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arXiv · 2610.15006

DisCo Algebra: Why Disintegration and Combination Are Not Inverses

Xuexing Lu
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Abstract

We present DisCo algebra---a compositional framework for joint probability built from a symmetric cooperad (disintegration) and a symmetric operad (combination) on the bare family $\mathcal{P}_n=\mathrm{Prob}([0,1]^n)$. No probability integral transform, no external summary functions, no enrichment of objects with conditional kernels. The disintegration side uses block-wise joint CDFs as intrinsic, measure-determined 1D summaries, yielding a lossy compression. The combination side reads the same summaries and reconstructs via regular conditional measures, yielding a lossless embedding. We prove that combination followed by disintegration is the identity on admissible inputs, but disintegration followed by combination is not---the information loss is exactly the conditional KL divergence between the true conditional distribution and the conditionally independent product. This asymmetry reveals a hard limit on how much cross-block dependence can be captured by scalar block summaries.

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Xuexing Lu. 2026-10-10. DisCo Algebra: Why Disintegration and Combination Are Not Inverses. https://arxiv.org/abs/2610.15006

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