{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:55:05Z","timestamp":1760151305285,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T00:00:00Z","timestamp":1646956800000},"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>In this paper, we focus on some aspects of the relation of spacetime and quantum mechanics and the study counterparts (in Set) of the categorical local symmetries of smooth 4-manifolds. In the set-theoretic limit, there emerge some exotic smoothness structures on R4 (hence the Riemannian nonvanishing curvature), which fit well with the quantum mechanical lattice of projections on infinite-dimensional Hilbert spaces. The method we follow is formalization localized on the open covers of the spacetime manifold. We discuss our findings in the context of the information paradox assigned to evaporating black holes. A black hole can evaporate entirely, but the smoothness structure of spacetime will be altered and, in this way, the missing information about the initial states of matter forming the black hole will be encoded. Thus, the possible global geometric remnant of black holes in spacetime is recognized as exotic 4-smoothness. The full-fledged verification of this proposal will presumably be possible within the scope of future quantum gravity theory research.<\/jats:p>","DOI":"10.3390\/e24030391","type":"journal-article","created":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T12:58:36Z","timestamp":1647003516000},"page":"391","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Categorical Smoothness of 4-Manifolds from Quantum Symmetries and the Information Loss Paradox"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7296-7355","authenticated-orcid":false,"given":"Jerzy","family":"Kr\u00f3l","sequence":"first","affiliation":[{"name":"Chair of Cognitive Science and Mathematical Modeling, University of Information Technology and Management, ul. Sucharskiego 2, 35-225 Rzeszw, Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4813-1010","authenticated-orcid":false,"given":"Torsten","family":"Asselmeyer-Maluga","sequence":"additional","affiliation":[{"name":"German Aerospace Center (DLR), Rosa-Luxemburg-Str. 2, D-10178 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,3,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"245010","DOI":"10.1088\/1361-6382\/aa945b","article-title":"Exotica and the status of the strong cosmic censor conjecture in four dimensions","volume":"34","author":"Etesi","year":"2017","journal-title":"Class. Quantum Gravity"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"104164","DOI":"10.1016\/j.geomphys.2021.104164","article-title":"Global solvability of the vacuum Einstein equation and the strong cosmic censor conjecture in four dimensions","volume":"164","author":"Etesi","year":"2021","journal-title":"J. Geom. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.dark.2017.12.002","article-title":"How to obtain a cosmological constant from small exotic R4","volume":"19","year":"2018","journal-title":"Phys. Dark Universe"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1785","DOI":"10.1088\/0264-9381\/11\/7\/015","article-title":"Localized exotic smoothness","volume":"11","author":"Brans","year":"1994","journal-title":"Class. Quantum Gravity"},{"doi-asserted-by":"crossref","unstructured":"Asselmeyer-Maluga, T. (2016). Smooth quantum gravity: Exotic smoothness and quantum gravity. At the Frontier of Spacetime, Springer. Fundamental Theories of Physics Vol 183.","key":"ref_5","DOI":"10.1007\/978-3-319-31299-6"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1007\/s10699-019-09646-z","article-title":"Quantum mechanics, formalization and the cosmological constant problem","volume":"25","year":"2020","journal-title":"Found. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1007\/s10699-019-09625-4","article-title":"On a possibly pure set-theoretic contribution to black hole entropy","volume":"25","author":"Etesi","year":"2020","journal-title":"Found. Sci."},{"doi-asserted-by":"crossref","unstructured":"Asselmeyer-Maluga, T., and Kr\u00f3l, J. (2019). Local external\/internal symmetry of smooth manifolds and lack of tovariance in physics. Symmetry, 11.","key":"ref_8","DOI":"10.3390\/sym11121429"},{"doi-asserted-by":"crossref","unstructured":"Kr\u00f3l, J., and Klimasara, P. (2020). Black holes and complexity via constructible universe. Universe, 6.","key":"ref_9","DOI":"10.3390\/universe6110198"},{"doi-asserted-by":"crossref","unstructured":"Kr\u00f3l, J., Bielas, K., and Asselmeyer-Maluga, T. (2022). Random World and Quantum Mechanics, under review.","key":"ref_10","DOI":"10.21203\/rs.3.rs-200271\/v3"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1023\/B:FOOP.0000019620.04821.a2","article-title":"Background Independence in Quantum Gravity and Forcing Constructions","volume":"34","year":"2004","journal-title":"Found. Phys."