The Code Underneath

An inverse-square probability mass function (PMF) is at the Newcomb–Benford law (NBL)’s root and ultimately at the origin of positional notation and conformality. $Pr\left(Z\right)={\left(2Z\right)}^{-2}$, where $Z\in {\mathbb{Z}}^{+}$. Under its tail, we find information as harmonic likelihood $\mathcal{L}\left(\left[s,t\right)\right)=H_{t-1}-H_{s-1}$, where $H_n$ is the nth harmonic number. The global $\mathbb{Q}$-NBL is $Pr\left(b,q\right)={\displaystyle \raisebox{1ex}{$\mathcal{L}\left(\left[q,q+1\right)\right)$}\!\left/ \!\raisebox{-1ex}{$\mathcal{L}\left(\left[1,b\right)\right)$}\right.}={\left(q{H}_{b-1}\right)}^{-1}$, where b is the base and q is a quantum ($1\le q<b$). Under its tail, we find information as logarithmic likelihood $\ell \left(\left[i,j\right)\right)=ln{\displaystyle \raisebox{1ex}{$j$}\!\left/ \!\raisebox{-1ex}{$i$}\right.}$. The fiducial $\mathbb{R}$-NBL is $Pr\left(r,d\right)={\displaystyle \raisebox{1ex}{$\ell \left(\left[d,d+1\right)\right)$}\!\left/ \!\raisebox{-1ex}{$\ell \left(\left[1,r\right)\right)$}\right.}={log}_r\left(1+{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$d$}\right.}\right)$, where $r\ll b$ 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 $\left[s,t\right)$ relative to b’s support. To encode the odds of quantum j against i locally, we multiply the prior odds ${\displaystyle \raisebox{1ex}{$Pr\left(b,j\right)$}\!\left/ \!\raisebox{-1ex}{$Pr\left(b,i\right)$}\right.}$ by a likelihood ratio, which is the NBL probability of bin $\left[i,j\right)$ relative to r’s support; the local Bayesian coding rule is $\tilde{o}\left(j:i|r\right)={\displaystyle \raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$j$}\right.}{log}_r{\displaystyle \raisebox{1ex}{$j$}\!\left/ \!\raisebox{-1ex}{$i$}\right.}$. The Bayesian rule to recode local data is $\tilde{o}\left(j:i|r^{\prime }\right)=\tilde{o}\left(j:i|r\right){\displaystyle \raisebox{1ex}{$lnr$}\!\left/ \!\raisebox{-1ex}{$ln{r}^{\prime }$}\right.}$. Global and local Bayesian data are elements of the algebraic field of “gap ratios”, ${\displaystyle \frac{A-B}{C-D}}$. 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 $\left\{x\in \mathbb{Q}|1\le \left|x\right|<b\right\}$, into the local Bayesian data of its logarithmic coding space, the ball $\left\{x\in \mathbb{Q}|\left|x\right|<1-{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$b$}\right.}\right\}$. The source’s conformal encoding function is $y={log}_r\left(2x-1\right)$, where x is the observed Euclidean distance to an object’s position. The conformal decoding function is $x={\displaystyle \frac{1}{2}}\left(1+r^y\right)$. Both functions, unique under basic requirements, enable information- and granularity-invariant recursion to model the multiscale reality.