{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:44:19Z","timestamp":1759970659460,"version":"build-2065373602"},"reference-count":87,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,30]],"date-time":"2025-01-30T00:00:00Z","timestamp":1738195200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>An inverse-square probability mass function (PMF) is at the Newcomb\u2013Benford law (NBL)\u2019s root and ultimately at the origin of positional notation and conformality. PrZ=2Z\u22122, where Z\u2208Z+. Under its tail, we find information as harmonic likelihood Ls,t=Ht\u22121\u2212Hs\u22121, where Hn is the nth harmonic number. The global Q-NBL is Prb,q=Lq,q+1L1,b=qHb\u22121\u22121, where b is the base and q is a quantum (1\u2264q&lt;b). Under its tail, we find information as logarithmic likelihood \u2113i,j=lnji. The fiducial R-NBL is Prr,d=\u2113d,d+1\u21131,r=logr1+1d, where r\u226ab is the radix of a local complex system. The global Bayesian rule multiplies the correlation between two numbers, s and t, by a likelihood ratio that is the NBL probability of bucket s,t relative to b\u2019s support. To encode the odds of quantum j against i locally, we multiply the prior odds Prb,jPrb,i by a likelihood ratio, which is the NBL probability of bin i,j relative to r\u2019s support; the local Bayesian coding rule is o\u02dcj:i|r=ijlogrji. The Bayesian rule to recode local data is o\u02dcj:i|r\u2032=o\u02dcj:i|rlnrlnr\u2032. Global and local Bayesian data are elements of the algebraic field of \u201cgap ratios\u201d, A\u2212BC\u2212D. The cross-ratio, the central tool in conformal geometry, is a subclass of gap ratio. A one-dimensional coding source reflects the global Bayesian data of the harmonic external world, the annulus x\u2208Q|1\u2264x&lt;b, into the local Bayesian data of its logarithmic coding space, the ball x\u2208Q|x&lt;1\u22121b. The source\u2019s conformal encoding function is y=logr2x\u22121, where x is the observed Euclidean distance to an object\u2019s position. The conformal decoding function is x=121+ry. Both functions, unique under basic requirements, enable information- and granularity-invariant recursion to model the multiscale reality.<\/jats:p>","DOI":"10.3390\/axioms14020106","type":"journal-article","created":{"date-parts":[[2025,1,30]],"date-time":"2025-01-30T05:40:46Z","timestamp":1738215646000},"page":"106","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The Code Underneath"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3980-5829","authenticated-orcid":false,"given":"Julio","family":"Rives","sequence":"first","affiliation":[{"name":"Department of Artificial Intelligence, Universidad Nacional de Educaci\u00f3n a Distancia (UNED), Paseo Senda del Rey, 7, 28040 Madrid, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"39","DOI":"10.2307\/2369148","article-title":"Note on the Frequency of Use of the Different Digits in Natural Numbers","volume":"4","author":"Newcomb","year":"1881","journal-title":"Am. J. Math."},{"key":"ref_2","first-page":"551","article-title":"The Law of Anomalous Numbers","volume":"78","author":"Benford","year":"1938","journal-title":"Proc. Am. Philos. Soc."},{"unstructured":"H\u00fcrlimann, W. (2006). Benford\u2019s law from 1881 to 2006. arXiv.","key":"ref_3"},{"doi-asserted-by":"crossref","unstructured":"Blair, D.E. (2000). Inversion Theory and Conformal Mapping, American Mathematical Society. Graduate Studies in Mathematics.","key":"ref_4","DOI":"10.1090\/stml\/009"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"952","DOI":"10.1119\/1.16838","article-title":"Benford\u2019s Law and Physical Constants: The Distribution of Initial Digits","volume":"59","author":"Burke","year":"1991","journal-title":"Am. J. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/s00283-010-9182-3","article-title":"Benford\u2019s Law Strikes Back - No Simple Explanation in Sight for Mathematical Gem","volume":"33","author":"Berger","year":"2011","journal-title":"Math. Intell."