DisCo Algebra: Why Disintegration and Combination Are Not Inverses
DisCo algebra creates a framework for joint probability using a symmetric cooperad (disintegration) and an operad (combination). It shows that while combining followed by disintegrating is reversible, disintegrating followed by combining is not. This reveals a fundamental limitation: compression via block summaries loses information about cross-block dependencies, meaning the format cannot perfectly reconstruct arbitrary high-dimensional distributions.