{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,24]],"date-time":"2025-10-24T07:48:03Z","timestamp":1761292083056,"version":"build-2065373602"},"reference-count":10,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,10,24]],"date-time":"2025-10-24T00:00:00Z","timestamp":1761264000000},"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 present a concise and self-contained extension of the Finite Ring Continuum (FRC) program, showing that symmetry-complete prime shells Fp with p=4t+1 exhibit a fundamental Euclidean-Lorentzian dichotomy. A genuine Lorentzian quadratic form cannot be realized within a single space-like prime shell Fp, since to split time from space one requires a time coefficient c2 in the nonsquare class of Fp\u00d7, but then c\u2209Fp. An explicit finite-field Lorentz transformation is subsequently derived that preserves the Minkowski form and generates a finite orthogonal group O(Q\u03bd,Fp2) of split type (Witt index 1). These results demonstrate that the essential algebraic features of special relativity\u2014the invariant interval and Lorentz symmetry\u2014emerge naturally within finite-field arithmetic, thereby establishing an intrinsic relativistic algebra within FRC. Finally, this dichotomy implies the algebraic origin of causality: Euclidean invariants reside within a space-like shell Fp, while Lorentzian structure and causal separation arise in its quadratic spacetime extension Fp2.<\/jats:p>","DOI":"10.3390\/e27111098","type":"journal-article","created":{"date-parts":[[2025,10,24]],"date-time":"2025-10-24T07:41:35Z","timestamp":1761291695000},"page":"1098","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Euclidean-Lorentzian Dichotomy and Algebraic Causality in Finite Ring Continuum"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-3836-8091","authenticated-orcid":false,"given":"Yosef","family":"Akhtman","sequence":"first","affiliation":[{"name":"Gamma Earth S\u00e0rl, 1162 Morges, Switzerland"},{"name":"Faculty of Space Technologies, AGH University of Krakow, 30-059 Krakow, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Akhtman, Y. (2025). Relativistic Algebra over Finite Ring Continuum. Axioms, 14.","DOI":"10.20944\/preprints202505.2118.v6"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Lev, F. (2020). Finite Mathematics as the Foundation of Classical Mathematics and Quantum Theory, Springer International Publishing.","DOI":"10.1007\/978-3-030-61101-9"},{"key":"ref_3","unstructured":"Smolin, L. (2013). Time Reborn: From the Crisis in Physics to the Future of the Universe, Houghton Mifflin Harcourt."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hawking, S.W., and Ellis, G.F.R. (1973). The Large Scale Structure of Space-Time. Cambridge Monographs on Mathematical Physics, Cambridge University Press.","DOI":"10.1017\/CBO9780511524646"},{"key":"ref_5","unstructured":"Lidl, R., and Niederreiter, H. (1997). Finite Fields. Encyclopedia of Mathematics and Its Applications, Cambridge University Press. [2nd ed.]."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Lam, T.Y. (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics, American Mathematical Society.","DOI":"10.1090\/gsm\/067"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1080\/00411457108231446","article-title":"Invariant variation problems","volume":"1","author":"Noether","year":"1971","journal-title":"Transp. Theory Stat. Phys."},{"key":"ref_8","unstructured":"Taylor, D.E. (1992). The Geometry of the Classical Groups. Sigma Series in Pure Mathematics, Heldermann Verlag."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Grove, L.C. (2002). Classical Groups and Geometric Algebra. Graduate Studies in Mathematics, American Mathematical Society.","DOI":"10.1090\/gsm\/039"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Akhtman, Y. (2025). Schr\u00f6dinger-Dirac Formalism in Finite Ring Continuum. Preprints.","DOI":"10.20944\/preprints202510.1486.v1"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/11\/1098\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,24]],"date-time":"2025-10-24T07:43:32Z","timestamp":1761291812000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/11\/1098"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,24]]},"references-count":10,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["e27111098"],"URL":"https:\/\/doi.org\/10.3390\/e27111098","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,24]]}}}