In 2016 I devised a proof that basically involved scanning lines and parabolas across hyperbolas, and showing that a certain parabola that always generated twins had infinite solutions. I still think it's valid, but the few people I tried to show it to had trouble following it. It was also written informally.
About 2 years later I came up with a revised version that no longer used the scanning, but used 2 surfaces instead. While the concept was sound, and much easier to follow, the proof was not rigorous enough for certain portions of its claims. It was more formal than the 1st, but still not to the standards that many expect, despite it offering value and solutions.
Then in 2020, I was able to explicitly show all the steps and logic of the 2nd attempt. It's not quite as easy to follow, but it's rigorous and much more formal.
The 3rd one has recently been added, and is the most formal, complete, and the best of the proofs. You should start there. I should still update the older material at some point if I get the chance. You can down load the pdf. or view the pages online. If you having trouble reading the pages, the pdf. version offers more scalability and is easier to read.
The older methods to prove the Twin Prime Conjecture are given and discussed below. The pictures currently cover 2 short papers. First is the original proof, sufficient to convey its workings to those familiar with the topic, but written somewhat informally, and second is an update, which furthers the formalization of the logic, yet is still not fully complete.
The Proof works generally as follows: 1 surface is assigned to the primes such that if you choose values not on that surface you generate a prime. A 2nd surface is assigned to values 2 away from a number that are prime such that it works like the 1st surface. The proof then shows how to generate an infinite number of values that meet both surfaces criterion.
* This proof below was found to have an error. Most of it, and the concept behind it are still correct, it just needed fixing in one place. It was an attempt to get the same results as the original proof, but more quickly and clearly. The original is still valid, and still has had no flaws found to this date.
A methods to prove the Twin Prime Conjecture are given and discussed. The pictures currently cover 2 short papers. First is the original proof, sufficient to convey its workings to those familiar with the topic, but written somewhat informally, and second is an update, which furthers the formalization of the logic.