},{"key":"ref_7","first-page":"132","article-title":"What is Benford\u2019s Law?","volume":"64","author":"Berger","year":"2017","journal-title":"Not. AMS"},{"unstructured":"Hossenfelder, S. (2018). Lost in Math: How Beauty Leads Physics Astray, Basic Books.","key":"ref_8"},{"unstructured":"Finch, S.R. (2018). Mathematical Constants II, Cambridge University Press. Encyclopedia of Mathematics and Its Applications.","key":"ref_9"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s40766-020-00004-5","article-title":"A Mini-introduction to Information Theory","volume":"43","author":"Witten","year":"2020","journal-title":"Riv. Del Nuovo C."},{"key":"ref_11","first-page":"370","article-title":"LII. An Essay Towards Solving a Problem in the Doctrine of Chances. By the late Rev. Mr. Bayes, F. R. S. Communicated by Mr. Price, in a Letter to John Canton, A. M. F. R. S","volume":"53","author":"Bayes","year":"1763","journal-title":"Philos. Trans. R. Soc. Lond."},{"unstructured":"Oldershaw, R.L. (2017). Nature is Startling\u2014Conformal Symmetry. Medium.","key":"ref_12"},{"doi-asserted-by":"crossref","unstructured":"Formann, A.K. (2010). The Newcomb-Benford Law in Its Relation to Some Common Distributions. PLoS ONE, 5.","key":"ref_13","DOI":"10.1371\/journal.pone.0010541"},{"unstructured":"Ariew, R. (1976). Ockham\u2019s Razor: A Historical and Philosophical Analysis of Ockham\u2019s Principle of Parsimony, University of Illnois at Urbana-Champaign.","key":"ref_14"},{"unstructured":"Planck Collaboration, Aghanim, N., Akrami, Y., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., Banday, A.J., Barreiro, R.B., and Bartolo, N. (2020). Planck 2018 results\u2014VI. Cosmological parameters. Astron. Astrophys., 641, A6.","key":"ref_15"},{"unstructured":"Helmenstine, A., and Composition of the Universe\u2014Element Abundance (2024, October 30). Science Notes. Available online: https:\/\/sciencenotes.org\/composition-of-the-universe-element-abundance\/.","key":"ref_16"},{"doi-asserted-by":"crossref","unstructured":"Mo, H., van den Bosch, F., and White, S. (2010). Galaxy Formation and Evolution, Cambridge University Press. Galaxy Formation and Evolution.","key":"ref_17","DOI":"10.1017\/CBO9780511807244"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1198\/tast.2009.0005","article-title":"A Simple Explanation of Benford\u2019s Law","volume":"63","author":"Fewster","year":"2009","journal-title":"Am. Stat."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1007\/s10701-007-9186-9","article-title":"The Mathematical Universe","volume":"38","author":"Tegmark","year":"2008","journal-title":"Found. Phys."},{"key":"ref_20","first-page":"1","article-title":"The Modular Group","volume":"20","author":"McCreary","year":"2018","journal-title":"Math. J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1080\/00107510500052444","article-title":"Power laws, Pareto distributions and Zipf\u2019s law","volume":"46","author":"Newman","year":"2005","journal-title":"Contemp. Phys."},{"unstructured":"Berger, A., Hill, T.P., and Rogers, E. (2023, February 21). Benford Online Bibliography. Available online: http:\/\/www.benfordonline.net.","key":"ref_22"},{"unstructured":"Miller, S.J. (2015). Benford\u2019s Law, Princeton University Press.","key":"ref_23"},{"unstructured":"Berger, A., and Hill, T.P. (2015). An Introduction to Benford\u2019s Law, Princeton University Press.","key":"ref_24"},{"unstructured":"de Finetti, B. (1972). Probability, Induction, and Statistics: The Art of Guessing, John Wiley.","key":"ref_25"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1214\/ss\/1177009869","article-title":"A Statistical Derivation of the Significant-Digit Law","volume":"10","author":"Hill","year":"1995","journal-title":"Stat. Sci."},{"unstructured":"Havil, J. (2003). Gamma\u2014Exploring Euler\u2019s Constant, Princeton University Press.","key":"ref_27"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1080\/00029890.1995.11990578","article-title":"The Significant-Digit Phenomenon","volume":"102","author":"Hill","year":"1995","journal-title":"Am. Math. Mon."