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.
, where
. Under its tail, we find information as harmonic likelihood
, where
is the
nth harmonic number. The global
-NBL is
, where
b is the base and
q is a quantum (
). Under its tail, we find information as logarithmic likelihood
. The fiducial
-NBL is
, where
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
relative to
b’s support. To encode the odds of quantum
j against
i locally, we multiply the prior odds
by a likelihood ratio, which is the NBL probability of bin
relative to
r’s support; the local Bayesian coding rule is
. The Bayesian rule to recode local data is
. Global and local Bayesian data are elements of the algebraic field of “gap ratios”,
. 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
, into the local Bayesian data of its logarithmic coding space, the ball
. The source’s conformal encoding function is
, where
x is the observed Euclidean distance to an object’s position. The conformal decoding function is
. Both functions, unique under basic requirements, enable information- and granularity-invariant recursion to model the multiscale reality.
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