On regularities in distributions of pairs of consecutive primes and in pairs of primes with a given prime pair gap

Key words: consecutive primes, unexpected biases, LemkeOliver, Soundararajan, Hardy and Littlewood, regularities, consecutive prime number formula, structure, randomness, geometric distribution, twin primes, prime pairs, grid, grid gap, Arithmetic Progression, primorial,

In 2016, Lemke Oliver and Soundararajan surprised—or even shocked?—number theorists by discovering unexpected biases in modular distributions of pairs of consecutive primes.  Quanta Magazine ran an article on their findings under a compelling title: “Mathematicians Discover Prime Conspiracy’. And Tao referred to the core observation of Lemke Oliver and Soundararajan in one of his blogs as ‘a surprising but now satisfactorily explained bias in the distribution of pairs of consecutive primes when reduced to a small modulus’.

Let me sketch another and broader view on  the subject, by highlighting characteristic properties of both modular and common (non-modular) distributions of pairs of consecutive primes, and how these emerge from specific subsets of such prime pairs. This view also encompasses a generalization of Hardy and Littlewoods conjecture on prime pairs and a more fundamental explanation than the one provided by Lemke Oliver & Soundararajan for the observed biases The next presentation is meant to illustrate this with two examples of mod 10 distributions. A more complete treatment of the subject can be found in
https://www.researchgate.net/publication/384562145_On_regularities_in_distributions_of_pairs_of_consecutive_primes_and_in_pairs_of_primes_with_a_given_prime_pair_gap.