The Code Underneath

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.