  A new variational method, the principle of least radix economy, is
formulated. The mathematical and physical relevance of the radix economy, also
called digit capacity, is established, showing how physical laws can be derived
from this concept in a unified way. The principle reinterprets and generalizes
the principle of least action yielding two classes of physical solutions: least
action paths and quantum wavefunctions. A new physical foundation of the
Hilbert space of quantum mechanics is then accomplished and it is used to
derive the Schr\"odinger and Dirac equations and the breaking of the
commutativity of spacetime geometry. The formulation provides an explanation of
how determinism and random statistical behavior coexist in spacetime and a
framework is developed that allows dynamical processes to be formulated in
terms of chains of digits. These methods lead to a new (pre-geometrical)
foundation for Lorentz transformations and special relativity. The
Parker-Rhodes combinatorial hierarchy is encompassed within our approach and
this leads to an estimate of the interaction strength of the electromagnetic
and gravitational forces that agrees with the experimental values to an error
of less than one thousandth. Finally, it is shown how the principle of
least-radix economy naturally gives rise to Boltzmann's principle of classical
statistical thermodynamics. A new expression for a general (path-dependent)
nonequilibrium entropy is proposed satisfying the Second Law of Thermodynamics.
