Discrete Reanalysis of a New Model of the Distribution of Twin Primes

Recently we have introduced a novel characterisation of the distribution of twin primes that consists of three essential elements. These are: that the twins are most naturally viewed as a subsequence of the primes themselves, that the likelihood of a particular prime in sequence being the first element of a twin is akin to a fixed-probability random event, and that this probability varies with $\pi_{1}$, the count of primes up to this number, in a simple way. Our initial studies made use of two unproven assumptions: that it was consistent to model this fundamentally discrete system with a continuous probability density, and that the fact that an upper-bound cut-off for prime separations exists could be consistently ignored in the continuous analysis. The success of the model served as a posteriori justification for these assumptions. Here we perform the analysis using a discrete formalism -- not passing to integrals -- and explicitly include a self-consistently defined cut-off. In addition, we reformulate the model so as to minimise the input data needed.