{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T15:13:31Z","timestamp":1783610011905,"version":"3.55.0"},"reference-count":36,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T00:00:00Z","timestamp":1783555200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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\u2013Madow 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&lt;10\u22127, and cross-modality testing on 4230 observations does not reject the interaction null at p&gt;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\u00b10.018. 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.<\/jats:p>","DOI":"10.3390\/e28070781","type":"journal-article","created":{"date-parts":[[2026,7,9]],"date-time":"2026-07-09T14:17:51Z","timestamp":1783606671000},"page":"781","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Signalling Entropy Across Measurement Scales: A Compositional Dilution Lemma and Cross-Modality Invariance for Information-Theoretic Analysis of Cancer Transcriptomes"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7061-2534","authenticated-orcid":false,"given":"\u00d6mer","family":"Akg\u00fcller","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mugla Sitki Kocman University, 48000 Mugla, T\u00fcrkiye"},{"name":"Department of Oncology, Institute of Health Sciences, Dokuz Eylul University, 35340 Izmir, T\u00fcrkiye"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1465-7153","authenticated-orcid":false,"given":"Mehmet Ali","family":"Balc\u0131","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mugla Sitki Kocman University, 48000 Mugla, T\u00fcrkiye"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ceren","family":"U\u00e7mako\u011flu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mugla Sitki Kocman University, 48000 Mugla, T\u00fcrkiye"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7317-4798","authenticated-orcid":false,"given":"Lucian","family":"Gaban","sequence":"additional","affiliation":[{"name":"Faculty of Economics, \u201c1 Decembrie 1918\u201d University of Alba Iulia, 510009 Alba Iulia, Romania"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,7,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1158\/2159-8290.CD-21-1059","article-title":"Hallmarks of cancer: New dimensions","volume":"12","author":"Hanahan","year":"2022","journal-title":"Cancer Discov."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2254","DOI":"10.1016\/j.cell.2025.12.049","article-title":"Hallmarks of cancer\u2014Then and now, and beyond","volume":"189","author":"Hanahan","year":"2026","journal-title":"Cell"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Brochet, T., Lapuyade-Lahorgue, J., Huat, A., Thureau, S., Pasquier, D., Gardin, I., Modzelewski, R., Gibon, D., Thariat, J., and Gr\u00e9goire, V. 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