Abstract  

Where does intelligence arise in artificial or natural systems? This work demonstrates that intelligence does not originate from biological complexity or the scale of artificial models, but from an arithmetically necessary architecture of signal processing. An intelligent system requires only four components: a linear projection, a linear reconstruction via the transpose, a nonlinearity, and a recursive loop. This minimal architecture inevitably produces signal completion, the automatic restoration of missing or noisy inputs. As a consequence, internal models, categorical states, and a cyclic transformation between elementary and complex forms emerge — the dynamic core of thinking. Modern AI architectures such as Transformers and State Space Models merely add a third linear mapping for pattern formation, yet follow the same underlying principle. The theory thus explains the operation of both natural brains and artificial systems and shows that intelligence is an unavoidable property of recursive, signal‑completing systems.