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 𝒫n=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 Rℐ=Combine∘Disintegrate. The information loss of this operator is exactly the probability curvature 𝒦ℐ(μ)---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.
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BibTeX
@misc{lu2026disco,
title = {{DisCo Algebra: Why Disintegration and Combination Are Not Inverses}},
author = {Xuexing Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15006},
eprint = {2610.15006},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15006},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - DisCo Algebra: Why Disintegration and Combination Are Not Inverses
AU - Xuexing Lu
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.15006
UR - https://arxiv.org/abs/2610.15006
AB - 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 𝒫_n=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_ℐ=Combine∘Disintegrate. The information loss of this operator is exactly the probability curvature 𝒦_ℐ(μ)---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.
ER -