The Expansion-Compression Loop: A Unified Framework for AI-Mediated Cognitive Decoupling

We introduce a unified formal framework for analysing what we term AI-mediated cognitive decoupling: the systematic separation of cognitive process from cognitive
product enabled by large language models (LLMs) operating as expansion and compression agents. We formalise two operators — the expansion operator E and the compression operator C — acting on semantic content, and characterise the composition C ◦ E as a lossy endomorphism on a semantic manifold S. We introduce the Decoupling Proposition, which asserts that under AI mediation the product of a cognitive act is no longer a function of its process, and formalise this as  conditional independence: Prod(A) ⊥ Proc(A) | θAI. We define three strata of semantic fidelity — structural, affective, and propositional — and show they decay at different rates under iterated C ◦ E-transforms. Finally, we introduce the concept of the Indecomposable Core K∗: a subset of cognitive value whose existence is ontologically tied to the process itself, such that C ◦ E(K∗) ⊊ K∗ regardless of model capacity.This framework provides the conceptual scaffolding and mathematical language forthe subsequent papers in the series.