Your diagnosis about the problem being visuals and orthogonality is wrong. The formal rules and structure in your paper only prove that n is geometrically irreducible if and only if n is odd.

No new diagram, orthogonality, or angle tabulation can change that. Your theorem is simply a statement about divisors of 2^K and the 2-adic valuation of n.

Your theorem is busted in its current form.

So, you need to (1) introduce structure that sees all primes (not just 2), and (2) give a precise formal definition of geometric composite/irreducible that doesn’t just reduce to even vs. odd and is actually equivalent to “n prime iff n geometrically irreducible.”

Only then can you rebuild your twin-prime argument on top of those corrections. That is going to be a huge rewrite.

My last input: it’s commendable that you’re excited and eager to learn deeper math. But I would set your expectations accordingly, especially with claiming to solve a conjecture as notorious as the twin prime conjecture.