Why phase?

A radio receiver does not record a single number per instant. It records two, in-phase and quadrature, which together form a complex number: an amplitude and a phase. Radar returns, MRI k-space measurements, coherent optical fields and ultrasound echoes are the same. In all of them the phase carries information: a frequency offset, a modulation, a delay, a velocity, a direction.

Standard deep learning is real-valued. Given complex data, it splits each sample into two unrelated real numbers and must learn, from data, how complex numbers multiply and how phase behaves. It can learn this, at a cost in data and model size, and with no guarantee that the structure is kept.

Computing in complex numbers

CNet, our open-source framework, keeps everything complex: weights, activations, attention, and the gradients that train them, which use the Wirtinger calculus. Its attention layer, Born attention, scores tokens with the squared magnitude of a complex inner product, the same rule quantum mechanics uses to turn amplitudes into measurement probabilities. A useful consequence is mathematical: that score has an exact finite feature map, so the attention can run as a recurrence with a fixed-size state. The model can then read a stream for as long as it runs without its memory growing.

An underexplored space

Complex-valued networks have been studied for decades, but the tooling is thin, and few complex models have been trained at the scale of modern language models. Our first results suggest the space is worth exploring: even on text, where phase carries nothing, an untuned complex model gets close to a mature Transformer recipe. Signals where phase does carry the information are where we expect the approach to matter most. We are starting with radio.