It is suggested that the long–standing conceptual difficulties of quantum theory (the measurement problem and associated concept of probability on the one hand and the notions of entanglement and non–locality on the other) arise fundamentally from the role played by the continuum field C of complex numbers in the axioms of standard quantum theory. An alternative representation of the 2Nth roots of unity is developed, based on a self–similar family P of permutation operators, acting on the digits and places of the binary expansion of a real number r0, generically Borel normal. Based on r0 and P, an analytically irregular real–valued function 0≤ r < 1 is constructed on S2, which plays the role of the Riemann sphere. As a result of its irregularity (or, equivalently, granularity), r is only uniquely definable on a countable set of (‘rational’) meridians of S2. P is further generalized to describe permutation–operator representations of the quaternions, from which transformations of r under rotations of the sphere can be defined.
Using P, a proposal for a constructive realistic deterministic single–world quantum theory is given. The ‘realistic’ roots–of–unity permutation operator encodes the apparent stochasticity of the complex phase function riEt/ħ in the standard quantum theoretic wave function; similarly, the quaternionic permutations encode ‘realistically’ the spatial entanglement relationships in the wave function. Based on this formalism, finite samples of values of r on the rational meridians are used to define a frequentist–based probability measure consistent with quantum–theoretic probability and correlation. The granularity of r is crucial in accounting for the theory's evasion of the Bell inequalities. Standard quantum phenomena are discussed from the perspective of the proposed theory. Some emphasis is given to the issues of contextuality and non–locality in the proposed theory.