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1511\/1998.31.358","article-title":"The First Digit Phenomenon","volume":"86","author":"Hill","year":"1998","journal-title":"Am. Sci."},{"unstructured":"Tao, T. (2024, October 30). Benford\u2019s Law, Zipf\u2019s Law, and the Pareto Distribution. Terrytao\u2019s Blog. Available online: https:\/\/terrytao.wordpress.com\/2009\/07\/03\/benfords-law-zipfs-law-and-the-pareto-distribution\/.","key":"ref_30"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1093\/biomet\/42.3-4.425","article-title":"On a Class of Skew Distribution Functions","volume":"42","author":"Simon","year":"1955","journal-title":"Biometrika"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"123129","DOI":"10.1016\/j.physa.2019.123129","article-title":"Several Common Probability Distributions Obey Benford\u2019s Law","volume":"540","author":"Fang","year":"2020","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"779","DOI":"10.1007\/s10260-020-00532-8","article-title":"The Mathematics of Benford\u2019s Law: A Primer","volume":"30","author":"Berger","year":"2021","journal-title":"Stat. Methods Appl."},{"unstructured":"Gonsalves, R.A. (2020). Benford\u2019s Law\u2014A Simple Explanation. Medium.","key":"ref_34"},{"unstructured":"(2021). DK. A Treatise On The Newcomb-Benford Law. Medium.","key":"ref_35"},{"unstructured":"(2022, April 01). Wikipedia Contributors, the Free Encyclopedia. Zipf\u2019s Law, 2022, Available online: https:\/\/simple.wikipedia.org\/wiki\/Zipf%27s_law.","key":"ref_36"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.shpsa.2020.08.006","article-title":"Structure, Scale and Emergence","volume":"85","author":"Hendry","year":"2021","journal-title":"Stud. Hist. Philos. Sci. Part A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1093\/bjps\/31.2.131","article-title":"Bayesian Conditionalisation and the Principle of Minimum Information","volume":"31","author":"Williams","year":"1980","journal-title":"Br. J. Philos. Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1016\/S0378-4371(01)00355-7","article-title":"Beyond the Zipf-Mandelbrot Law in Quantitative Linguistics","volume":"300","author":"Montemurro","year":"2001","journal-title":"Phys. A Stat. Mech. Its Appl."},{"unstructured":"(2022, April 04). Wikipedia Contributors, the Free Encyclopedia. Zeta Distribution, 2021, Available online: https:\/\/en.wikipedia.org\/wiki\/Zeta_distribution.","key":"ref_40"},{"unstructured":"Finch, S.R. (2003). Mathematical Constants, Cambridge University Press. Encyclopedia of Mathematics and Its Applications.","key":"ref_41"},{"doi-asserted-by":"crossref","unstructured":"Held, L., and Bov\u00e9, D.S. (2013). Applied Statistical Inference: Likelihood and Bayes, Springer.","key":"ref_42","DOI":"10.1007\/978-3-642-37887-4"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1103\/PhysRevLett.59.381","article-title":"Self-organized criticality: An explanation of the 1\/f noise","volume":"59","author":"Bak","year":"1987","journal-title":"Phys. Rev. Lett."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1007\/s002490100143","article-title":"Pink-noise Behaviour of Biosystems","volume":"30","author":"Szendro","year":"2001","journal-title":"Eur. Biophys. J."},{"unstructured":"Mattey, G.J. (1997). Lecture Notes on \u201cCritique of Pure Reason\u201d: Appearance and Thing in Itself, UC Davis Philosophy Department. Philosophy 175 Home Page.","key":"ref_45"},{"unstructured":"Stone, J.V. (2015). Information Theory: A Tutorial Introduction.","key":"ref_46"},{"unstructured":"Fechner, G.T., Boring, E.G., Adler, H.E., and Howes, D.H. (1966). Elements of Psychophysics, Holt, Rinehart and Winston. Henry Holt Edition in Psychology.","key":"ref_47"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1016\/S0167-7152(97)00142-9","article-title":"On Benford\u2019s Law to Variable Base","volume":"37","author":"Schatte","year":"1998","journal-title":"Stat. Probab. Lett."