  This paper is the first one in the series devoted to the calculation of
particle mass spectrum in Topological GeometroDynamics. TGD Universe is
critical at quantum level and an attractive idea to realize criticality is via
conformal invariance. Ordinary real numbers do not allow this but if one
assumes that in long length scales p-adic topology replaces real topology as
effective topology situation changes. The existence of square root in the
vicinity of p-adic real axis implies 4-dimensional algebraic extension of
p-adic numbers ($p>2$), which can be regarded as padic counterpart of light
cone and consists of convergence-cubes of p-adic square root function. Later
work has demonstrated that convergence cubes of square root function serve as
natural quantization volumes in p-adic field theory limit of TGD.
