Understanding how complex systems perform computations, i.e. how they store, transmit, and transform information across their components, is a major goal across the natural sciences. However, characterising genuinely collective computations across multiple inputs and outputs that cannot be reduced to contributions from individual components remains an open problem. In this work, we introduce $Φ\text{ID}^{IG}$, a geometric theory of information dynamics that captures irreducible collective phenomena and enables a complete decomposition of multi-input multi-output operations into interpretable computational modes, providing a complete informational profile of a system. We first validate our framework to identify and quantify distinct computational modes in discrete and Gaussian systems with known ground truth. We then apply it to uncover the computational architecture of complex systems across synthetic and real-world scenarios, including chaotic maps, where distinct universality classes are associated with characteristic modes of computation; brain anatomy of over 100 mammalian species, revealing an evolutionary shift towards more distributed computation in the human neocortex; and financial markets, where collective computations recover the modular organisation of market sectors. Overall, our results show how complex systems can be quantitatively investigated through multidimensional computational portraits derived from the geometry of their underlying probability distributions.

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