Preprint posted on Preprints.org.
ISSN/ISBN: Not available at this time. DOI: 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. Pr (Z) = (2Z)−2, where Z ∈ Z+. Under its tail, we find information as harmonic likelihood L ([s, t)) = Ht−1 − Hs−1, where Hn is the nth harmonic number. The global Q-NBL is Pr (b, q) = L([q,q+1))/L([1,b)) = (qHb−1)−1, where b is the base, and q is a quantum (1 ≤ q < b). Under its tail, we find information as logarithmic likelihood ℓ ([i, j)) = ln j/i. The fiducial R-NBL is Pr (r, d) = ℓ([d,d+1))/ℓ([1,r)) = logr (1 + 1/d), where r < b is the radix of a local complex system. 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 two numbers, s and t, by a likelihood ratio that is the NBL probability of bucket [s, t) relative to b’s support. The local Bayesian rule is ˜o (j : i|r) = i/j logr j/i. 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, which is the NBL probability of bin [i, j) relative to r’s support. This two-factor structure is recurrent under arithmetic operations. The Bayesian rule to recode local data is ˜o (j : i|r′) = ˜o (j : i|r) ln r/ln r′. A particular case of Bayesian data produces the algebraic field of "referential ratios", A−B A−C . 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, the annulus {x ∈ Q| 1 ≤ |x| < b}, into its logarithmic coding space, the ball {x ∈ Q| |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.
Bibtex:
@misc{,
doi = {10.20944/preprints202411.0594.v1},
url = {https://doi.org/10.20944/preprints202411.0594.v1},
year = 2024,
month = {November},
publisher = {Preprints.org},
author = {Julio Rives},
title = {The Code Underneath},
}
Reference Type: Preprint
Subject Area(s): Number Theory, Statistics