},{"doi-asserted-by":"crossref","unstructured":"Kr\u00f3l, J., Asselmeyer-Maluga, T., Bielas, K., and Klimasara, P. (2017). From quantum to cosmological regime. The role of forcing and exotic 4-smoothness. Universe, 3.","key":"ref_12","DOI":"10.3390\/universe3020031"},{"doi-asserted-by":"crossref","unstructured":"Moerdijk, I., and Reyes, G.E. (1991). Models for Smooth Infinitesimal Analysis, Springer Science + Business Media.","key":"ref_13","DOI":"10.1007\/978-1-4757-4143-8"},{"doi-asserted-by":"crossref","unstructured":"Gompf, R., and Stipsicz, A. (1999). 4-Manifolds and Kirby Calculus.","key":"ref_14","DOI":"10.1090\/gsm\/020"},{"unstructured":"Hamkins, D.J., and Seabold, D.E. (2012). Well-founded Boolean ultrapowers as large cardinal embeddings. arXiv.","key":"ref_15"},{"key":"ref_16","first-page":"267","article-title":"On \u2207-model of set theory","volume":"13","year":"1965","journal-title":"Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys."},{"doi-asserted-by":"crossref","unstructured":"Bell, J.L. (2005). Set Theory. Boolean-Valued Models and Independence Proofs, Clarendon Press. [3rd ed.]. Oxford Logic Guides: 47.","key":"ref_17","DOI":"10.1093\/acprof:oso\/9780198568520.003.0008"},{"unstructured":"Viale, M., Audrito, G., and Steila, S. (2021, June 16). Iterated Forcing, Category Forcings, Generic Ul- Trapowers, Generic Absoluteness. Available online: http:\/\/www.logicatorino.altervista.org\/matteo_viale\/book.pdf.","key":"ref_18"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/0003-4843(71)90017-9","article-title":"The theory of Boolean ultrapowers","volume":"2","author":"Mansfield","year":"1971","journal-title":"Ann. Math. Logic."},{"unstructured":"Jech, T. (2003). Set Theory, the Third Millenium Edition, Springer. Springer Monographs in Mathematics.","key":"ref_20"},{"doi-asserted-by":"crossref","unstructured":"Kadison, R.V., and Ringrose, J.R. (1997). Fundamentals of the Theory of Operator Algebras II.","key":"ref_21","DOI":"10.1090\/gsm\/016"},{"unstructured":"De Groote, H.F. (2005). Observables I: Stone spectra. arXiv.","key":"ref_22"},{"unstructured":"Goldblatt, R. (1988). Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, Springer. Graduate Texts in Mathematics 188.","key":"ref_23"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2015.10.007","article-title":"Black hole remnants and the information loss paradox","volume":"603","author":"Chen","year":"2015","journal-title":"Phys. Rep."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"092002","DOI":"10.1088\/1361-6633\/aa778e","article-title":"Information loss","volume":"80","author":"Unruh","year":"2017","journal-title":"Rep. Prog. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1007\/s10714-015-1872-x","article-title":"How to include fermions into general relativity by exotic smoothness","volume":"47","author":"Brans","year":"2015","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2825","DOI":"10.1007\/s10714-012-1419-3","article-title":"On the geometrization of matter by exotic smoothness","volume":"44","year":"2012","journal-title":"Gen. Relativ. Gravit."},{"doi-asserted-by":"crossref","unstructured":"Asselmeyer-Maluga, T., and Kr\u00f3l, J. (2014, January 14\u201319). Higgs boson, magnetic monopoles and exotic smoothness in 4D. Proceedings of the 12th Asia Pacific Physics Conference (APPC12), Makuhari, Japan.","key":"ref_28","DOI":"10.7566\/JPSCP.1.013019"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1950097","DOI":"10.1142\/S0217732319500974","article-title":"A topological approach to Neutrino masses by using exotic smoothness","volume":"34","year":"2019","journal-title":"Mod. Phys. Lett. A"},{"doi-asserted-by":"crossref","unstructured":"Asselmeyer-Maluga, T., Kr\u00f3l, J., and Miller, T. (2020). Does our universe prefer exotic smoothness?. Symmetry, 12.","key":"ref_30","DOI":"10.3390\/sym12010098"},{"doi-asserted-by":"crossref","unstructured":"Ellis, G.F.R., Maartens, R., and Maccallum, M.A.H. (2012). Relativistic Cosmology, Cambridge University Press.","key":"ref_31","DOI":"10.1017\/CBO9781139014403"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1767","DOI":"10.1023\/A:1020788611031","article-title":"Letter: Cosmological anomalies and exotic smoothness structures","volume":"34","author":"Brans","year":"2002","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1472","DOI":"10.1109\/TMI.2007.899173","article-title":"Clinical DT-MRI estimation, smoothing, and fiber tracking with log-Euclidean metrics","volume":"26","author":"Fillard","year":"2007","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"171","DOI":"10.4171\/qt\/5","article-title":"Man and machine thinking about the smooth 4-dimensional Poincar\u00e9 conjecture","volume":"1","author":"Freedman","year":"2010","journal-title":"Quantum Topol."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/3\/391\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:34:59Z","timestamp":1760135699000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/3\/391"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,11]]},"references-count":34,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["e24030391"],"URL":"https:\/\/doi.org\/10.3390\/e24030391","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,3,11]]}}}