Semantic Entropy and Structural Invariance in LLM-Mediated ExpansionCompression Loops

We develop a quantitative information-theoretic account of semantic decay in large-language-model (LLM) mediated Expansion--Compression (EC) loops.Building on the unified framework of DECO Paper~0 \citep{deco_paper0},we introduce semantic entropy $\HS(X)$ as the differential entropy of a random variable distributed over a semantic manifold, and prove that each application of the EC-transform $T = \C \circ \E$ is a strictly entropy-reducing operation in expectation (Semantic Entropy Collapse
Theorem).

We derive closed-form bounds on the mutual information $I\!\left(K_A;\,\hat{K}_B\right)$ between the originating ideas $K_A$ and the recipient's extraction $\hat{K}_B$ as functions of expansion ratio $\rho_E$, compression ratio $\rho_C$, LLM temperature $\tau$, and iteration count $n$. We introduce the \emph{Semantic Gravity Well} model: a potential landscape on the semantic manifold in which the LLM training centroid acts as an attractor, and original content resides at unstable high-curvature saddle points.

Under this model, we prove the Differential Decay Theorem: propositional fidelity $F_{\mathrm{prop}}$ decays exponentially in $n$, affective fidelity $F_{\mathrm{aff}}$ decays sub-exponentially, and structural fidelity $F_{\mathrm{str}}$ converges to a strictly positive constant (the causal skeleton invariant) under mild conditions.Finally, we characterise the Semantic Channel Capacity of the EC-loop and show it is strictly less than the Shannon capacity of the raw linguistic channel, with the gap determined by the curvature of the semantic manifold at the seed point. These results quantify the information-theoretic cost of AI-mediated cognitive decoupling and provide empirically testable predictions for the subsequent papers in the series.