DisCo Algebra: Why Disintegration and Combination Are Not Inverses

Abstract

We present DisCo algebra---a compositional framework for joint probability built from a symmetric cooperad (disintegration) and a magma-operad (non-associative partial combination, restricted to the image of disintegration) 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 on admissible inputs. We prove that combination followed by disintegration is the identity on admissible inputs, but disintegration followed by combination is not---it defines a renormalization operator $R_{\mathcal{I}}=\text{Combine}\circ \text{Disintegrate}$. The information loss of this operator is exactly the probability curvature ${\mathcal{K}}_{\mathcal{I}}(\mu )$---the conditional KL divergence between the true conditional distribution and the conditionally independent product. This curvature measures the obstruction to lossless decomposition and reveals a hard limit on how much cross-block dependence can be captured by CDF-based scalar block summaries.

arXiv comment12 pages