The set of conjectures regarding cardinality of two-element prime constellations can be reduced, for each specific constellation, to a question of whether an example constellation can be found that is "large enough." A proof of this was proposed this past June, and is known to have passed some rounds of peer review; the current paper resides at

The primary argument of this paper is that a sieve can be constructed which isolates the twin primes from the rest of the natural numbers, and this sieve can further adhere to a property described as the Leapfrog Lemma. In short, the purpose of this lemma is to encapsulate the Odd-Even Theorem and other properties of sieves in such a way so that an induction or other proof of infinity can be established, and then any sieve which has all the properties matching those found in this lemma is known to produce an infinite set of whatever object is under scrutiny.

While the Leapfrog Lemma made its first formal appearance only as recently as September 12th, 2017, the concept was certainly around almost as long as the Odd-Even Theorem, which in turn was discovered as early as February, 2001, and posted here the following October.

There is some speculation on a method for transforming the work in the above paper in order to apply the concepts of the main lemma to a wider selection of sieves. One of the current contenders is to change the function definitions so that the functions accept a vector input and return a set of values as the output. This particular approach could allow non-single-function sieves to be evaluated using the lemma, but brings about the question of whether sieves can be combined and retain valued properties, and if so, what limitations on this process might exist.