Essay
Version 1
This version is not peer-reviewed
The Code Underneath
Version 1
: Received: 7 November 2024 / Approved: 7 November 2024 / Online: 8 November 2024 (11:59:31 CET)
How to cite: Rives, J. The Code Underneath. Preprints 2024, 2024110594. https://doi.org/10.20944/preprints202411.0594.v1 Rives, J. The Code Underneath. Preprints 2024, 2024110594. https://doi.org/10.20944/preprints202411.0594.v1
Abstract
An inverse-square probability mass function (PMF) is at Newcomb-Benford Law’s (NBL) root and ultimately at the origin of positional notation and conformality. Under its tail, we find information as harmonic likelihood, leading to the global NBL ruled by the base B. Under its tail, we find information as logarithmic likelihood, leading to the local (fiducial) NBL ruled by the radix R. In the framework of bijective numeration, we prove that the set of Kempner’s series conforms to the global NBL and that the local NBL is length- and position-invariant.
The global Bayesian rule multiplies the correlation between numbers, s and t, by a likelihood ratio that is the NBL probability of bucket [s;t) relative to B’s support. To encode the odds of quantum j against i locally, we multiply the prior odds Pr(B;j)/Pr(B;i) by a likelihood ratio that is the NBL probability of bin [i;j) relative to R’s support. This two-factor structure is recurrent under arithmetic operations.
A particular case of Bayesian data produces the algebraic field of "referential ratios". The cross-ratio, the central tool in conformal geometry, is a ratio of referential ratios. A one-dimensional coding source reflects the harmonic external world into its logarithmic coding space, the ball {x∈Q| abs(x)< 1-1/B}. The source’s conformal encoding function is y=logR(2x-1), where x is the observed Euclidean distance to an
object’s position. The conformal decoding function is x = ½ (1 + ry). Both functions, unique under basic requirements, enable information- and granularity-invariant recursion to model the multiscale reality.
Keywords
Keywords Probability Mass Function (PMF), Inverse-Square Law (ISL), Newcomb-Benford Law (NBL), positional Notation (PN), radix economy, harmonic series, trigamma function, likelihood, curious series,
Subject
Computer Science and Mathematics, Other
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Comments (0)
We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.
Leave a public commentSend a private comment to the author(s)
* All users must log in before leaving a comment
