Signalling Entropy Across Measurement Scales: A Compositional Dilution Lemma and Cross-Modality Invariance for Information-Theoretic Analysis of Cancer Transcriptomes.

We develop a unified information-theoretic framework for the analysis of cancer transcriptomic dysregulation across measurement modalities. Three functionals capture distributional, network-aware, and joint-dependence aspects of expression: the Shannon entropy with a Miller-Madow correction, the signalling entropy rate over the protein interaction graph, and the Gaussian total correlation on a principal-component projection. A closed-form algebraic expression yields a linear-time algorithm for the signalling entropy rate. A Compositional Dilution Lemma decomposes bulk entropy into intrinsic and compositional contributions, and a Cross-Modality Invariance Proposition provides an empirically falsifiable null hypothesis. Validation uses 700,202 single cells and 3942 bulk samples across five cancer types. Pan-cancer tumour elevation is significant at p<10-7, and cross-modality testing on 4230 observations does not reject the interaction null at p>0.5. The invariance conclusion is corroborated by cancer-level paired sign-flip permutation, cancer-block bootstrap, and empirical distribution function tests, and the prognostic Cox regressions satisfy proportional-hazards diagnostics with cross-validation concordance of 0.696±0.018. Immune deconvolution against the LM22 signature validates cell-type-specific predictions, partitioning cancers into myeloid-driven and lymphoid-driven classes. Breast cancer Cox regressions instantiate the predicted orthogonality of distributional and network-aware functionals after immune adjustment.