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/978-94-017-1586-7_6","article-title":"The Logic of Bayesian Probability","volume":"Volume 24","author":"Corfield","year":"2001","journal-title":"Foundations of Bayesianism"},{"unstructured":"Zalta, E.N. (2019). Bayes\u2019 Theorem. The Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University. [Spring ed.].","key":"ref_50"},{"unstructured":"Walker, S., and Cronin, L. (2024, October 30). Time Is an Object. Aeon. Available online: https:\/\/www.oasis.cnrs.fr\/news\/aeon-time-is-an-object.","key":"ref_51"},{"unstructured":"Wikipedia Contributors, the Free Encyclopedia (2021, June 04). Odds Algorithm, 2021. Available online: https:\/\/en.wikipedia.org\/wiki\/Odds_algorithm.","key":"ref_52"},{"doi-asserted-by":"crossref","unstructured":"Girdhar, Y., and Dudek, G. (2009, January 25\u201327). Optimal Online Data Sampling or How to Hire the Best Secretaries. Proceedings of the Canadian Conference on Computer and Robot Vision, Kelowna, BC, Canada.","key":"ref_53","DOI":"10.1109\/CRV.2009.30"},{"unstructured":"(2024, December 12). Wikipedia Contributors, the Free Encyclopedia. Golomb Coding, 2024, Available online: https:\/\/en.wikipedia.org\/wiki\/Golomb_coding.","key":"ref_54"},{"unstructured":"Milne, J.J. (1911). An Elementary Treatise on Cross-Ratio Geometry: With Historical Notes, The University Press.","key":"ref_55"},{"unstructured":"(2024, February 04). Wikipedia Contributors, the Free Encyclopedia. Doppler Effect, 2024, Available online: https:\/\/en.wikipedia.org\/wiki\/Doppler_effect.","key":"ref_56"},{"unstructured":"Tipler, P.A., and Llewellyn, R. (2012). Modern Physics, W. H. Freeman.","key":"ref_57"},{"unstructured":"Krakauer, D. (2024, October 30). Complexity, Agency, and Information. Sean Carroll\u2019s Mindscape. Available online: https:\/\/www.preposterousuniverse.com\/podcast\/2023\/07\/10\/242-david-krakauer-on-complexity-agency-and-information\/.","key":"ref_58"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/0024-3795(84)90131-9","article-title":"Linear fractional transformations in rings and modules","volume":"56","author":"Young","year":"1984","journal-title":"Linear Algebra Its Appl."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1080\/00029890.2001.11919727","article-title":"Modular Miracles","volume":"108","author":"Stillwell","year":"2001","journal-title":"Am. Math. Mon."},{"unstructured":"Poincar\u00e9, H. (2016). Science and Hypothesis (1905), Read Books Limited.","key":"ref_61"},{"unstructured":"(2023, January 20). Wikipedia Contributors, the Free Encyclopedia. Poincar\u00e9 Disk Model, 2023. Available online: https:\/\/en.wikipedia.org\/wiki\/Poincar%C3%A9_disk_model.","key":"ref_62"},{"unstructured":"Wikipedia Contributors, the Free Encyclopedia (2024, January 29). Real Projective Line, 2023. Available online: https:\/\/en.wikipedia.org\/wiki\/Real_projective_line.","key":"ref_63"},{"key":"ref_64","first-page":"47","article-title":"Sobre Funciones de Negacion en la Teoria de Conjuntos Difusos","volume":"3","author":"Trillas","year":"1979","journal-title":"Stochastica"},{"doi-asserted-by":"crossref","unstructured":"Abramowitz, M., and Stegun, I.A. (1965). Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Dover Publications.","key":"ref_65","DOI":"10.1063\/1.3047921"},{"doi-asserted-by":"crossref","unstructured":"Richeson, D.S. (2019). Euler\u2019s Gem: The Polyhedron Formula and the Birth of Topology, Princeton University Press. Princeton Science Library.","key":"ref_66","DOI":"10.1515\/9780691191997"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1080\/00029890.1936.11987804","article-title":"Restricted Theory of Relativity in Terms of Hyperbolic Functions of Rapidities","volume":"43","author":"Karapetoff","year":"1936","journal-title":"Am. Math. Mon."