集中于功能空间的紧致、低变异性子集，使得从少量样本快速泛化成为可能。幼儿仅需极有限接触便能识别狗，正是因为“狗”的感知流形具有紧致性、结构性和低维特性。我们通过确定性功能拓扑框架将这一原理形式化：物理过程产生的有效实现集合构成巴拿赫空间的紧致子集，具有稳定不变量、有限豪斯多夫半径及诱导的连续感知泛函。这种几何结构为知识边界提供了明确限制，为可识别性提供了条件，并为稀疏证据下的泛化提供了保证——这些特性对自然智能与人工智能皆至关重要。在机电、电化学及生理学领域中，我们证明现实世界过程持续生成具有相同几何特征的紧致感知流形。即使控制方程未知，随着采样增加，经验半径趋于饱和时，其边界仍能以完全自监督的方式被发现。这些结果表明确定性功能拓扑为感知、表征与世界模型构建提供了统一的数学基础。它从几何角度解释了为何生物学习者和自监督人工智能系统能从少量观察中泛化，并将紧致感知流形确立为未来人工智能架构的基本构建模块。最后，本研究将生物感知与现代自监督模型统一于单一几何原理之下：二者皆从现实世界感知流形的紧致性与不变性中获得泛化能力。