},{"unstructured":"Russell, N. (2024, October 30). Framing Lorentz Symmetry. CERN Courier. Available online: https:\/\/cerncourier.com\/a\/framing-lorentz-symmetry\/.","key":"ref_68"},{"unstructured":"Ford, K.W. (1963). Information Theory and Statistical Mechanics. Statistical Physics, Benjamin.","key":"ref_69"},{"unstructured":"Wikipedia Contributors, the Free Encyclopedia (2024, September 10). Differential Entropy, 2024. Available online: https:\/\/en.wikipedia.org\/wiki\/Differential_entropy.","key":"ref_70"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1007\/BF02757029","article-title":"Black Holes and the Second Law","volume":"4","author":"Bekenstein","year":"1972","journal-title":"Lett. Nuovo C. (1971\u20131985)"},{"unstructured":"Wikipedia Contributors, the Free Encyclopedia (2024, January 31). Tetration, 2024. Available online: https:\/\/en.wikipedia.org\/wiki\/Tetration.","key":"ref_72"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"e1","DOI":"10.1017\/dsj.2016.16","article-title":"Model Granularity in Engineering Design, Concepts and Framework","volume":"3","author":"Maier","year":"2017","journal-title":"Des. Sci."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"062149","DOI":"10.1103\/PhysRevE.102.062149","article-title":"Coarse Graining of Biochemical Systems Described by Discrete Stochastic Dynamics","volume":"102","author":"Seiferth","year":"2020","journal-title":"Phys. Rev. E"},{"unstructured":"Weinan, E. (2011). Principles of Multiscale Modeling, Cambridge University Press.","key":"ref_75"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1145\/62.322429","article-title":"Beyond Floating Point","volume":"31","author":"Clenshaw","year":"1984","journal-title":"J. Assoc. Comput. Mach."},{"unstructured":"Peltzer, A.R. (2019). The Riemann Zeta Distribution. [Ph.D. Thesis, University of California].","key":"ref_77"},{"unstructured":"Mitzenmacher, M., and Upfal, E. (2017). Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis, Cambridge University Press.","key":"ref_78"},{"unstructured":"Ken-Iti, S. (1999). L\u00e9vy Processes and Infinitely Divisible Distributions, Cambridge University Press. Cambridge Studies in Advanced Mathematics.","key":"ref_79"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1080\/0025570X.1988.11977350","article-title":"The Ubiquitous Pi","volume":"61","author":"Castellanos","year":"1988","journal-title":"Math. Mag."},{"unstructured":"Billingsley, P. (1995). Probability and Measure, Wiley.","key":"ref_81"},{"unstructured":"Mathar, R.J. (2011). Survey of Dirichlet Series of Multiplicative Arithmetic Functions. arXiv.","key":"ref_82"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/0001-8708(70)90029-0","article-title":"Bernoulli Shifts with the Same Entropy Are Isomorphic","volume":"4","author":"Ornstein","year":"1970","journal-title":"Adv. Math."},{"unstructured":"Peterson, P., Baker, E., and McGaw, B. (2010). Discrete Probability Distributions. International Encyclopedia of Education, Elsevier. [3rd ed.].","key":"ref_84"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"123661","DOI":"10.1016\/j.jmaa.2019.123661","article-title":"Four Parametric Linear Euler Sums","volume":"484","author":"Alzer","year":"2020","journal-title":"J. Math. Anal. Appl."},{"doi-asserted-by":"crossref","unstructured":"Grove, L.C., and Benson, C.T. (1985). Finite Groups in Two and Three Dimensions. Finite Reflection Groups, Springer.","key":"ref_86","DOI":"10.1007\/978-1-4757-1869-0"},{"key":"ref_87","first-page":"1124","article-title":"Magic, Mystery, and Matrix","volume":"45","author":"Witten","year":"1998","journal-title":"N. Am. Math. Soc."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/106\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:38:45Z","timestamp":1759919925000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/2\/106"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,30]]},"references-count":87,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["axioms14020106"],"URL":"https:\/\/doi.org\/10.3390\/axioms14020106","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,1,30]]}}}